Qualitative Theory of Control Systems

Translations of Mathematical Monographs 141

Qualitative Theoryof Control Systems

Translations of

MATHEMATICALMONOGRAPHS

Volume 141

Qualitative Theoryof Control SystemsA. A. Davydov

American Mathematical Society, Providence, Rhode Islandin coorperation withMIR Publishers, Moscow, Russia

A. A. AasbIuoB

KALIECTBEHHASI TEOPHSI YHPABJI$IEMbIX CHCTEM

Translated by V. M. Volosov from an original Russian manuscriptThe present translation is published under an agreement

between MIR Publishers and the American Mathematical Society.

1991 Mathematics Subject Classification. Primary 34C20, 93C 15, 94D20;Secondary 34A34, 49J17.

ABSTRACT. This book is devoted to the analysis of control systems using results from singularitytheory and the qualitative theory of ordinary differential equations. In the main part of the book,systems with two-dimensional phase space are studied. The study of singularities of controllabilityboundaries for a typical system leads to the classification of normal forms of implicit first-orderdifferential equations near a singular point. Several applications of these normal forms are indicated.The book can be used by graduate students and researchers working in control theory, singularitytheory, and various areas of ordinary partial differential equations, as well as in applications.

Library of Congress Cataloging-in-Publication DataDavydov, A. A.

Qualitative theory of control systems/A. A. Davydov.p. cm. - (Translations of mathematical monographs, ISSN 0065-9282; v. 141)

ISBN 0-8218-4590-X1. Control theory. I. Title. II. Series.

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Table of Contents

Introduction 1

Chapter 1. Implicit First-Order Differential Equations 5

§1. Simple examples 5

§2. Normal forms 12

§3. On partial differential equations 17

§4. The normal form of slow motions of a relaxation type equation on thebreak line 19

§5. On singularities of attainability boundaries of typical differential inequal-ities on a surface 21

§6. Proof of Theorems 2.1 and 2.3 24§7. Proof of Theorems 2.5 and 2.8 26

Chapter 2. Local Controllability of a System 29§1. Definitions and examples 29§2. Singularities of a pair of vector fields on a surface 36§3. Polydynamical systems 43§4. Classification of singularities 60§5. The typicality of systems determined by typical sets of vector fields 71

§6. The singular surface of a control system 72§7. The critical set of a control system 77§8. Singularities of the defining set and their stability 88§9. Singularities in the family of limiting lines in the steep domain 92

§10. Transversality of multiple 3 -jet extensions 99

Chapter 3. Structural Stability of Control Systems 103§ 1. Definitions and theorems 103§2. Examples 109§3. A branch of the field of limiting directions 111

§4. The set of singular limiting lines 113

§5. The structure of orbit boundaries 120§6. Stability 124§7. Singularities of the boundary of the zone of nonlocal transitivity 130

Chapter 4. Attainability Boundary of a Multidimensional System 135

§ 1. Definitions and theorems 135

vii

viii TABLE OF CONTENTS

§2. Typicality of regular systems 138§3. The Lipschitz character of the attainability boundary 139§4. The quasi-Holder character of the attainability set 140

References 145

Introduction

Many of the processes around us are controllable. They develop in different waysdepending on actions that affect them. As a rule, what can affect a specific processis limited by the characteristics of the process itself and by the special features ofthe controller. The analysis of the controllability of a process, i.e., of the possibilityto obtain a desirable development by means of feasible actions, is one of the mainproblems in the theory of control systems. In the present book this problem is solvedusing results of the theory of singularities and of the qualitative theory of ordinarydifferential equations.

The book consists of four chapters. The main part (Chapters 2 and 3) is devotedto the controllability of systems with two-dimensional phase space (i.e., systems whosestate can be described by a point on a surface, e.g., on the two-dimensional sphere, orthe torus, or the plane). In Chapter 4 the controllability (attainability) boundaries ofmultidimensional systems are investigated. In Chapter 1, normal forms of a genericimplicit first-order differential equation in a neighborhood of a singular point arefound. Now let us discuss in more detail Chapters 2, 3, 4, and 1 in that order.

As we have already noted, in the last three chapters we study control systems. Itis assumed that the evolution of the system is described by an ordinary differentialequation with the vector field that depends on the control parameter. This vector fieldand the range of the control parameter characterize the technical capabilities of thesystem.

The control objectives can be diverse. Chapter 2 deals with the local controllabilityof systems on smooth surfaces. The regions in the phase space consisting of points withthe same controllability properties are described for a typical system (i.e., for almostevery system in the space of systems). Contrary to many interesting and sophisticatedinvestigations on the necessary and sufficient conditions for the controllability of asystem in the neighborhood of an individual point (see, e.g., Petrov [Pe], Agrachev andGamkrelidze [AG], Sussmann [S2], the latter containing an extensive bibliography),we shall study not only the system controllability in the neighborhood of an individualpoint, but also the entire above-mentioned regions themselves. We shall show that for atypical system these regions are stable with respect to small perturbations and differ onefrom another only at some individual points on their boundaries. In particular, theseregions have the same closure. In the generic case, at each point of the complement tothis closure the positive linear hull of the set of feasible velocities does not contain thezero velocity and is bounded by an angle smaller than 180°. The sides of this angledetermine the limiting directions of the feasible velocities at that point. We classify thesingularities of the limiting direction field of a typical system. Such singularities werestudied by Filippov [F2] for an analytical polydynamic system, and by Baitman [B1,B2] for a typical pair of smooth vector fields on a surface. In addition to the results

I

2 INTRODUCTION

published in [D3], Chapter 2 contains a detailed analysis of the local controllabilityof a typical system in the neighborhood of a singular point of its limiting directionsfield. The results of this analysis partly overlap with the results of Goncalves [G1],who studied a different class of systems.

In Chapter 3 nonlocal controllability of a typical system on a closed orientablesurface is studied. The main result is the theorem on the structural stability for thefamily of orbits of points of a typical system on a compact orientable surface. Thistheorem is an analog of the classical Andronov-Pontryagin-Baggis-Peixoto result onthe structural stability of a typical vector field on a sphere or on a closed orientablesurface. Moreover, we study nonlocal transitivity zones of a typical system, i.e., openregions in the state space, each of which coincides with the intersection of the positiveand the negative orbit of any of its points. Any two states belonging to such a zone canbe transformed into each other by a suitable control action. It is shown that the numberof nonlocal transitivity zones of a typical system is finite. We list typical singularities ofthe boundaries of these zones and describe the structure of the boundary for a typicalsystem and also the structure of the boundary of the positive (negative) orbit of anypoint. In essence, the investigations in this chapter are close to those by Lobry [L1]and Sussmann [Si] on the structural stability of the complete controllability of atypical system, by Sieveking [Si] and Colonies and Kliemann [CK] on limiting sets andcontrollability sets of systems, and by Baitman [B1] on nonlocal transitivity zones ofa typical bidynamical system on the plane.

Chapter 4 is devoted to the controllability (attainability) boundary of a systemwith state space of an arbitrary dimension. It is shown that for a typical system thisboundary is a locally Holder hypersurface in the phase space. The analysis in thischapter is closely related to the study of lower bounds for the interior of the set ofstates attainable from a point within a short time and are based on similar ideas (see,e.g., Hermes [He]). For instance, Agrachov and Sarychev [AS] and Gershkovich [Ge]showed that for some classes of systems the Holder index can be better than in ourresults. The presentation in this chapter essentially follows [D5, D6].

Chapter 1 includes the results on the normal form near a singular point fora typical implicit first-order differential equation which is not solved with respectto the derivative. Originally, these results were obtained as a byproduct of studiesof singularities of controllability (attainability) boundaries for a typical system ona surface. However, implicit first-order differential equations are important in thedescription of some phenomena distantly related to control theory, and therefore aseparate chapter is devoted to these normal forms and their applications. Singularitiesof implicit first-order differential equations were chosen as a topic for the competitionsponsored by King Oscar II of Sweden in 1885. The four topics were selected by thejudges (the judges included Weierstrass, Hermite, and Mittag-Leffler; see [AM]). Thethird topic involved finding the normal forms of an equation in the neighborhood ofa singular points. In 1932 Cibrario obtained the first of these forms (dy/dx)2 = x inthe neighborhood of a regular singular point of an implicit equation [Ci]. Dara [Da]rediscovered this normal form and showed that nonregular singular points of a genericimplicit equation can be divided into five types, namely folded saddles, folded nodes,folded foci, and elliptic and hyperbolic cusps (or pleats or gathers) (it should be notedthat the three folded singularities were distinguished earlier) [SP, PF]. We show thatnormal forms of the folded singularities are as simple as those of the singular pointsof typical vector fields in the plane found by Poincare.

INTRODUCTION 3

It turns out that normal forms of folded singular points have very many appli-cations. They are encountered in the theory of mixed partial differential equations(M. Cibrario), in the investigation of relaxation type equations (V. I. Arnol'd andF Takens), in applications to plasma physics (A. D. Pilija and V. I. Fedorov), in theanalysis of the behavior of the net of asymptotic lines on a smooth surface (R. Thom),and in the study of the attainability boundaries of control systems (A. A. Davydov).The normal forms of the mixed partial differential equations presented in Chapter 1are published here for the first time (except for lecture notes on the theory of singu-larities [D7]). For various physical considerations (or for the sake of convenience)these normal forms were used earlier in [PF]. Together with the wave equation and theLaplace and Cibrario equations they form a complete list of normal forms of typicallinear partial differential equations of the second order on the plane.

When preparing this book, a particular effort was made to present the materialso that it would be accessible for a broad readership in diverse areas of mathematicsand allied fields. The results of each chapter can be understood without reading theothers. The proofs only require familiarity with the basic notions and theorems of thequalitative theory of differential equations and singularity theory. This material canbe found in [P2, A3, Ha, AGV, GG].

To conclude the introduction, I wish to express my warmest gratitude toV. I. Arnol'd. He drew my attention to problems in the theory of singularities ofcontrol systems; his attention to my work and discussions with him facilitated progressin their solution. I am also grateful to D. V. Anosov, A. A. Agrachev, A. F. Filip-pov, and A. M. Leontovich for valuable discussion and to my wife Lidiya for help inpreparing the manuscript.

A. Davydov

CHAPTER 1

Implicit First-Order Differential Equations

In this chapter the normal forms are found for a typical differential equation notsolved with respect to the derivative in neighborhoods of its singular point. The mainresult is that in a neighborhood of each singular point for which the discriminant curveis smooth, the equation is reduced to the normal form y = (dy/dx + kx)2 under adiffeomorphism of the plane of the variables x, y (by using homeomorphisms it ispossible to obtain k = -1, 1/9, or 1/4). In § 1 we give some examples motivating ourinvestigation and define the basic notions of the theory of implicit equations. In §2 weformulate the main theorems on normal forms. In §3-5 we describe some applicationsof these normal forms. The concluding sections of this chapter are devoted to proofsof the main theorems.

§ 1. Simple examples

We start with examples of three phenomena where implicit first-order differentialequations play an important role and then we define the key notions of the theory ofimplicit first-order differential equations.

1.1. One-dimensional mechanical system. Consider a point mass moving in a lineunder the action of two forces. One force possesses a smooth (i.e., of class C°°)potential U that depends on the position x of the point on the line. The other isthe force of friction, which is proportional to the velocity with proportionality factork > 0 that also depends on the position of the point on the line. By Newton's secondlaw, the equation of motion can be written as

mz = -Ux(x) - k(x). ,

where m is the mass of the particle, Ux = aU/ax, and x = dx/dt. The system isdissipative because as the point moves, the total energy decreases (due to friction). Wederive the equation for this process:

E _ (m12/2+ U(x))r = mzz + Ux(x)z

_ (mz + Ux(x))z = -k(x)z2 = -2k(x)(E - U(x))/m.

Hence, E _ -2k(x)(E - U(x))/m. From the equation for the total energy we findx2 = 2(E - U(x))/m. Consequently, the family of system trajectories in the plane ofthe variables x, E (the energy balance plane) coincides with the family of the integralcurves of the implicit first-order equation

(1.1) (dE/dx)2 = 2k2(x)(E - U(x))/m.

As can be easily seen, this equation cannot be smoothly solved with respect to thederivative in a neighborhood of any point of the graph of the function E = U(x).

5

6 1. IMPLICIT FIRST-ORDER DIFFERENTIAL EQUATIONS

(a) (b)

FIGURE 1.1

(c)

The phase plane is transformed into the energy balance plane under the foldingmapping, i.e., (x, x)' (x,,=n.z2/2 + U(x)). The set of critical points of this mapping(i.e., all the points for which the rank of the mapping derivative is not maximal)coincides with the axis in the phase plane, and the set of critical values (i.e., thevalues of the mapping at its critical points) coincides with the graph of the functionE = U(x) in the energy balance plane. It is easy to see that each critical point ofthe folding mapping of the phase plane is a Whitney fold (i.e., in a neighborhood ofthis point the mapping has the form (u, v) H (s = u, r = v2) in the appropriatecoordinates u, v and s, r in the source and target spaces with origins at that point andits image, respectively).

Each critical point xo of the potential U has a corresponding singular point(xo, 0) of the velocity field (z, - Ux (x) /m - k(x)x/m) of the system in the phaseplane. Under the folding mapping of the phase plane the singular point goes into afolded singular point of equation (1.1) . For Uxx (xo) < 0, 0 < 4m Uxx (xo) < k2 (X0),or k2(xo) < 4m Uxx (xo) this singular point is a saddle, a node, or a focus, respectively.

For example, to the critical points -1, 0, and 2 of the potential U(x) = x4 -4x3/3 - 4x2 + 11 there correspond three singular points (-1, 0), (0, 0) and (2, 0) ofthe system velocity field in the phase plane, and three folded singular points (-1, 28/3),(0, 11), and (2, 1/3) in the energy balance plane, respectively. For the mass m = 1and the constant coefficient of friction k = 8 these singular points are, respectively,a node, a saddle, and a focus. The behavior of the family of system trajectories in aneighborhood of folded singular points of these types is illustrated in Figures 1.1 a-c,respectively. The double line is the graph of the function E = U(x); the dashed andsolid lines represent the images, under the folding mapping of the phase plane, ofthe parts of the system's phase trajectories lying in the upper and lower half planes,respectively; the singular point itself is represented by a small circle.

1.2. The net of characteristics of a mixed equation. Consider a second-order partialdifferential equation in the plane of the variables x, y:

(1.2) a(x,y)UYx+2b(x,y)uxy+c(x,y)uy}, +F(x,y,u,ux,uy)=0,

where a, b, and c are differentiable functions, F is a given function, and u is theunknown function. The regions where the function 0 = b2 - ac is negative andpositive are called, respectively, the ellipticity and the hyperbolicity regions of theequation. The implicit differential equation

a(x,y)dy2-2b(x,y)dxdy+c(x,y)dx2 = 0

(in a form that is symmetric with respect to dx and dy) is called the characteristicequation of equation (1.2). In a neighborhood of every point of the hyperbolicity region

§1. SIMPLE EXAMPLES 7

the characteristic equation decomposes into two first-order equations that describe twosmooth branches of the field of characteristic directions. The integral curves of thisfield are called characteristics. They play an important role in the theory of partialdifferential equations.

In the general case, the gradient of the function A is nonzero at all points wherethe function itself vanishes. Thus, the zero level line of the function is a smooth (moreprecisely, smoothly embedded) curve in the plane. This is the line of type changefor equation (1.2) in that the ellipticity region lies on one side of the line, and thehyperbolicity region on the other side. Consequently, (1.2) is a mixed equation in aneighborhood of each point of this line. In the generic case the functions a and c donot vanish simultaneously at any point on the line of type change because otherwisethe gradient of the function A at this point would also vanish. Consequently, in aneighborhood of such a point the characteristic equation can be reduced to a quadraticequation with respect to the derivative dy/dx or dx/dy by dividing it by dx2 or dy 2,

respectively. Hence, we obtain an implicit first-order equation. Near the type-changeline the equation no longer decomposes into two smooth first-order equations andcannot be smoothly solved with respect to derivative. When approaching the line,the two characteristic directions tend to each other. On the line itself they coincideand determine a smooth field of straight lines on it. Generally, the field rotates whenmoving along the line and consequently, it may touch the line at some points with thefirst order of contact. At the point of tangency the vector (-A , A) determines thecharacteristic direction and hence satisfies the equation

a(x,y)A + 2b (x, y)AxA), +c(x,y)A = 0.

In the general case, the family of characteristics of equation (1.2) has a folded singularpoint at the point of tangency. The singularity may be a saddle, a node, or a focus(Figures 1.la-c respectively; but in contrast with the trajectories of the mechanicalsystem no direction of motion is defined on the characteristics of (1.2)).

For instance, for the equation

uxx + (kx2 - y)u)), + F(x, y, u, ux, up) = 0

zero is a folded saddle, a folded node, or a folded focus for k < 0, 0 < k < 1/16,or 1/16 < k, respectively. It will be shown in §3 that almost every equation (1.2) isreducible to this form (with some k) in a neighborhood of its folded singular points.

1.3. The net of limiting lines of a differential inequality. Imagine that a waterflow with velocity field (-x, -fly), /3 > 0, in a planar sea is carrying a swimmer tozero. The swimmer can move in standing water in any direction with a velocity notexceeding 1. The possible paths the swimmer can take are described by the differentialinequality (z+x)2+(.v+fy)2 < 1. Theinequality x2+f2y2 > 1 determines the steepdomain where the swimmer cannot resist the flow. At each point of the steep domainthe directions of the swimmer's admissible velocities at this point form an angle notexceeding 180°. The sides of the angle are called the limiting directions at this point.Thus, a two-valued fields of limiting directions is defined in the steep domain. Theintegral curves of the field are called limiting lines. It is easy to show that (1) theselines are in fact the integral curves of the implicit first-order differential equation

(x dy - fly dx)2(x2 + fl2y - 1) = (x dx + ly dy)2


FIGURE 1.2

(which is symmetric with respect to dx and dy) and (2) in a neighborhood of eachboundary point of the steep domain this equation cannot be smoothly solved withrespect to the derivative (dy/dx or dx/dy).

When approaching the boundary of the steep domain from inside, the angleformed by the limiting directions tends to the straight angle. At the boundary itselfthe limiting directions determine a smooth field of straight lines. Generally, as in theprevious example, this field rotates when we move along the boundary. Consequently,in the general case, it can also have first-order contact with the boundary at somepoints. In the general case, at each point of tangency the net of limiting lines has afolded singular point: a saddle, a node, or a focus.

Figure 1.2 represents the net of limiting lines of the differential inequality underconsideration for some ,8 > 2. The double line in the figure is the boundary of thesteep domain, and dashed and solid lines are the limiting lines of the two branchesof the field of limiting directions. We can clearly see the folded singular points: thesaddles are at (±1, 0) and the nodes at (0, +1/fl).

As will be shown in Chapter 3, the limiting lines are important in studying theboundaries of attainability sets. For example, in Figure 1.2 the set of points thatthe swimmer can attain from zero is an open domain. It is bounded by the closureof the union of the four limiting lines entering the folded nodes, which are outgoingseparatrices of the folded saddles.

1.4. Key notions in the theory of implicit first-order differential equations. In allthree examples mentioned above we obtained an implicit equation describing a two-valued direction field. Generally speaking, an implicit equation

(1.3) F(x,y,p)=0,

where p = dy/dx and F is a smooth function, determines a multivalued directionfield.

ExAMPLE 1. For the equation p3 - 3x p - y = 0 the direction field is three-valuedin the region y2 < 4x3, two-valued on the part of semicubical parabola y2 = 4x3 lyingin the right half plane, and single-valued in the remaining part of the plane.

We identify the space of implicit equations with the space of smooth functionsF and endow it with a fine C3 Whitney topology. The proximity of two functions in


FIGURE 1.3

this topology means the proximity of their derivatives up to the third order inclusiveat all points in the space of the variables x, y, p. In this case the proximity iscontrolled arbitrarily well at infinity. A typical or generic implicit equation is anequation belonging to an open everywhere dense set of this space in the given topology.

For a typical equation (1.3) the gradient of the function F is nonzero at all pointswhere the function itself vanishes. Indeed, the simultaneous vanishing of both thefunction and its gradient imposes four independent conditions on the point in thethree-dimensional space, and therefore this phenomenon is not observed for a typicalequation. Hence, a generic implicit equation determines a smooth surface in the spaceof the variables x, y, p. This space is called the space of 1 jets of functions y (x), andthe surface is called the surface of the equation.

A folding mapping of an implicit equation is the projection of the equation surfaceon the plane of the variables x, y along the axis p. A point on the equation surface issaid to be regular if it is not a critical point of the folding of the equation. Other pointsof the equation surface are said to be singular; singular points form the criminant of theequation. The image of the criminant under the folding of the equation is called thediscriminant curve. For a typical implicit equation, every critical point of the equationfolding (i.e., a point on the criminant) is either a Whitney fold or a Whitney cusp (orpleat, or gather). (In a neighborhood of a critical point which is a Whitney pleat, themapping can be written in the form (u, v) H (r = u, s = v3 - uv) in suitable smoothlocal coordinates u, v and r, s in the source and target spaces with origins at thatpoint and its image, respectively.) In particular, the criminant itself is a smooth (i.e.,smoothly embedded) curve in the space of 1 jets.

EXAMPLE 2. For the equation p3 - 3xp - y = 0 the criminant is determinedby the equations x = p3, y = -2p3, and the discriminant curve coincides with thesemicubical parabola y2 = 4x3 (Figure 1.3). The point (0, 0, 0) is the Whitney cuspof the equation folding. Other points of the criminant are the Whitney folds of theequation folding. The remaining points of the equation surface are regular points ofthe equation.

It is often more convenient to study the direction field of an implicit equation notin the plane of the variables x, y, but on the equation surface. The direction field onthe equation surface is cut by the field of contact planes, which is defined in the space

10 I. IMPLICIT FIRST-ORDER DIFFERENTIAL EQUATIONS

of 1 jets of functions by the 1-form a = dy - p dx. The contact plane at a point in thisspace consists of all vectors applied at the point on which the form vanishes. It canbe easily seen that the contact plane always contains the direction of the axis p, i.e., isvertical.

The direction field cut by the contact structure on the equation surface is smoothin a neighborhood of every point where the contact plane is not tangent to the surface.In particular, this condition is always fulfilled at each regular points of the equation.It is clear that in the x, y plane, outside the discriminant curve, the image of thecut-out field under equation folding coincides with the multivalued direction field ofthe equation.

For a typical equation the contact plane and the tangent plane to the equationsurface coincide only at certain points of the criminant, which, in addition, are Whitneyfolds of the equation folding. Indeed, the criminant of a typical equation is a smoothcurve in the space of 1 jets. At each point of this curve there are two vertical planes:the tangent plane to the equation surface and the contact plane. Therefore, two fieldsof vertical planes are defined on the criminant. Generally, when moving along thecriminant, these two fields rotate around the vertical direction belonging to them.Consequently, for a typical equation these fields may also touch each other with first-order contact at a point that is not a Whitney cusp of the equation folding. Such pointsof tangency will be called folded singular points.

Thus, the direction field cut on the surface of a typical equation by the contactstructure is smooth everywhere except at the folded singular points. We shall investigatethe behavior of this field in a neighborhood of a folded singular point. The coordinatesystem in the x, y plane with origin at the image of that point under equation foldingis chosen so that the image of the criminant in a sufficiently small neighborhood of thepoint belongs to the abscissa axis. With this choice of coordinate system the foldedsingular point under consideration coincides with the point (0, 0, 0) in the space of 1 jetsof functions, and the tangent plane to the equation surface at zero becomes the planey = 0. However, for a typical equation (1.3) the gradient of the function F is nonzeroat all points where the function itself vanishes. Consequently, for a typical equation inthis coordinate system we have F3, (0, 0, 0) L 0. By the implicit function theorem, in aneighborhood of zero the equation is equivalent to the equation y = f (x, p), wheref is a smooth function and f (0, 0) = 0 = f, (0, 0) = f ( 0 , 0 ) due to the choice ofthe coordinate system. Hence, in a neighborhood of zero x and p can be taken aslocal coordinates on the equation surface. In these coordinates the direction field underconsideration coincides in a neighborhood of zero with the direction field of the smoothequation p dx = f, (x, p) dx + fp (x, p) dp or (f, (x, p) - p)dx + fp (x, p)dp = 0.This equation has a singular point at zero because f, (0, 0) = fn (0, 0) = 0. Thus,in a neighborhood of the folded singular point the direction field cut on the equationsurface by the contact structure is the direction field of a smooth differential equationon this surface for which this point is singular. For a typical equation this singularpoint is nondegenerate in the sense that it is a nondegenerate singular point of thevector field (f f, (x, p), p - f, (x, p)) and, consequently, it can be a saddle, a node, ora focus. Moreover, in the case of a saddle or a node the linearization eigenvectors ofthis vector field at that point are transversal to both the criminant of the equation andthe kernel of the derivative of the folding mapping at the point, and the correspondingeigenvalues have different moduli.


//V.; -

FIGURE 1.4

Hence, the folded singular points of a typical equation can be classified as saddles,nodes, and foci. These three types of singular points are shown in Figures 1.4aC,respectively. The vertical arrow in the figures symbolizes the equation folding mapping.The upper diagrams represent the families of integral curves of the direction field of atypical implicit equation on its surface. The lower diagrams demonstrate the imagesof these families under the equation folding mapping: the solid lines are the images ofthe parts of the integral curves from one layer of the covering, and the dashed lines arethose from the other. The double line represents the criminant and the discriminantcurve (cf. Figures 1.1 a-c).

A singular point of equation (1.3) is said to be regular if the criminant is smoothat the point, i.e., if the rank of the mapping ((x, y, p) H (F, Fr)) equals 2 and thecriminant is not tangent to the contact plane at this point. It is clear that the foldedsingular points belong to the class of nonregular singular points. Whitney cusps ofthe equation folding are also nonregular singular points. These critical points will becalled cusped (or pleated, or gathered, or composite) singular points. For a typicalequation the direction field in a neighborhood of a cusped singular point is smooth,and the image of the family of the integral curves under equation folding may havetwo differential forms (Figures 1.5a,b; the notation as in the previous figure withthe dotted lines representing integral curves from the third layer of the covering).Following Dara [Da], these two types of cusped singularities will be called hyperbolicand elliptic cusps, respectively. In Section 2.3 we shall present an analytical conditionthat can discriminate between these two types of singular points.

Dara [Da] showed that a typical implicit equation can have only five types ofnonregular singular points: a folded saddle, a folded node, a folded focus, and ellipticand hyperbolic cusps. He also conjectured that a typical equation in a neighborhoodof its singular point is topologically (i.e., in a suitable continuous coordinate system)equivalent to the equation y = (p2 + 6Xx2)/2 with x < 0, 0 < X < 1/4, and 1/4 < Xfor a folded saddle, a folded node, and a folded focus, respectively, to the equationx = p3 - yp for an elliptic cusp, and to the equation x = p3 + yp for a hyperboliccusp. In Section 2.4 we show that the equivalence to the three normal forms of thefolded singularities does in fact take place, and a C°°-equation being reducible to them(under the ordinary additional conditions that are imposed on the eigenvalues of thelinearization of the direction field on the equation at the singular point; these conditionsare formulated after the remark to Theorem 2.3) by means of a C°°-diffeomorphism


FIGURE 1.5

of the x, y plane. The topological equivalence eliminates the parameter X in each ofthese three normal forces. Hence, the folded singular points of an equation solvedwith respect to the derivative have a single modulus under the diffeomorphisms, and,likewise the singular points of ordinary equations, are structurally stable with respectto the homeomorphisms. In Section 2.5 we show that Dara's hypothesis is not true forthe cusped singularities because their topological normal forms must contain functionmoduli.

1.5. Germ and singularity. Two objects of the same nature (sets, vector fields,families of curves, mapping, etc.) are said to be equivalent at a point if they coincide ina neighborhood of the point. The equivalence class of an object at a point is called itsgerm at this point.

EXAMPLE 3. The functions of one variable gi (x) = x and g2(x) = (x + Ix1)/2have a common germ at each point of the positive x-halfaxis and different germs ateach of the other points.

Two germs (of objects of the same nature) are said to be Ck-diffeomorphic if thereexists a germ of a Ck-diffeomorphism that transforms one of the germs into the other.The class of Ck-diffeomorphic germs is called a Ck-singularity or, simply, a singularity.

REMARK. A Ck-diffeomorphism is a one-to-one mapping which together with itsinverse is k times continuously differentiable, a C°-diffeomorphism is called a homeo-morphism.

ExAMPLE 4. The set y = 1x2 -1 in the plane has the same singularity at the points(-1, 0) and (1, 0) as the set y = I x I at zero.

§2. Normal forms

Here we formulate the basic theorems on normal forms. Unless otherwise stipu-lated, we shall only consider smooth (i.e., of class C°°) objects.

§2. NORMAL FORMS 13

2.1. Good involutions. A direction field on a surface is said to be smooth if in aneighborhood of each point on the surface it is the direction field of a smooth differ-ential equation a (u, w )du + b (u, w) dw = 0, where u and w are local coordinates. Thepoints where the coefficients a and b simultaneously vanish are called singular pointsof the direction field. A singular point of a direction field is said to be nondegenerate ifthe functions a and b can be chosen so that each of the eigenvalues of the linearizationof the vector field (-b, a) at that point is nonzero and the ratio of the eigenvalues is not± 1. The directions of the corresponding eigenvectors will be called the eigendirectionsof the direction field.

Let v be a direction field having a nondegenerate singular point at zero. Aninvolution having a line of fixed points passing through zero is said to be compatiblewith the field v if on this line, and on this line only, the directions of the field and of itsimage under the involution coincide. An involution compatible with a field v is said tobe v-good if the eigendirections of the field v and of the derivative of the involution atzero are pairwise distinct.

EXAMPLE 1. Let us take x and p as coordinates on the surface of the equation2y = p2 + Xx2, 0 ; X 1/4. Zero is a nondegenerate singular point of the directionfield v of this equation. The involution (x, p) H (x, - p) of this surface is v-good.

Two objects (germs of involutions or curves, directions at points, etc.) are saidto be equivalent along a field v or v-equivalent if they can be transformed into eachother by a C°°-diffeomorphism of the plane such that each integral curve of the fieldis mapped into itself.

We now fix a direction field v with a nondegenerate singular point at zero.

THEOREM 2.1. The germs at zero of two v-good involutions are v-equivalent if andonly if the tangents at zero to the fixed lines of these involutions can be joined in the spaceof directions at zero with a continuous curve not passing through the eigendirections ofthe field v at zero.

Theorem 2.1 is proved in Section 6.1. It immediately implies Theorem 2.2. Thenumber of v-equivalence classes of germs at zero of v-good involutions is equal to two(one) if zero is a saddle or a node (accordingly, a focus) of the fixed field v.

REMARK. The set of v-good involutions is open in C 1-topology and everywheredense in C°°-topology in the space of involutions compatible with the field v.

2.2. Normal singular points. The exponent of a nondegenerate singular point of adirection field is defined as the ratio of the eigenvalue with maximum modulus of thelinearization of the corresponding vector field to that with minimum modulus for asaddle or a node and as the modulus of the ratio of the imaginary part of the eigenvalueto the real part for a focus; the exponents are preserved under diffeomorphisms.

A nondegenerate singular point of a direction field is said to be Ck-normal if thegerm at this point of the family of integral curves of the field is Ck-diffeomorphic tothe germ at zero of the family of phase trajectories of the linear vector field v2, v2 orv3 for a saddle, a node, or a focus, respectively, where

(2.1) v2(x,Y) =0

(2.2) v3(x,Y) =1 1 ) (x

C Y


and a is the exponent of this singular point. The symbols v2 and v3 will also be used todenote the direction fields determined by the differential equations with these vectorfields.

It is easy to show that the involution 01: (x, y) F--> (((a + 1)x - 2ay)/(a - 1),(2x - (a + 1)y)/(a - 1)) is v2-good and the involution 02: (x, y) H (x - 2y/a, -y)is v3-good.

Let zero be a C°°-normal singular point of a field v with exponent a.

THEOREM 2.3. The germs at zero of the direction field v, of the family of its integralcurves, and of the v-good involution are simultaneously reduced by a C°°-dfeomorphismof the plane to the germs at zero of the direction field v2 (v3), of the family of its integralcurves, and of the involution 01 (02) for a saddle or a node (accordingly, a focus).

Theorem 2.3 is proved in Section 6.2.

REMARKS. The conditions of C°°-normality required in Theorem 2.3 are almostalways fulfilled, namely:

1. According to the Siegel theorem, a saddle is C°°-normal if (1, a) is a point ofthe type (M, v) (i.e., min{I1 - m1 - MA I, ja - ml - m2al} > M11mI" for all integralvectors m = (m I, m2) with nonnegative components, m1 + m2 > 2). The measure ofthe set of points that are not points of the type (M, v) for any M > 0 is equal to zeroif v > 1 [A2].

2. A node is C°°-normal if its exponent is not a natural number. For a smoothvector field in the plane belonging to a set in the space of such fields (in a fine Whitneytopology), which is open in C I -topology and everywhere dense in C°°-topology, thiscondition is fulfilled at each of the nodes of the field.

3. A nondegenerate focus is always C°°-normal. Using homeomorphisms (i.e.,C°-diffeomorphisms) it is also possible to "eliminate" the exponent a of a singularpoint without requiring the C°°-normality of the point. Let zero be a nondegeneratesingular point of a field v.

THEOREM 2.4. The germs at zero of the direction field v, of the family of its integralcurves, and of the v-good involution are simultaneously reduced by a homeomorphismof the plane to the germs at zero of the direction field v2 (v2 or v3), of the family ofits integral curves, and of the involution 01 (01 or 02) for a = -2 (accordingly, a = 2,a = 1) for a saddle (accordingly, a node and a focus).

This theorem is an immediate consequence of Theorems 2.5 and 2.8 proved above,and therefore we omit its proof.

2.3. More on folded and cusped singularities. The folding mapping of equa-tion (1.3) determines the folding involution of the equation in a neighborhood ofits critical point which is a Whitney fold. On the surface of the equation the involutionpermutes points whose images under the following mapping of this equation coincide.

A nonregular singular point of equation (1.3) at which the equation folding hasa critical point which is a Whitney fold is called a folded saddle, a folded node, or afolded focus if (1) the direction field v of the equation has at this point a nondegeneratesaddle, a nondegenerate node, or a nondegenerate focus, respectively, and (2) thefolding involution of this equation (which is defined locally in a neighborhood of thepoint) is v-good. These three types of singular points will be called folded singularpoints.

In Example 1 in Section 2.1 we had a folded saddle, a folded node, and a foldedfocus at zero for X < 0, 0 < X < 1/4, and 1 /4 < X, respectively.

§2. NORMAL FORMS 15

The germ of the folding involution at a folded singular point of equation (1.3) isgood for the direction field of this equation. The converse is also true.

THEOREM 2.5. The germ at zero of the pair (direction field v with a nondegener-ate singular point at zero, v-good involution) is C°°-diffeomorphic to the germ at thefolded singular point of the pair (a direction field, a folding involution) of a suitableequation (1.3).

This theorem is proved in Section 6.1.A nonregular singular point of equation (1.3), which is also a Whitney pleat of

equation folding, will be called a cusped singular point or a cusped singularity of thisequation. The germ of the surface of equation (1.3) at a cusped singular point of theequation coincides with the germ at zero of the surface x = p f (x, p), where f is asmooth function, f (0, 0) = fn (0, 0) = 0 < fr,r, (0, 0), for a suitably chosen coordinatesystem in the x, y plane. A cusped singular point is said to be elliptic (hyperbolic) iffy, (0, 0) < 0 (accordingly, f), (0, 0) > 0). It is easy to show that the ellipticity and thehyperbolicity of a cusped singular point do not depend on the choice of the coordinatesystem.

REMARK. We noted in Section 1.4 that Dara [Da] proved that a typical equa-tion (1.3) has only nonregular cusped and folded singular points.

2.4. Normal folded singularities. A folded singular point of equation (1.3) is saidto be C°°-normal if it is a C°°-normal singular point of the direction field of theequation. Theorem 2.3 immediately implies

THEOREM 2.6. The image of the germ of the family of integral curves of equation (1.3)at a C°°-normal folded singular point, which is a saddle, a node or a focus, under thefolding mapping of this equation is C°°-dii feomorphic to the germ at zero of the familyof curves

(2.3) Ix ± vI -a (x/a + /) = c, c c R,

(2.4) 1z f Vly- I -a(x/a f ,fy_) = c) U (x f Vly- = 0), c E R,

or

5R sin (a In R + c),

(2.5) x+ =Rcos(a1nR+c),0<c<27r,

respectively, where R = F± j)2 + a2 y; here a is the exponent of the singular point

(the indexing of the curves in the image can be made identical).

The germ of equation F = 0 at a point on the surface of the equation is said to beC'-diffeomorphic to the germ of the equation Fi = 0 at a point on the surface of thelatter equation if there is a Ck-difeomorphism of a neighborhoods of the projectionsof these points on the x, y plane which transforms the germs of the families of phasetrajectories of these equations into each other (0 < k; for k = 0 we shall say thatthese germs are topologically equivalent). The smooth (analytic, in the analytical case)normal form p2 = x of the germ of a typical equation (1.3) at its regular singular pointwas first found by Cibrario [Ci] and then rediscovered by Dara [.Da] and Brodskii [A2].Brodskii used the form p2 = xE(x, y), where E is the smooth function obtained byThom [Th]. Theorem 2.6 directly implies the following theorem on normal forms.


THEOREM 2.7. The germ of equation (1.3) at a C°°-normal folded singular pointis C°°-diffeomorphic to the germ at zero of the equation (p + kx)2 = y for k =a(a + 1)-2/2 (k = (1 + a2)/8), where a is the exponent of the singular point, for asaddle or a node (accordingly, a focus).

REMARKS. 1. Remarks in the two previous sections show that the conditions ofTheorems 2.6 and 2.7 are almost always fulfilled. For instance, all folded nodes andfoci of a typical equation (1.3) are C°°-normal.

2. The change of variables z = x, y = 2(y + kx2/2) reduces the normal form(p + kx)2 = y to the Dara normal form y = (p2 + Xx2)/2 with X = 2k, where k < 0,0 < k < 1/8, and 1/8 < k for a saddle, a node, and a focus, respectively.

3. The differential equation of the family of curves (2.3) or (2.4) (accordingly(2.5)) is reduced to the normal form indicated in Theorem 2.7 by the stretchingx = ax, y = ay with a = 4(a + 1)2a-2 (accordingly, a = 16(1 + a2)-2).

Theorem 2.4 directly implies

THEOREM 2.8. The germ of equation (1.3) at a folded singular point of the equation istopologically equivalent to the germ at zero of the equation (p-x)2 = y ((p+x/9)2 = yor (p + x/4)2 = y) for a saddle (accordingly, a node or a focus).

REMARKS. 1. For the topological normal form of an implicit equation in a neigh-borhood of its folded singular point any value of k belonging to the interval (-oo, 0)((0, 1/8) or (1/8, oo)) can be taken as a saddle (accordingly, a node or a focus).

2. The three topologically different cases of the behavior of the family of integralcurves of an implicit equation in a neighborhood of its singular point (a saddle, a node,and a focus) were found in [SP], [PF], [Ta], and [Ku].

2.5. Elliptic and hyperbolic cusps. We shall show that in a neighborhood of anelliptic cusp equation (1.3) has the modules of functions with respect to the topologicalequivalence (for a hyperbolic cusp the argument is the same).

Let the equation have an elliptic pleat at zero. In a neighborhood of zero the surfaceof this equation is determined by equation x = pf (y, p) in a suitable coordinatesystem, where f is a smooth function, f y (0, 0) < 0 = f (0, 0) = f p (0, 0) < f ( 0 , 0).Consider the perturbed equation x = p f I (y, p), where f I is a smooth functioncoinciding with f for (pf (y, p))p > 0. Let us compare the phase diagrams of thetwo equations in a neighborhood of zero in the x, y plane. The germ at zero of thephase diagram of the original equation is topologically equivalent to the one shown inFigure 1.6 (the notation is the same as in Figure 1.5).

The perturbed equation possesses the same solid and dashed lines but, generally,has some other dotted lines. For the family of curves in this figure to be preservedunder a homeomorphism, the homeomorphism must be coordinate-wise in the sectorx/2 < y < 2x: x = a(x), y = b(y), and for the image of the criminant to bepreserved the conditions b (y) = 2a (y/2), a (4x) = 4a (x) must be fulfilled for x, y > 0.Consequently, all the possible homeomorphisms are restricted to the sector x/2 < y <2x by monotone continuous mappings a: (R+, 0) H (ll8+, 0) possessing the propertya(4x) = 4a(x). However, the family of dotted lines can be "spoiled" by a function oftwo variables. Consequently, even relative to topological equivalence, equation (1.3)has the moduli of functions in a neighborhood of an elliptic cusp.

REMARK. The presented argument proves Bruce's hypothesis in the smooth case.Using the results of [A5], it is shown in [Br] that the family of phase trajectories of

§3. ON PARTIAL DIFFERENTIAL EQUATIONS 17

x

FIGURE 1.6

equation (1.3) in a neighborhood of the projection of a pleated singular point can beobtained as the image, under the projection to the plane, of the family of sections ofthe standard swallowtail in 1E83 by the level surfaces of the standard function on R3.Bruce's hypothesis asserts that the pair of mappings (the projection and the function)appearing here has the moduli under diffeomorphisms that preserve the swallowtail.

2.6. The real analytical case and the case of finite order of smoothness. The argu-ment in Section 2.5 does not apply to the analytical case. However, Bruce's hypothesisitself is true in this case as well. The modules appearing in this situation are describedin [HIIY].

The assertions of the theorem also remain true in the real analytical case and inthe case of a finite but sufficiently high order of smoothness (of class Ck, k > 3). Inthe latter case a normalizing change of variables of class Ck-2 can be chosen.

§3. On partial differential equations

In this section we use the results of §2 to complete the classification of typicalsecond-order partial differential equations in the plane.

3.1. Elliptic and hyperbolic types. We continue our investigation of equation (1.2)from Section 1.2:

a (x, y) uxx + 2b (x, y) uxy + c(x, y)uyy + Ft (x, y, u, ux, u3,) = 0,

where a, b, and c are differentiable functions, Fl is a given function, and u is anunknown function. The Ck-typical equation (1.2) is an equation with coefficientvector (a, b, c) that belongs to an open and everywhere dense set in the space of suchvectors in a fine Ck Whitney topology. We set A = b2 - ac. The following obviouslemma holds.

LEMMA 3.1. For a C1-typical equation (1.2) we have d A ; 0 whenever A= 0.

For a typical function A this lemma is easily proved using Sard's theorem [GG].However, in the case under consideration the function A itself is calculated fromthree other functions, and the lemma must be proved using the strong transversalitytheorem [AGV].


Thus, for a C I -typical equation (1.2) the line of type change (in the theory ofpartial differential equations it is usually called the parabolic line or the line of typedegeneracy) is a smooth curve in the plane. On one side of this line the equation iselliptic (where A < 0), and on the other side the equation is hyperbolic (where A > 0).We state the following well-known theorem.

THEOREM 3.2. In a neighborhood of any point belonging to the ellipticity and hyper-bolicity regions equation (1.2) reduces to one of the equations

(3.1) u" + u, + Ft (x, Y, u, u. , u y) = 0,

U" - uyy + F1 (x, y, u, u., u)') = 0,

respectively, by using a suitably chosen smooth coordinate system with origin at that pointand by multiplying by a smooth positive function.

Here the function F1 is of the same class of smoothness as the function F. Thetheorem is proved in various textbooks on partial differential equations (e.g., see[CH],[P1]).

3.2. The Cibrario normal form. Given Lemma 3.1, the line of type change of aC I -typical equation (1.2) is a smooth curve in the plane. As was found in Section 1.2,the field of characteristic directions of equation (1.2) determines a smooth field ofstraight lines on this curve. A point on the line of type change is called a regular pointof equation (1.2) if the field of straight lines is not tangent to the line at that point, andis called a singular point of the equation otherwise.

THEOREM 3.3 (Cibrario). A C I -typical equation (1.2) is reduced in a neighborhoodof each of its regular points (on the line of type change) to the equation

(3.3) Yuxx + uyy + Fl (x) Y, u, u, , uy) = 0

by selecting a suitable smooth coordinate system with origin at that point and multiplyingby a smooth positive function.

As in Theorem 3.2, the function F1 in (3.3) is of the same class of smoothness asthe function F. The theorem is proved in Cibrario's paper [Ci]. This theorem can alsobe obtained as a consequence of the theorem on the normal form p2 = x of an implicitfirst-order differential equation in a neighborhood of its regular singular point; thelatter theorem is proved in [A2], [Da]. For F - 0 equation (3.3) is a special case of theChaplygin equation K(y)u,,x + u3,,, = 0, where K is a function satisfying the conditionyK(y) > 0 for y 0. Equations of this form are important in the description oftransonic gas flow [Be].

3.3. Normal form in a neighborhood of folded singular points. Similarly to foldedsingular points of typical implicit first-order differential equations, a singular point ofa C2-typical equation (1.2) may be a folded saddle, a folded node, or a folded focus.A singular point of equation (1.2) is said to be Ck -normal if the corresponding foldedsingular point of its characteristic equation is Ck -normal. The exponent of the singularpoint of equation (1.2) is equal to the exponent of the corresponding folded singularpoint. A consequence of Theorem 2.7 is

§4. THE NORMAL FORM OF SLOW MOTIONS 19

THEOREM 3.4. A C 1-typical equation (1.2) in a neighborhood of its C°°-normalfolded singular point with exponent a is reduced to the equation

(3.4) uY, + (kx2 - y)ujj, + Ft (x, y, u, u.,, uy) = 0,

where k = a(a + 1)-2/4 for a folded saddle and a folded node and k = (1 + a2)/16 fora folded focus, by selecting a suitable smooth coordinate system with origin at that pointand multiplying by a smooth positive function.

REMARKS. 1. All folded nodes and foci of a C2-typical equation (1.2) are C°°-normal. As to the C°°-normality of a folded saddle, here, similarly to the case of animplicit first-order differential equation, it is sufficient for (1, a) to be a point of thetype (M, v) for some numbers M > 0 and v (see Remark 1 on Theorem 2.3). Themapping a H k = a(a + 1)-2/4 is a diffeomorphism of the interval (-00, -1) tothe interval (-oo, 0), and the measure of the set of values of a that are not pointsof the type (M, v) for any M > 0 is zero if v > 1. Consequently, equation (3.4) isthe normal form of a C2-typical equation (1.2) in a neighborhood of one of its foldedsingular points having a typical exponent (or a typical parameter k in equation (3.4)corresponding to this exponent). Under the above conditions, equations (3.1), (3.2),(3.3), and (3.4) form the complete list of normal forms of the generic equation (1.2).

2. Given certain natural physical assumptions an equation of type (3.4) describesthe transformation of electromagnetic waves into plasma waves in cold anisotropicplasma with two-dimensional inhomogeneity [PF].

3. Transonic gas flow in a Laval nozzle is described by a quasilinear second-order equation (i.e., equation (1.2) whose coefficients a, b, and c depend both on thevariables x and y and on the unknown function and its first derivatives) [Be]. Thisequation changes its type on the sonic line where the gas velocity is equal to the speedof sound. In this case the family of characteristics depends on the solution and hasa folded saddle (no examples of transonic flows with folded nodes or foci are knownto me). Consequently, a solution describing a "smooth" gas flow is also a smoothsolution to an equation (1.2) with a folded saddle.

4. Consider a C°°-normal singular point of equation (2.1). In the x, y coordinatesystem with origin at that point, the parameter k in normal form (3.4) can be calculatedby the formula

k = (D.. (0, 0)/(D2 (0, 0),

where c _ =(b2 - ac)/a2, if the line of type change is tangent to the abscissa axis atthat point.

§4. The normal form of slow motions of a relaxation type equation on the break line

In this section we apply the results on normal forms of implicit first-order differ-ential equations to derive the normal forms of families of trajectories of slow motionsof a relaxation type equation.

Consider a relaxation type equation with two-dimensional slow variable q andone-dimensional fast variable p:

(4.1) q=eQ(q,p)+..., P=P(q,p)+eR(q,p)+...,

where Q, P, and R are smooth functions, - is a small parameter, and the dots replaceterms of order e2. By a system we mean the vector (Q, P, R). A generic system is a

20 I. IMPLICIT FIRST-ORDER DIFFERENTIAL EQUATIONS

point belonging to an open and everywhere dense set in a vector space with a fine C3Whitney topology.

The equation P(q, p) = 0 defines the slow surface of the system. This surface is theset of rest points of the system (singular points of equation (4.1)) for e = 0. For thegeneric system the differential dP is nonzero wherever the function P itself vanishes.For such a system the slow surface is a smooth two-dimensional submanifold in thespace of the variables q, p. Moreover, the folding of the system, i.e., the restriction ofthe projection (q, p) F--> q to the slow surface, has singular points only of the type ofthe Whitney fold and cusp.

In a generic system, the vector fields (0, 1) and (Q, R) are collinear on a curve thatcuts transversally the slow surface at the regular points of the system folding. Outsidethis curve we consider the field of the planes of the system spanned by these two vectorfields. This field of planes cuts out on the slow surface the direction field of slowmotion whose integral curves are called the integral curves of slow motion. We shallstudy the singularities of families of these curves in neighborhoods of break points,i.e., the critical points of the system folding. The field of planes of generic systemis a contact structure (i.e., field of planes which is locally C°°-diffeomorphic to thefield of zeros of the 1-form dy - pdx in the space of 1 -jets of functions) except ona smooth two-dimensional surface of degeneracy. Moreover, for such a system thissurface intersects transversally the line of critical points of the system folding at pointsthat are not cusp points and does not touch the kernel of the derivative of the folding atthe intersection point. Such an intersection point will be called a point of degeneracy.Consequently, for a generic system the family of the integral curves of slow motionmay have exactly the same singularities on the break line outside the degeneracy pointsas the family of integral curves of a typical implicit first-order differential equation ata singular point. More exactly, the following theorem holds.

THEOREM 4.1. For a generic system the germ of the pair (field of planes of the system;slow surface of the system) at a point on the break line that is not a point of degeneracyis reduced by a C°°-diffeomorphism fibered over the space of slow variables to the germof the pair (surface of typical equation (1.3); field of zeroes of the 1 form dy - pdx) ata singular point of the equation.

We note that for a typical system this theorem not only classifies the singularitiesof slow motion on the break line outside the degeneracy points but also normalizesthe field of planes of the system in a neighborhood of these singularities.

The normal form of the family of the integral curves of slow motion of a typicalsystem in a neighborhood of a degeneracy point was found by V. I. Arnol'd.

THEOREM 4.2 ([A4], [D2]). For a generic system the image of the germ at a point ofdegeneracy of the family of the integral curves of slow motion under the system folding isC°°-diffeomorphic to the germ at zero of the family of images of level lines of the functionf (u, v) = u + uv3 + v5 under the mapping (u, v) 1-4 (x = u, y = v2) of the Whitneyfold.

In other words, on the surface of slow variables the trajectories of slow motion canbe written in a suitable local system of smooth coordinates in the form x±xy3/2± y5l2 =

const in a neighborhood of the singularity in question.

REMARKS. 1. The relationship between implicit equations and relaxation typeequations was discovered by Takens [Ta]. V. I. Arnol'd found the lists of typical

§5. ON SINGULARITIES OF TYPICAL DIFFERENTIAL INEQUALITIES 21

singularities on the break line for both the slow motion of the system and the systemitself, see [A4], [AAIS].

2. We note that, in contrast with implicit equations, on the trajectories of slowmotion the direction of motion is defined in a natural way. To this end the vector (P,Q) at a regular point of the system folding should be projected along the fast variableon the tangent plane to the slow surface at this point. The vector resulting from theprojection is tangent at that point to the trajectory of the slow motion passing throughit. It is this vector that indicates the direction of motion along the trajectory. Forinstance, on the slow surface the families of trajectories of slow motion correspondingto folded singular points differ from those in Figures 1.1 a-c in the change of thedirection of motion on either the dashed lines or the solid lines.

The results on relaxation type equations are surveyed in [AAIS], [SZ].

§5. On singularities of attainability boundariesof typical differential inequalities on a surface

In Section 1.3 we observed the appearance of folded saddles and nodes as singu-larities of families of limiting lines of a differential inequality on the boundary of itssteep domain. In Chapter 2 we shall show that these singularities (including the foldedfocus) are typical singularities of the family of limiting lines of a control system on asurface. For a typical control system such singularities appear only on the boundary ofits steep domain (or on the boundary of its zone of complete controllability, which isthe same for typical systems). In contrast to control systems, folded singular points ofthe family of limiting lines of a typical differential inequality can also be encountered(besides points on the boundary of the zone of complete controllability) both at pointsin the steep domain and at points on the boundary of the domain of definition of theinequality. In the present section we give two examples of the appearance of foldedsingularities.

5.1. Definitions. A differential inequality F(z, i) < 0, where z = (x, y) is a point inthe plane, is determined by a smooth function F such that at each point z in the planethis inequality has a bounded (in the tangent plane) set of solutions. We identify the setof inequalities with the space of these functions and endow it with a fine C4 Whitneytopology. A typical differential inequality or a differential inequality in general positionis an inequality belonging to an open and everywhere dense set in this space relativeto the indicated topology. A velocity v c TT M is said to be feasible at a point z ifF(z, v) < 0. By the domain of definition of a differential inequality we mean the set ofpoints in the plane with at least one feasible velocity. The steep domain of a differentialinequality consists of all the points where the positive linear hull of the set of feasiblevelocities does not contain the zero velocity.

ExAMPLE 1. For the differential inequality

(5.1) (x - v(x, y))2 + (.v - w(x, y))2 <- f (x, y)

the domain of definition is determined by the inequality f (x, y) > 0, and the steepdomain is described by the inequalities v2(x, y) + w2(x, y) > f (x, y) > 0.

In the steep domain of a differential inequality a two-valued field of limitingdirections is defined, whose integral curves are called limiting lines (see Section 1.3).


5.2. Folded singularities on the boundary of the domain of definitions. Consider aswimmer in a planar sea (with coordinates x, y) carried by a current with velocity field(v, w). Let the swimmer's ability to resist the flow (the swimmer's power) depend ona point in the plane. More precisely, assume that the square of its maximum velocity(in standing water and in any direction) at this point does not exceed the value of asmooth function f at the point. Hence, the swimmer's possible motion is describedby differential inequality (5.1). We consider the space of inequalities of this form withtopology induced by the space of inequalities. The notion of typicalily is defined in asimilar way.

For a typical inequality (5.1), the differential of the function f is nonzero at allpoints where the function itself vanishes. Consequently, the boundary of the domain ofdefinition of a typical inequality is a smooth curve in the plane. Moreover, for a typicalinequality the functions v, w, and f do not vanish simultaneously. In other words,the water field has no singular points on the boundary of the domain of definition. Ateach boundary point there is a unique feasible velocity-the value of the water fieldin this point. Generally, this field rotates when moving along the boundary of thedomain of definition. Therefore, in the case of a typical inequality it may touch theboundary at some points with a first-order contact. Such tangency points are calledsingular points of the boundary of the domain of definition; the other points of theboundary are said to be regular.

THEOREM 5.1. For a typical differential inequality (5.1) the germ of the familyof the limiting lines at a point z on the boundary of its domain of definition is C°°-diffeomorphic to the germ at zero (1) of the family of integral curves of the equation(y')2 = x if z is a regular point of the boundary and (2) of the family of integral curvesof equation (y' + a (x, y))2 = yb(x, y), where a and b are smooth functions, b (O, 0) = 1,a (0, 0) = 0 # a, (0, 0) # 1/8, if z is a singular point of the boundary.

At a singular point on the boundary of the domain of definition, the family oflimiting lines has a folded saddle, a folded node, and a folded focus for a, (0, 0) < 0,0 < a, (0, 0) < 1/8, and 1/8 < a, (0, 0), respectively (also see Theorem 4.10 inChapter 2). We do not present the proof of Theorem 5.1. It is based on the calculationof the field of limiting directions of inequality (5.1). This field turns out to be thedirection field of an implicit first-order differential equation. The proof is completedby applying the general theory of such equations.

The swimmer can move along the limiting lines with a velocity belonging to the fieldof limiting velocities. Thus, there is a natural direction of motion on these lines. Figures1.1a-c will demonstrate the behavior of the family of limiting lines in a neighborhoodof the folded singular points under study if the direction of motion is changed to theopposite on either the dashed or solid lines. The folded saddles and nodes may leadto singularities on the attainability boundary of a differential inequality that are stablewith respect to small perturbations of both the inequality and the start set.

ExAMPLE 2. For (v, w) = (1, -kx), where k E R, and f (x, y) = y, inequality(5.1) takes the form

(5.2) (X - 1)2 + (,v + kx)2 < Y.

The domain of definition of this inequality coincides with the closure of the upper halfplane, and the boundary of the domain is the abscissa axis. Except for zero, all thepoints on the abscissa axis are regular points of the boundary; zero is a singular point.

§5. ON SINGULARITIES OF TYPICAL DIFFERENTIAL INEQUALITIES 23

FIGURE 1.7

The family of limiting lines of differential inequality (5.2) has a folded saddle, a foldednode, and a folded focus for k < 0, 0 < k < 1/8, and 1/8 < k, respectively. In thecase of a saddle or a node this point lies on the attainability boundary if the line y = 1is taken as the start set. The attainability boundary has singularities at that point, i.e.,is not smooth. It can easily be seen that the observed phenomenon is stable relative tosmall perturbations of both the start set and the differential inequality.

5.3. Folded singularities inside the steep domain.

ExAMPLE 3. Consider a smooth function yr on the line, which is equal to one onthe interval [-1, 1] and to zero outside the interval [-2, 2] and is strictly monotone oneach of the intervals [-2, -1] and [1, 2]. At each of the points in the plane the set offeasible velocities of the inequality

(5.3)w(x2 + (y - 100 - 2(kx)2 -2Y2)2)[x2 + (1' - 100 - 2(kx)2 -2 Y2)2 - 1]

+ (1 - ,(X2 + (1' - 100 - 2(kx)2 -2Y2)2))[(x - 1)2 + (Y + kx)2 - y] < 0

is the union of the sets of feasible velocities at the point of the two inequalities

(5.4) )C2 + (y - 100 - 2(kx)2 -2Y2)2 < 1,

(5.5) (x - 1)2 + (v - kx)2 < Y.

Clearly, zero belongs to the steep domain of inequality (5.3). Consider the familyof limiting lines of the "minimal" direction (as usual, the angles in the plane arecounted counterclockwise). This family coincides with the family of limiting lines ofthe minimal direction of inequality (5.4) in the lower half plane and of inequality (5.5)in the closure of the upper half plane. Hence, in a neighborhood of zero in the upperhalf plane the family of limiting lines of the minimal direction of the unified inequalityin a neighborhood of zero has "half" a folded singularity if 0 k 1/8. Figure 1.7demonstrates the behavior, in a neighborhood of zero, of the family of limiting lines ofthe unified inequality for 0 < k < 1/8, that is, in the case of a folded node. The solidand dashed lines in the figure represent the limiting lines of the minimal and maximaldirections, respectively, and the double line is the abscissa axis.

We note that during a motion at a velocity belonging to the velocity field ofthe minimal direction, there is a naturally determined sliding regime on the positivex-halfaxis.


Take a point A on the negative x-semiaxis lying sufficiently close to zero. Theboundary of the positive orbit of any point which is sufficiently close to A passesthrough zero and has a singularity there. This phenomenon is stable with respect to tosmall perturbations of both the differential inequality and the start point.

REMARKS. 1. The problem of studying the attainability sets and the family oflimiting lines of typical differential inequalities was formulated by Myshkis [My]. Wesolve this problem for typical control systems on a surface in Chapters 2 and 3. Inthese studies folded singularities are important.

2. We also mention the application of normal forms of implicit differential equa-tions to the analysis of the behavior of the family of asymptotic lines in a neighborhoodof parabolic points of a generic surface. For example, the germ of a typical surface atsome points on the line of its parabolic points is reduced by a projective mapping (sucha mapping preserves the family of asymptotic lines) to the germ at zero of the surfacez = y2 + yx2 + Ax4 + o((x2 + y2)2), where A is a number [La]. On the latter surfacezero is a folded saddle, a folded node, or a folded focus of the family of asymptoticlines for A < 1/4, 1 /4 < A < 25/96, or 25/96 < A, respectively.

§6. Proof of Theorems 2.1 and 2.3

6.1. Proof of Theorem 2.1. Lemmas 6.1, 6.2, and 6.3 are needed to prove thetheorem and are themselves proved in Sections 6.3, 6.4, and 6.5. Let v be a smoothdirection field having a nondegenerate singular point at zero.

LEMMA 6.1. The germs at zero of two v-good involutions having the same lines offixed points are v-equivalent.

LEMMA 6.2. The germs at zero of two smoothly embedded curves tangent to eachother at zero are v-equivalent if none of the eigendirections of the field v is tangent tothese curves at zero.

LEMMA 6.3. Two different directions at zero are v-equivalent if and only if they canbe joined (in the space of directions at zero) with a continuous curve not passing throughthe eigendirections of the field v at zero.

Assume that the tangent directions at zero to the lines of fixed points of two v-goodinvolutions can be joined in the space of directions at zero with a continuous curve notpassing through the eigendirections of the field v at zero. Then, by Lemma 6.3, thesetangents are v-equivalent. According to Lemma 6.2, the germs at zero of the lines offixed points of involutions are v-equivalent. By Lemma 6.1, it follows that the germsat zero of the two involutions are v-equivalent.

Conversely, if the germs at zero of two v-good involutions are v-equivalent, thenthe germs at zero of their lines of fixed points and their directions at zero are alsov-equivalent. By Lemma 6.3, these directions can be joined with a continuous curvenot passing through the eigendirections of the field v at zero.

6.2. Proof of Theorem 2.3. For a focus the theorem follows from Theorems 2.1and 2.2. In the case of a saddle or a node Theorem 2.3 is also implied by Theorems 2.1and 2.2 and the fact that the involution (x, y) H (-x, y) transforms the family of phasetrajectories of field (2.1) into itself, and the two connected components in Theorem 2.2are transformed into each other under the involution.

§6. PROOF OF THEOREMS 2.1 AND 2.3 25

6.3. Proof of Lemma 2.1. By the field of infinitesimal deformation of an involutiona is meant a vector field whose value at a point a(.) is the velocity of the point undera change of the finite involution. We have the following obvious lemmas.

LEMMA 6.4. A vector field h is the field of infinitesimal deformation of an involutiona if and only if a*h = -h.

LEMMA 6.5. If g is the deformation of an identical diffeomofphism at a rate h, thenthe involution a is deformed at a rate h - a*h.

(Under the diffeomorphism g the involution a goes into gag-1.)We now prove Lemma 6.1. Let a1 and a2 be v-good involutions with the same line

of fixed points. Consider a smooth function cp, cp(0) = 0, having nonzero derivativesat zero with respect to each of the eigendirections of the derivative of the involutional at zero. Locally, in a neighborhood of zero, in the coordinates x = cp + a* cp,y = W - a* W the involutions al and a2 have the form a1: (x, y) H (x, -y), a2:(x, y) F_+ (x + y2r(x, y), -y + y2s(x, y)), where the functions r and s are smoothsince al and a2 have the same line of fixed points. The derivatives of these involutions onthis line are the same for small x and y since both al and a2 are v-good. Consequently,there exist the coordinates = x + y2R(x, y) and q = y + y2S(x, y), where R and Sare smooth functions, in which the involution a2 has the form a2: (c, rl) i -rl).

Locally in a neighborhood of zero we consider the smooth deformation y,:ii,) H -It) that transforms the involution al into a2, where , = - x +

ty2R(x, y), q, = y + ty2S(x, y). We have yo = al, y1 = a2. Denote by V, therate of this deformation.

We take the smooth vector field v that determines the direction field in questionand has a nondegenerate singular point at zero. Lemma 6.1 will be proved if we willbe able to represent the deformation rate in a neighborhood of the axis t in the form

(6.1) V, = f,v - (Yr ,/ ,)Y,*v,

where f, is a smooth function of the variables x, y depending smoothly on t. Letus show that this representation does in fact exist. The solvability of the homologicalequation (6.1) with respect to f, is based on the fact that the field v and its imageunder the involution y, are nonlinear outside the line of the fixed points.

As can easily be seen, the deformation rate V (we omit the subscript t) has atwo-fold zero on the curve y = 0 (rl = 0). By Lemma 6.4, we have y* V = - V.Consequently,

(6.2) V rl) = r13P(c,112)a/aa + r12)a1 0,j1,

where p and q are smooth functions.On the line of fixed points of the involution y we have y*v = -v. Consequently,

(6.3) rl) = rll rl)a/ac +

m are smooth functions. Representing f in the form f i12) = t12) +rlw r12), where u and w are some functions, and substituting this expression for fand also expressions (6.2) and (6.3) for V and v into (5.6) we arrive at the followingsystem with respect to u and w:

U17 (1 (c, 17) + l -r1)) + wr12(l ( , rl) - l (c, -11)) = p2);

q) + -v)) + -q)) = 1/2).


Cancelling out #7 in the first equation we obtain a linear system with respect to uand w whose determinant has the form 82(41(0, 0)m,, (0, 0) + H(c, #2)), where H isa smooth function; H(0, 0) = 0 since zero is a singular point of the field v, and, inparticular, we have m(0, 0) = 0; 1(0, 0)m,, (0, 0) 0 0 because this singular point isnondegenerate. Now, since the right-hand side of the system is divisible by 72, weconclude that in a neighborhood of the axis t there exists a smooth solution u, w tothe system. Lemma 6.1 is proved.

6.4. Proof of Lemma 6.2. Let v be a vector field determining the direction field vand having a nondegenerate singular point at zero. Denote by g' the mapping of thephase flow of this field at time t.

Let us perform the sigma-process with center at zero. Two curves in question willbe transformed into two smooth curves passing through a point on a pasted projectiveline transversal to both of them. The vector field is regularly extended to this pointand is tangent to the pasted line. Therefore, the time it takes to move along the fieldfrom one of the curves to the other is smooth function z of the point on the first curve.The desired v-equivalence is where T is the smooth extension of the functionz to the plane. Lemma 6.2 is proved.

6.5. Proof of Lemma 6.3. The diffeomorphism that transforms each integral curveof the field v into itself transforms into itself each of the open sectors into which thedirection space at zero (this is a one-dimensional projective space) is split by theeigendirections of the field v at zero.

Conversely, two directions belonging to a sector of a field v = Ax + ... having anondegenerate singular point at zero are transformed into each other by the mappingeAt for a suitable value of t and, consequently, by one of the transformations of thephase flow of the field v" as well. Lemma 6.3 is proved.

§7. Proof of Theorems 2.5 and 2.8

7.1. Proof of Theorem 2.5. Let a smooth direction field v have a nondegeneratesingular point at zero and let an involution a be v-good. We take a coordinate systemx, y in a neighborhood of zero with origin at zero that normalizes the involutiona: (x, y) (x, -y). Let v(x, y) = (yA(x, y2) + y2B (x, y2), C (X, y2) + yD (x, y2))be a smooth vector field determining the direction field and having a nondegeneratesingular point at zero. In a neighborhood of zero in the plane the differentialequation of the family of images of the integral curves of the field v under the mapping(x, y) H ( = x, q = y2) satisfies the conditions of the theorem. This equation iswritten as

B

Theorem 2.5 is proved.

7.2. Proof of Theorem 2.8 (for a saddle; the arguments are the same for a nodeand a focus). Although Theorem 2.8 follows from Theorem 2.4, we shall prove itindependently; Theorem 2.4 is a direct consequence of Theorems 2.5 and 2.8.

We first consider a C°°-normal folded saddle with exponent P. Without lossof generality, we can assume that the family of curves (2.3) with a = /3 is underconsideration. Let C1 (x, y) and C2(x, y) be the indices of the phase trajectories of thefamily that pass through a point (x, y), y > 0 (namely, C1 (x, y) = (x/b + /) I x +

§7. PROOF OF THEOREMS 2.5 AND 2.8 27

V/y-I-I and C2(x,y) = (x//3 - Vly-)I x - IyJ-I -J'). We define a continuous mapping WQ

of the plane of the variables x, y onto the plane of the variables C1, C2:

pa : (x,y) H (CI (x, Iy1)sgn(x + Iyl), C2 (x, IYI)sgn(x - lyl))

This mapping is a homeomorphic transformation of the closure of the first (fourth)quadrant into the set C> > C2, while the closure of the second (third) quadrantgoes into the set C1 < C2 under the transformation. Consequently, the mapping DQdefined by the formula cp = cp=2 x cpfl on the closure of each of the quadrants is ahomeomorphism of the plane. The definition of cpp implies that this homeomorphismmaps the family of curves (2.3) with a = 8 onto the same family with a = -2.

To construct a similar homeomorphism for an arbitrary folded saddle with ex-ponent /3, it is necessary to number in a suitable manner the phase trajectories in aneighborhood of zero. We choose a smooth coordinate system in the plane of thevariables x, y so that the image of the germ at the singular point in question on thesurface of the equation F = 0 (of the closure of the union of the separatrices of thesaddle) under the equation folding coincides with the germ at zero of the set y > 0(accordingly, (y = x2) U (y = x2//32)). We assign the index C = 0 to the imageunder the equation folding of the separatrices of the saddle under study; and assign theindex C = x Ix I -Q//3 to the phase trajectory passing through the point (x, 0) for smallx. For small y > 0 there are two phase trajectories passing through the point (0, y).We assign the index C = -y(1-Q)/2 (C = y(1-fl)V2) to the phase trajectory passingthrough this point and having a greater (accordingly, smaller) angle of inclination tothe abscissa axis. The mapping 1 p is now defined locally in a neighborhood of zero inthe same way as for a normal folded saddle. This mapping transforms the germ at zeroof the family of phase trajectories of the equation into the germ at zero of the familyof curves (2.3) for a = -2.


CHAPTER 2

Local Controllability of a System

In this chapter some results on the local controllability of a typical system will beobtained as a consequence of the classification of singularities of the limiting directionfield of the system. The classification is presented in §4. In § 1 we define the class ofsystems under study. In §2 we consider the simplest case when the number of differentvalues of the control parameter is equal to two. In §3 we show what qualitative changesoccur in the field of limiting directions of a typical system with increasing number ofdifferent values of the control parameter. The concluding sections (§§5-10) are devotedto the proofs of the main results from §4.

§1. Definitions and examples

In this section the basic notions of Chapter 2 are defined and illustrative examplesare presented.

1.1. The class of control systems. We assume that the phase space of a controlsystem (i.e., its state space) is a smooth real surface without the boundary endowedwith a Riemann metric. Here and henceforth the term "smooth" means "infinitelydifferentiable" or, which is the same, "of class C°°". The control system itself isdetermined in the vicinity of each point on the surface (i.e., in a sufficiently smallneighborhood) by an equation

(1) z = f (z, u),

where z is a point on the surface, a = dz/dt, u is a control parameter running over aunion U of the finite number of pairwise disjoint compact smooth manifolds, and fis a smooth function with respect to the set of the variables.

DEFINITION. The set f (z, U) is called the velocity indicatrix of the point z, andevery velocity belonging to it is called a feasible velocity at that point.

EXAMPLE 1. A swimmer in the current [My]. The swimmer is moved in the (x, y)plane by the water current with velocity field (v(x, y), w(x, y)). The swimmer himselfcan swim in standing water in any direction with unit velocity. His possible motion inthe stream is described by the control system

z =v(x,y)+cosu, y =w(x,y)+sinu,

where u is a circular angle, which defines the direction of the swimmer's motion instanding water.

29

30 2. LOCAL CONTROLLABILITY OF A SYSTEM

EXAMPLE 2. Ship drifting. An "inertialess" ship in a sea in the xy plane loosespower and control. It is alternately drifted by a water current with velocity field(vI (x, y), v2(x, y)) and near-surface wind (an air flow) with velocity field (WI (x, y),W2 (x, y)). In this situation the ship's possible motion is described by the control system

x=uv,(x,y)+(1-u)w,(x,y), y=uv2(x,y)+(1-u)w2(x,y),

where u can take the values 0 or 1.

The representation of a control system in the form of equation (1) only gives itstechnical potential. The possible motion of the system in the phase space is determinedby the class of feasible controls.

DEFINITION. By a feasible control we mean a piecewise continuous function of timewith a range in the set U.

Selecting a feasible control and substituting it into equation (1), we obtain theordinary differential equation

(2) z = f (z, u(t))

This equation satisfies the conditions of the existence and uniqueness theorem becauseits right-hand side is a differentiable function in the variable z and a piecewise continu-ous function in the variable t. Consequently, on choosing the starting time of motionand the starting point (the initial state of the system) we can find, in accordance withequation (2), the corresponding feasible motion z(t) of the system, at least for thevalues of time sufficiently close to the starting time. The trajectory of this motion iscalled a feasible trajectory of the system.

DEFINITION. A point z2 is said to be attainable from a point zI in time T if thereexists a feasible motion z(t), 0 < t < T, such that z(0) = zi and z(T) = z2, and issaid to be attainable from the point zI if there exists at least one such value of T.

The choice of the zero starting time in the latter definition is not essential becausethe beginning of the motion can be transferred to any specified time to by a delay inthe feasible control: u(t) = u(t - to). The set of points attainable from a point z intime t will be denoted A, (z); for negative values of t this set consists of all points fromeach of which the point z is attainable in time It I.

DEFINITION. The set of all points attainable from a point z is called its positiveorbit, and the set of all points from which point z is attainable is called its negativeorbit.

EXAMPLE 3. Consider the following simple motion in the plane: X2 + y2 = 1 or,which is the same, z = cos u, y = sin u, where u is a circular angle. The set of pointsattainable from a point (x, y) in time t is the circle of radius Its centered at (x, y). Thepositive (negative) orbit of each point coincides with the whole plane.

EXAMPLE 4. The swimmer's motion in a current with the velocity field (2, 0) isdescribed by the control system x = 2 + cos u, y = sin u, where u is a circularangle. The set of points attainable in time t from a point (xo, yo) is a circle of radiusItI centered at (xo + 2t, yo). The union of these circles over all t > 0 (t < 0) isthe positive (negative) orbit of the point and is the cone x - xo > (y - yo) v >(x - xo), respectively, (-(x - xo) > (y - yo)/ > (x - xo)) (Figure 2.1).

§ 1. DEFINITIONS AND EXAMPLES 31

FIGURE 2.1

REMARK. Below controls (control actions) are taken only from the class of feasiblecontrols.

1.2 Steep domain, local transitivity zone, and rest zone.

DEFINITION. For a control system, by the cone of a point we mean the positivelinear hull of the velocity indicatrix of the point. The steep domain of a control systemis the set of all points in the phase space whose cones do not contain the zero velocity.

EXAMPLE 5. We come back to Examples 3 and 4 from the previous section. Inthe first of them the cone of each of the points in the plane coincides with the entiretangent plane, and, consequently, the steep domain is empty. In the other example thecone of each point in the plane consists of the velocities (u, v), u > IvIv > 0, andhence does not contain the zero velocity. Consequently, the steep domain coincideswith the whole plane.

If the cone of the point does not coincide with the entire tangent plane, then thedirections of the velocities belonging to it form an angle not exceeding 180°. The sidesof the angle are called the limiting directions at that point. Hence, a two-valued fieldof limiting directions is defined in the steep domain. Its integral curves are called thelimiting lines.

EXAMPLE 6. The swimmer's motion in the stream with the velocity field (-x, -fly),fl > 1 is described by the control system x = -x + cos u, y = -fly + sin u, where uis a circular angle. The steep domain of this system is determined by the inequalityx2 + fl2y2 > 1. The field of limiting directions is defined in the closure of the domain.It can easily be shown by calculation that the limiting lines are exactly the integralcurves of the differential equation (x dy - fly dy)2(x2 + fl2y2 - 1) = (x dx + fly dy)2(also see Section 1.3 in Chapter 1).

DEFINITION [My]. By the zone of local transitivity of a control system we meanthe set of all points in the phase space such that each of them and any point lyingsufficiently close to it are attainable from each other in a small time.

DEFINITION. A point in the phase space is called a rest point of a control system ifthe system can stay in any preassigned time in an arbitrary neighborhood of this pointfor the initial point of motion taken at that point and a suitably chosen control action.The set of all rest points is called the rest zone of the control system.

THEOREM 1.1 (on the rest zone). The rest zone of a control system contains the localtransitivity zone and does not intersect the steep domain.


g(.)=g(Z)+2

g(.) = g(z)

FIGURE 2.2

COROLLARY 1.2. The local transitivity zone and the steep domain of a control systemare disjoint.

PROOF OF THE THEOREM. We start with proving the second assertion of the theorem.It suffices to show that each point in the steep domain is not a rest point of the system.The velocity indicatrix of every point of the phase space is a closed set, and the cone of apoint belonging to the steep domain does not contain the zero velocity. Consequently,at a point z of the steep domain in the tangent plane there is a straight line such thatthe zero velocity and the velocity indicatrix I (z) of the point lie on different sides ofthat line. However, by the definition of the class of systems under consideration, thefield of indicatrices is continuous (i.e., the indicatrices at two points lying close to eachother are close in the Hausdorff metric [Fl]). Hence, there is a smooth function gon the phase space whose derivative along any feasible velocity at an arbitrary pointbelonging to the closure of a neighborhood V of the point z is positive. The minimumof the derivative over all points in that closure and over all velocities belonging to theindicatrices of these points is equal to a positive constant c.

Consequently, any feasible motion trajectory starting from the point z does notleave the neighborhood V in a sufficiently short time t > 0 but passes from the level lineg (z) of the function g to a higher level g (z) + ct/2 (see Figure 2.2; the double line inthe figure is the velocity indicatrix of the point z, the set A, (z) is shaded, and the levellines of the function g are shown in solid lines). Thus, for any choice of the controlaction the system leaves a sufficiently small neighborhood of the point z in a timeless than t. Consequently, the point z is not a rest point of a control system. By thearbitrariness in the choice of the point belonging to the steep domain, we conclude thatthis region and the rest zone of the control system are disjoint. The second assertionof the theorem is proved.

The first assertion of the theorem immediately follows from Lemma 3.1. In anyneighborhood of a point belonging to the local transitivity zone control system mayremain indefinitely long if the initial point of motion is taken at this point and if thecontrol action is suitably chosen.

The theorem is proved modulo the lemma.

PROOF OF THE LEMMA. Consider an arbitrary neighborhood V of a point z in thelocal transitivity zone. In a probably smaller neighborhood V, V c V, the moduli ofthe feasible velocities are bounded by a constant. Consequently, if a point zi, zl L z,

§1. DEFINITIONS AND EXAMPLES 33

FIGURE 2.3

lies sufficiently close to the point z, then the two points are attainable from each otheralong the arcs of feasible trajectories lying entirely inside V. Pasting together the arcgoing from the point z to the point z1 and the arc going from the point z1 to thepoint z we obtain a closed feasible trajectory lying in this neighborhood (Figure 2.3).The periodic motion along this trajectory makes it possible for the system to stayindefinitely long in the neighborhood V and hence in V as well. The lemma is proved.

THEOREM 1.4 (on the boundaries). The boundary of the convex hull of the velocityindicatrix at each point on the boundary of the steep domain (accordingly of the localtransitivity zone and of the rest zone) contains the zero velocity.

PROOF. According to the choice of the class of control systems under study, thevelocity indicatrix of each point is bounded and closed, and the field of indicatrices iscontinuous. Consequently, the convex hull of the velocity indicatrix is also boundedand closed, and the field of the convex hulls of the indicatrices is continuous. Inparticular, if at a point z in the phase space the zero velocity lies outside (inside) theconvex hull of the velocity indicatrix of that point, then the closure of a neighborhoodof the zero velocity lies outside (inside) the convex hull of the velocity indicatrix ofany point lying sufficiently close to the point z. Consequently, z is an interior point ofthe steep domain (accordingly, of the local transitivity zone [Pe] and, by the previoustheorem, of the rest zone as well). The theorem is proved.

REMARK. The dimension of the phase space is not essential in the proofs of The-orems 1.1, 1.4, and Lemma 1.3. These theorems and the lemma are true for controlsystems whose phase spaces have dimensions higher than two.

Generally speaking, the rest zone of a control system is "broader" than the localtransitivity zone.

EXAMPLE 7. The rest zone of the control system in the problem of a swimmer ina stream with velocity field (1, 0) coincides with the whole plane whereas the localtransitivity zone is empty.

REMARK. As will be seen in §4, the two zones typically differ by a discrete set ofpoints.

1.3. Ship drift.

EXAMPLE 8. Let us determine the rest zone and the local transitivity zone in theproblem of ship drift in the current with velocity field (2, 0) and near-surface wind


FIGURE 2.4

with velocity field (-1, y - x2) (see Example 2 in Section 1.1). The ship's motion isdescribed by the control system

z=3u-1, y=(1-u)(y-x2),where u assumes one of two values 0 or 1. The water and wind velocity fields arecollinear on the parabola y = x2 and determine a field of straight lines on it. The fieldis tangent to the parabola only at the point (0, 0) with the first order of contact; thisproperty is unremovable under a C2-small perturbation of the wind and water fields.For a pair of wind and water fields that are close to each other, their line of collinearityis close to the parabola, and the field of straight lines on this curve is tangent to it withthe first order contact at a point lying close to zero.

PROPOSITION 1.5. For the system in question the steep domain coincides with thecomplement of the parabola y = x2, the rest zone coincides with the parabola itself, andthe local transitivity zone is the part of the parabola in the region y > 0.

PROOF. We first show that all the points of the parabola y = x2 in the upper halfplane belong to the local transitivity zone. For definiteness, consider a point z on theright branch of the parabola. At every point on the parabola lying near z the windand water fields have opposite directions and the phase trajectories of these fields aretangent with the first order of contact. Consequently, there exists an arbitrarily smallcycle composed of parts of the wind and water phase trajectories that encloses thepoint z (Figure 2.4). From this cycle it is possible to attain any point in the regionbounded by it by drifting with the water field at a point on the arc CA with the sameordinate. From each point in this region this cycle can also be attained by driftingwith the water field. Consequently, going from point to point in this region can bedone using no more than two switchings. The transition time does not exceed two ABlengths. Hence, the point z belongs to the local transitivity zone and, by Theorem 1.1,to the rest zone as well.

The origin does not belong to the local transitivity zone because it is impossibleto attain zero from any point A on the negative semiaxis of ordinates for any choiceof the control action (Figure 2.5; the positive orbit of point A is shaded, and the solidand dashed lines are the phase trajectories of the wind and water fields, respectively).

We now show that the origin does not belong to the rest zone. Consider a neigh-borhood of the origin and a sufficiently small number 6 > 0 such that the circlex2 + y2 < 62 lies in this neighborhood. It suffices to show that for any time T >> 1

§ 1. DEFINITIONS AND EXAMPLES 35

FIGURE 2.5

there is a feasible motion z(t), 0 < t < T, z(0) = (0,0), during which the systemremains in this circle and hence in the chosen neighborhood. Take a positive numbere << 1 and consider the following motion of the system. It starts from zero under thedrift by the wind field and then alternately uses the wind field during time a and thewater field during time e/2. The trajectory of motion will lie within the vertical linesx = -e and x = 0. When attaining the left vertical line, the system will switch to thewater field and when attaining the right vertical line, it will switch to the wind field.We realize n such cycles (that is, 2n - 1 switchings) and attain the axis of ordinates.

LEMMA 1.6. The absolute value of the ordinate of the point of attainment of the axisOy after 2n - 1 switchings does not exceed the expression e3 (q + q2 + + q"), whereq = exp e.

Let us apply the lemma. We have

e3(q + q2 + ... + q") = e3q(q" - l)/(q - 1) < e2q»+1

Consequently, the trajectory of motion of the system lies within a circle of radius(e2 +e 4 q 2n+2)1/2 < e(1 + q2,t+2)1/2 with center at zero. The radius is definitely lessthan 6, and the time of motion 3ne/2 exceeds T if we choose e = 6/(1 + e2T+4)1/2 andthe number of cycles n satisfying the relation T + 1 < (n + 1)e < T + 2. Consequently,the origin is a rest point of the system. The proposition is proved modulo the lemma.

PROOF OF LEMMA 1.6. When the system moves with the water field, the ordinate ofsystem's state does not change. It decreases each time the wind field is used (Figure 2.5),i.e., when moving along the phase trajectories of the system of differential equationsz = -1, y = y - x2 during time e from the point (0, a), a < 0, of the latest attainmentof the axis of ordinates. The general solution to the equation y = y - x2 has theform y = x2 - 2x + 2 + C exp(-x), C E R. The value C = a - 2 correspondsto the phase trajectory of this equation passing through the point (0, a). This phasetrajectory intersects the vertical line x = -e at a height

(3)

e2+2e+2-2q+aq=e2+2e+2-2(1+e+e2/2+e3/3!+...)+aq_ -2(e3/3!+...)+aq> -e3q+aq_ (a - 6 3)q


We now prove the lemma by induction on the number of cycles. For n = 1 theassertion of the lemma is true because the motion starts at the point a = 0 and, afterthe switching on the vertical x = -e, attains a point on the axis Oy with ordinatewhose absolute value is less thane 3q. Assume that the assertion of the lemma is truefor n = k, i.e., after 2k - 1 switchings (k cycles) the motion attains a point on the axisOy with ordinate a whose absolute value is less than b = -'(q + q2+. + q"). Let usshow that this is also true for n = k + 1. Indeed, of the two phase trajectories of thewind field passing through the points (0, a) and (0, -b), respectively, the former liesabove the latter. Consequently, the latter intersects the vertical x = 6 - e at a pointwith a smaller ordinate than the former. Whence, by formula (3), we conclude thatafter k + 1 cycles the absolute value of the ordinate of the point of attainment of theaxis Oy does not exceed I C - b - e3)I = e3(q + q3 + + qk+I), which is what weintended to prove.

REMARKS. 1. The above example shows that the terms "rest point" and "restzone" are conditional. They must be interpreted from a purely practical viewpointin the sense that even with high resolution equipment and after observing the systemfor a finite period of time (e.g., a year) we shall fail to detect the system leaving aneighborhood of such a point.

2. In this example the control system starts moving from zero and leaves in a finitetime any bounded neighborhood of zero for an arbitrary choice of the control action.

§2. Singularities of a pair of vector fields on a surface

Here we study singularities of the families of phase trajectories of a typical pair ofvector fields on a surface. These singularities are important when studying the fieldsof limiting directions of both a bidynamical system and a typical control system.

2.1. Definitions. For a pair of vector fields v = (VI, v2) and w = (WI, W2) in theplane Oxy we set p = VIW2 - v2wl.

DEFINITION. A pair of vector fields in the plane is said to be typical if the followingthree conditions are fulfilled:

(1) every singular point of each of the two fields is nondegenerate (in particular,the eigenvalues of the linearization of each of the fields at its singular point,which is a node, are different);

(2) zero is not a critical value of the function p, i.e., dp 54 0 if p = 0;(3) if the derivative of the function p at a point belonging to the line p = 0 along

each of the fields v and w is zero, then the second derivative of that functionalong each of the fields is nonzero.

DEFINITION. For a typical pair of vector fields in a plane, a point z in that plane iscalled a regular point if p(z) 0, a saddle (accordingly, nodal and focal) zero passingpoint if z is a singular point, which is a saddle (accordingly, a node and a focus),a passing point (a 0 -passing point) if p(z) = 0, the derivative of the function p atthat point along each of the fields is nonzero, and the directions of these fields at thepoint are the same (opposite), and a turning point (a 0-turning point) if p(z) = 0, thederivative of the function p at the point along each of the fields is equal to zero, andthe directions of the fields at this point are the same (opposite).

REMARK. A typical pair of vector fields on a surface is defined in a similar way,and the same names are used for the points of the pair.

§2. SINGULARITIES OF A PAIR OF VECTOR FIELDS ON A SURFACE 37

EXAMPLE 1. The pair of vector fields (x, -y) and 1, 2x - 1 in the plane is typical.For this pair the origin (0, 0) is a saddle zero-passing point, (1/3, 1/9) is a turningpoint, the other points of the parabola y = x (1 - 2x) are 0-passing points for x < 0and passing points for x > 0, and the rest of the points in the plane are regular.

2.2. The singularities of a pair of fields.

PROPOSITION 2.1. For a typical pair of vector fields on a surface each point of thesurface belongs to one of the following six types: a regular point, a zero passing point, apassing point, a a-passing point, a turning point, or a a-turning point.

A typical pair of vector fields at the points of these types has the following singu-larities.

THEOREM 2.2 (on a pair of vector fields). For a typical pair of vector fields v and won a surface the germ of the families of their phase trajectories at each point on surface isCk-dii feomorphic to the germ of the families of phase trajectories of one of the followingeleven pairs of vector fields:

(i) the following six pairs of vector fields: (1) (1, 0) and (0, 1), (2) (1, X) and (1, -x),(3) (1, x) and (-1, x), (4) (1, y - x2) and (1,x2 - y), and (5) either (l' y - x2) and(-1, y - x2) or (-1, x2 - y) and (1, x2 - y) if the given point is, respectively, (1) aregular point, (2) a passing point, (3) a a -passing point, (4) a turning point, and (5) a0-turning point; in this case k = oo;

(ii) the following five pairs of vector fields: (1) (1, 1) and (-x, y), (2) either (1,1)and (-x, -2y) or (1, 1) and (x, 2y), and (3) either (1, 0) and (x - y, x) or (1, 0) and(-x - y, x) if the point in question is (1) a saddle zero passing point, (2) a nodalzero passing point, and (3) a focal zero passing point, respectively; in this case k = 0.

The eleven types of singularities of a typical pair of vector fields indicated in thetheorem are illustrated in Figures 2.6a-k, respectively. Shown in this figure is the familyof limiting lines of a control system defined in a neighborhood of the singularities ofthe corresponding pair of vector fields. The thin solid and dashed lines are the integralcurves (the limiting lines) of the first and second branches of the field of limitingdirections, respectively; the double line is the set of passing points; the thick line is theset of 0-passing points; each of the turning points, 0-turning points, and zero-passingpoints is encircled; and the coordinate axes are omitted.

2.3. Proofs of the proposition and the theorem. In a neighborhood of a point z onthe surface we select a system of local coordinates x, y with origin at that point. Ifp(z) 54 0, then this is a regular point. In this case each of the vector fields v and w hasa nondegenerate smooth first integral in a neighborhood of the point. The values ofthese integrals determine a smooth coordinate system in a neighborhood of the pointz. Consequently, the pair of the fields has a singularity (il) at the point z (Figure 2.6a).

Let p(z) = 0. Then one of the following three cases is possible: (1) both thederivatives at the point z of the function p along the fields v and w are nonzero, (2)one of them is zero and the other is not, and (3) both are zero. We consider these threecases in turn.

The two derivatives are nonzero. At the point z both fields are nontangential tothe line p = 0 because otherwise at least one of the derivatives would be zero. Inparticular, the values of the two fields at this point are nonzero. Consequently, z iseither a passing point or a 0-passing point if these values have the same or oppositesigns, respectively.


-s+

(a) (b) (c)

y=x2

(d)

(S)

(e)

(h)

(f)

(i)

(j) (k)

FIGURE 2.6

In a neighborhood of a nonsingular point a smooth field is smoothly rectifiable[A3]. Consequently, it can be assumed that one of the fields is the field (1, 0). Near thepoint z the line p = 0 can be described by an equation x = g(y), where g is a smoothfunction, because the field (1, 0) is not tangent to this line at zero. We make the changeof variables z = x - q (y), y = y preserving the rectified field to transform the line


p = 0 into the axis of ordinates (in what follows the tilde over the new coordinates isdropped).

The second of the fields is collinear to the first one on the coordinate axes and isnonzero at zero. Consequently, after multiplication by a smooth positive function (thisoperation preserves the phase trajectories) this field takes the form (e, xa (x, y)), wheree is equal to either 1 or -1 if z is a passing point or a 0-passing point, respectively, anda is a smooth function. By the definition of a typical pair of vector fields, dp 0 ifp = 0. Consequently, a (0, 0) 0, and the second field has a first integral of the formof y + x2A(x, y), where A is a smooth function, A(0, 0) $ 0. In a neighborhood ofzero the smooth change of variables

x = xIA(x,y)/211/2, Y = ysgnA(0,0) +z2/2(sgn A is equal to 1 if A > 0 and to -1 if A < 0) transforms the families of phasetrajectories of these fields into the families of phase trajectories of the pair of fields (i2)for e = 1 and (i3) for e = -1 (see Figures 2.6b and 2.6c, respectively).

One of the derivatives is equal to zero. The value of one of the fields at the point zis nonzero because otherwise both the derivatives of the function p at the point z alongthese fields would be equal to zero. However, the point z is singular for the other fieldbecause the values of the fields at this point are collinear and one of these derivatives isequal to zero. By the definition of a typical pair of vector fields, this singular point isnondegenerate. Consequently, z is a saddle or nodal or focal zero-passing point. Weconsider these three cases in turn using

LEMMA 2.3 (on a zero-passing point). For a typical pair of vector fields, the valueof one of them at each singular point is not an eigenvector of the linearization of the otherfield.

PROOF. For definiteness, let z be a singular point of the field v = (v I, v2). Assumethe contrary, namely let w (z) be an eigenvector of the linearization of the field v at thepoint z belonging to an eigenvalue A. This means that the following two relations aresatisfied:

((vl,x - 2)w1 + v1,yw2)(Z) = 0, (v2,,wl + (v2,y - 2)w2)(Z) = 0.

(As usual, the symbol of a variable in the subscript of a function means differentiationof thefunction with respect to this variable, e.g., g., = 19g/ax.) Given these relationsand the condition v (z) = 0, we calculate at the point z the derivative of the functionp along the field w:

[(vlw2 - V2W1)xwl + (v1w2 - v2w1)yw2](Z)77= [(vl,xw2 - V2,xwl)Wl + (vl,yw2 - V2,yw1)w21(Z)

= [(vl,xw2 - (v2,y - 2)w2)wl - ((vl,,i- - 2.)w1 +V2,yw1)W2](Z)

= [wlw2(vl,, + v2,), - 2 - vl,, + 2 - v1,),)](z) = 0.

Thus, the derivative is equal to zero. However, the derivative at the point z of thefunction p along the field v is also equal to zero because v(z) = 0. Consequently,both derivatives are equal to zero. This contradicts the conditions in the subcase underconsideration and proves Lemma 2.3.

We come back to the investigation of the subcases. For definiteness, assumethat v(z) = 0. To study the first two subcases (a saddle and a node) we take a


coordinate system of class C2 with origin at the point under consideration so that (1)the coordinate semiaxes are phase trajectories of the field v and (2) the axis of ordinatescorresponds to the greater (in absolute value in the case of a node) eigenvalue of thelinearization of the field v at the point z and the field w becomes the field (1, 1). It isclear that such a coordinate system exists.

A saddle zero-passing point. In the selected coordinates the field v has the formv(x, y) = (-xa(x, y), yb(x, y)), where a and b are positive differentiable functionsin a neighborhood of zero. The collinearity line of the pair of fields passes throughzero and is located in the second and fourth quadrants. Any point on this line lyingin the second quadrant is a passing point, and any point in the fourth quadrant is ae-passing point. To prove the assertion of the theorem in the subcase under study itsuffices to transform the family of phase trajectories of the field v into the family ofhyperbolas xy = c while preserving the phase trajectories of the field w (the straightlines x - y = c). The pairs consisting of preimage curves and image hyperbolas aredetermined in the first and third quadrants by the common point at which these linesintersect the diagonal y = x and in the second and fourth quadrants by the commontangent line (of the form of y - x = c) of the lines. After simple calculations weconclude that in a neighborhood of zero the corresponding mapping is determined bythe formulas

(x, y) ((x - y + ec)/2, (y - x + ec)/2),

where c = [(x - y)2 + (-1)k4g2]1/2, e = sgn(q + x), k is the index of the quadrant,and q is determined from the phase trajectory of the field v (passing through the point(x, y)) in the following way. In the first and third quadrants q is the abscissa of thepoint at which the trajectory intersects the diagonal y = x, and in the second andfourth quadrants q is equal to half the difference between the ordinate and the abscissaof the point where the trajectory intersects the collinearity line. In a neighborhoodof zero this mapping is a homeomorphism. It preserves the phase trajectories of thefield w and transforms the family of phase trajectories of the field v into the familyof hyperbolas xy = c. Hence, a typical pair of vector fields has a singularity (iil)(Figure 2.6g) at a saddle zero-passing point.

A nodal zero passing point. In the chosen coordinates the field v has the formv(x, y) = (xa(x, y), yb(x, y)), where a and b are differentiable functions having thesame sign, lb (0, 0) I > Ia (0, 0) I > 0. The collinearity line of the fields passes throughzero and is located in the first and third quadrants. A point on this line lying in thefirst (third) quadrant is a 8-passing point, and a point belonging to the third (first)quadrant is a passing point if the node is stable (unstable).

To prove the theorem in the subcase under consideration it suffices to transformthe family of phase trajectories of the field v in a neighborhood of zero into the familyof branches of the parabolas y = cx2 while preserving phase trajectories of the field w(the straight lines y - x = c). In the vicinity of zero we index the phase trajectoriesof the field v (the branches of the parabolas y = cx2) with the points where theyintersect the circle x2 + y2 = e for a sufficiently small & > 0. For a sufficiently small ,5we assign to the straight line y - x = 6 the index of the phase trajectory of the fieldv (of the branch of the parabola) with which this line has contact near zero. Thus,in a neighborhood of zero we obtain two indexing systems for the points: the first ofthem relates to the phase trajectories of the field v and the straight lines y - x = c,and the other corresponds to the same straight lines and the branches of the parabolasy = cx2. The mapping transforming the points with the same indexes one into another


is defined in a neighborhood of zero and, as can easily be seen, is a homeomorphismof some neighborhoods of zero. The homeomorphism preserves the family of straightlines y - x = c and transforms the family of phase trajectories of the field v into thefamily of branches of the parabola y = cx2. Hence, a typical pair of vector fields hasa singularity (ii2) (Figures 2.6h and i) at a nodal zero-passing point.

A focal zero-passing point. For definiteness, assume that the focus is stable (thecase of an unstable focus reduces to the former by the time reversion). We choosesmooth coordinates x, y near the point z with the origin at z such that (1) the fieldw becomes the field (1, 0) and (2) the axis of ordinates is the collinearity line of thefields v and w (in this case (p(x, y) = xp(x, y)) and consists of 0-passing points inthe upper half plane and of passing points in the lower half plane. It is clear that sucha coordinate system exists. We reduce the mapping of the axis of ordinates into itselfdetermined by the phase trajectories of the field v corresponding to the "half-coil" tothe same mapping of the field (-x - y, x) by means of change of variable of the formof y = g(y) (the tilde in the new coordinate y will be omitted). We now considerthe arcs of phase trajectories of the fields v and (-x - y, x) that correspond to the"half-coil" and join the identical points on the axis of ordinates; the former trajectoriesare transformed into the latter while preserving the ordinates of the points.

Near zero, the constructed mapping is a homeomorphism preserving the familyof phase trajectories of the field w (the family of the horizontal lines y = c) andtransforming the family of the phase trajectories of the field v into the family of thephase trajectories of the field (-x - y, x). It follows that a typical pair of vector fieldshas a singularity (ii3) (Figures 2.6j and k) at a focal zero-passing point.

Both derivatives are equal to zero. The values of the two fields v and w at the pointz are nonzero because by the definition of a typical pair of fields the second derivativeof the function p at this point along each of the fields is nonzero. Consequently, z iseither a turning point or a 0-turning point if the directions of these values of the fieldsare the same or opposite, respectively. For definiteness, assume that z is a turningpoint (in the case when z is a 9-turning point the arguments are similar).

We choose a smooth coordinate system in a neighborhood of the point z with originat z so that the field w, becomes the field (1, 0). In these coordinates we have p, (0, 0) =0 p,, (0, 0) because the first derivative of the function p at that point z along thefield v is equal to zero whereas the second derivative is not. Furthermore, py, (0, 0) 0because, by the definition of a typical pair of fields, d p ; 0 when p = 0. By the Matherdivision theorem, in a neighborhood of zero we have p(x, y) = H(x, y) (y + x2g(x)),where H and g are smooth functions, H (0, 0) 0 [GG], and g (0, 0) 0 becausep (0, 0) 0. Consequently, in a neighborhood of zero the transformation of thevariable z = x Jg (x) J1 /2, possibly with change of sign in y, reduces the collinearityline of the fields to the parabola y = x2 (the tilde in the new variables is dropped).Take the smooth first integral I of the field v whose value on a phase trajectory isequal to the ordinate of the point where the trajectory intersects the axis of ordinates.The integral has the form I (x, y) = y + xJ(x, y), where J is a smooth function. Wehave J (0, 0) = J, (0, 0) = 0 J (0, 0) because p (0, 0) = p,, (0, 0) = 0 p,,, (0, 0) .Consequently, J}, (0, 0) # 0 because the derivative k, on the collinearity line y = x2of the fields is everywhere equal to zero: J(x, x2) + xJ, (x, x2) - 0. Therefore,x2 + xJ(x, x2) = x2 + x3a (x), where a is a smooth function, a (0) 0. By theDefour theorem, the germs at zero of the functions x2 and x2 + x3a (x), a (0) 0, arereduced to the germs at zero of the functions z2 and z2 +2 z3 sgn a (0), respectively, by


a smooth transformation of the variable x preserving the direction of the coordinateaxis [JD] (the coefficient 2 is introduced for convenience of further calculations). Withthe new coordinate z, the field's collinearity line is determined by equation yq (y) = z2,where q is a smooth function, q(0) > 0. Using the change of variable y = yq(y) weagain reduce it to the equation of the parabola y = x2 (the tilde in the new variablesis dropped). We note that, as before, in the new coordinates the family of phasetrajectories of the field w is a family of horizontal lines.

There are two possible subcases here: a (0) > 0 and a (0) < 0. The latter is reducedto the former by interchanging the notation of the fields v and w in the pair (at thebeginning of the above argument). We consider the first subcase.

In this subcase the restriction of the integral I to the collinearity line y = x2 is equalto x2 + 2x3 and coincides with the restriction of the function y + 3xy - x3 to this line.Consequently, I (x, y) = y+3xy-x3+(y-x2)Q(x, y), where Q is a smooth function.We have Q(x, x2) - 0 because It (x, x2) 0 and I,- (x, x2) = -2xQ(x, x2). Therefore,Q(x, y) = (y - x2)S(x, y) and I(x, y) = y + yx2 - x3 + (y - x2)2S(x, y), where S isa smooth function. The assertion of the theorem in the case under consideration nowfollows from the facts presented below.

LEMMA 2.4. The function y + 3xy - x3 + (y - x2)2S(x, y), where S is a smoothfunction, is reduced to the function y + 3xy - x3 by a transformation of variables of theform of x = . + (y - . 2)P(z, y), y = y, where P is a smooth function.

The lemma immediately implies

COROLLARY 2.5. For a typical pair of vector fields the germ at a turning point ofthe families of their phase trajectories is C°°-diffeomorphic to the germ at zero of thefamilies of phase trajectories y = c, and y + 3xy - x3 = c2, where c,, c2 E R, and themotion along the trajectories is from left to right.

In particular, the germ at zero at the turning point (0, 0) of the families of phasetrajectories of the pair of fields (1, y - x2) and (1, x2 - y) is also C°°-diffeomorphicto the normal form indicated in the corollary. Consequently,' a typical pair of vectorfields has a singularity (i4) (Figure 2.6d) at each of its turning points.

The proof of the theorem is completed modulo the lemma.

PROOF OF LEMMA 2.4. It suffices to show that there is a smooth function P forwhich the indicated change of variables transforms the first function in the lemma intothe second one. We substitute the expression for the variables x and y in terms of xand y into the first function and transform the resulting expression (the arguments ofthe functions P and S and the tilde in the new variables are omitted):

y + 3(x + (y - x2)P)y - (x + (y - x2)P) + (y - (x + (y - x2)P))S

= y + 3xy - x3 + 3y(y - x2)P - 3x2(y - x2)P - 3x(y - x2)2P2

- (y - x2)3P3 + (y - x3)3(1 - 2xP - (y - x2)P2)S

=y+3xy-x3+(y-x2)x [3P - 3xP2 - (y - x2)P3 + (1 - 2xP - (y - x2)P2)2S].

It suffices to find a function P for which the expression in square brackets is identicallyequal to zero. We interpret this expression as a function H in three variables x, y, andP: H = H(x, y, P). We have H(0, 0, -S(0, 0)/3) = 0 and Hp (0, 0, -S(0, 0)/3) = 3(and, generally, Hp (0, 0, P) - 3). Consequently, by the implicit function theorem, the

§3. POLYDYNAMICAL SYSTEMS 43

equation H(x, y, P) = 0 possesses a smooth solution P = P(x, y) in a neighborhoodof the point (0, 0, -S(0, 0)/3). The lemma is proved.

REMARKS. 1. The analytic normal form of the germ at zero of the pair of analyticfunctions x2 and x2 + x3a(x), a(0) 0, has a functional modulus ([Vol, [Es]).Consequently, the analytic normal form of the germ at a turning point or at a a-turning point of families of phase trajectories or limiting lines contains a functionalmodulus.

2. For k > 1 the assertion (ii) of the theorem is false. This is because the germ ofthe family of phase trajectories of a vector field at a nondegenerate singular point of thefield has a numerical parameter that is preserved under differentiable transformationsof coordinates. This parameter is the exponent of the singular point. It is definedas the ratio of the eigenvalue of the linearization of the field at that point having thegreatest absolute value to that with the smallest absolute value for a saddle and a nodeand as the modulus of the ratio of the imaginary part of the eigenvalue to its real partfor a focus (see Section 2.2 in Chapter 1).

§3. Polydynamical systems

In this section we study control systems determined by a finite set of smoothvector fields (feasible velocity fields). We begin with the simplest case when the controlsystem is determined by a typical pair of vector fields on a surface and then pass to thecase # U > 3 (# U denotes the number of different values the control parameter mayassume).

3.1. The simplest case (# U = 2).

THEOREM 3.1 (on a bidynamical system). For a control system determined by atypical pair of vector fields on a surface the following three assertions are true:

(i) the complement of the steep domain coincides with the rest zone and is the set ofall zero passing points, 0 -passing points, and 0-turning points of the pair;

(ii) the local transitivity zone is the set of all 0 -passing points of the pair;(iii) the germ of the family of limiting lines at each of the points z in the phase space is

Ck-diffeomorphic to the germ at zero of the control system determined by one ofthe six (five) pairs of vector fields in assertion (i) (resp. (ii)) of Theorem 2.2 ona pair of fields if k = oo (k = 0) and the point z is not (resp. is) a zero passingpoint.

PROOF. Assertion (iii) on a bidynamical system immediately follows from thetheorem on a pair of fields. It is illustrated in Figures 2.6a-k, where the correspondingfamilies of limiting lines in normal form are shown.

By Proposition 2.1, for a typical pair of vector fields in a plane each of the pointsin the plane belongs to one of the following six types: a regular point, a zero-passingpoint, a passing point, a 0-passing point, a turning point, and a0-turning point. It isclear that each of the regular points, passing points, and turning points (zero-passingpoints, 0-passing points, and a-turning points) belongs (does not belong) to the steepdomain of the control system determined by this pair of fields. Consequently, tocomplete the proof of the theorem it suffices to show that the rest zone and the localtransitivity zone of the system contain the zero-passing points, the a-passing points,and the a-turning points. We shall do this in turn.


FIGURE 2.7

A zero-passing point belongs to the rest zone of the system because the zero velocityis feasible at that point. This point does not belong to the local transitivity zone becausefor the corresponding normal forms of pairs of fields in Theorem 2.2 the points in theleft half plane either are all unattainable from zero (in the case of a saddle or nodalzero-passing point) or are attainable from zero in a time exceeding a positive constant(the case of a focal zero-passing point).

A 0-passing point. We shall show that such a point belongs to the local transitivityzone of the system and, consequently, by Theorem 1.1, belongs to the rest zone of thesystem. In accordance with Theorem 2.2, we select a system of smooth coordinatesin a neighborhood of the point with origin at the point so that the family of phasetrajectories of the pair of fields becomes the family of phase trajectories of the fields (1,x) and (1, -x) (i.e., the families of the parabolas 2y = x2 + ci and 2y = -x2 + c2).For a small number E > 0 we consider a cycle composed by arcs of the parabolasy = x2/2 - e and y = -x2/2 + e, Ix I < 2e (Figure 2.7). Any two points belongingto the closed region bounded by this cycle are attainable from each other along afeasible trajectory not leaving the region and having no more than two switchingsfrom one field to the other (here the same phenomenon is observed as in Example 8in Section 1.3; see Figure 2.4). The time of motion over this trajectory does notexceed 3 2e /m (e ), where m(,-) is the minimum value in this region of the moduliof the components along the x axis of the feasible velocity fields. Fore -* 0+ thisvalue tends to a positive constant because these fields are continuous and the value ofeach of them at zero has a nonzero component along the x axis. Consequently, wehave 3v/2__e/m(e) -* 0 for e - 0+, and hence a 0-passing point belongs to the localtransitivity zone of the system in question.

A a-turning point does not belong to the local transitivity zone because for thecorresponding pairs of fields belonging to the normal forms in Theorem 2.2 the pointson both the positive and negative semiaxes of abscissas are unattainable from zero(Figures 2.6e and f). This point belongs to the rest zone of the system. Indeed, inthe case shown in Figure 2.6f the system in a neighborhood of the 0-turning pointis transferred to any a-passing point lying sufficiently close to the former point. Bywhat was proved above, a a-passing point belongs to the local transitivity zone of thesystem, and, consequently, by Lemma 1.3, the system can stay in a neighborhood ofthis point indefinitely long for a suitably chosen control action.


In the case illustrated in Figure 2.6e the proof that a 0-turning point belongs tothe rest zone is analogous to the proof in Section 1.3 that the origin belongs to the restzone. In a neighborhood of such a point a smooth coordinate system with origin atthe point can be chosen so that the families of phase trajectories of the pair of fieldsbecome the families of lines y = cl and y + 3xy - x3 = c2; cI, c2 C R (in accordancewith Corollary 2.5), and the motion along the trajectories is from left to right in thefirst family and from right to left in the other family. Take a positive number e << 1and consider the following feasible motion of the system. It starts from zero along atrajectory belonging to the first family and then alternately uses trajectories belongingto the second and first families with switchings on the verticals x = e and x = 0,respectively. On completing n such cycles (accordingly, with 2n - 1 switchings) wearrive at the axis of ordinates.

LEMMA 3.2. The ordinate of the point of attainment of the axis of ordinates after2n - 1 switchings is equal to -E 3 (1 + q + q2 + + q"- I), where q = 1 + 3E.

We use the lemma to complete the proof of the theorem. We have

E3(l + q + q 2 +... +q,:-I) = e2(-l +q")/3 < e2exp(3ne - 1).

Consequently, the trajectory of motion in question lies in a circle of radius (e2 +E4 exp(6nE - 2)1/2 < E (1 + exp (6n6 - 2)) 1/2 with center at zero. The radius will be lessthan any given number 6 > 0, and the time of motion will be greater than any givennumber T > 0 if we take e = 6[1 + exp(3m(T + 1) - 2)]-I/2 and the natural numbern satisfying the inequality mT < 2nE < m (T + 1). Here m is the maximum value inthe 6-neighborhood of zero of the moduli of the components along the x axis of thefeasible velocity fields. Hence, a 9-turning point belongs to the rest zone in the caseshown in Figure 2.6e as well.

The theorem on a bidynamical system is proved modulo Lemma 3.2. The lemmawill be proved by induction on the number of cycles n.

For n = 1 the assertion of the lemma is true because after one cycle the ordinateof the point of attainment of the axis of ordinates is found from the equation y +3xy - x31,=o = -E3 and is equal to -E3. Assume that after k cycles the ordinate ofthe point of attainment of the axis of ordinates is yk = -E3(1 + q + q2 + ... , qk-1)

Then the ordinate of the point of attainment of the axis of ordinates after k + 1cycles is found from the equation y + 3xy - x31,,-o = yk + 3eyk - E3 and is equal to-e3 (1 + q + q2 + + qk ). Induction on the number of cycles proves the lemma fork = n.

3.2. A tridynamical system.

DEFINITION. A set of vector fields on a surface is said to be typical if the followingthree conditions are fulfilled:

(1) every pair of fields belonging to the set is a typical pair;(2) the collinearity lines of any two different pairs of fields belonging to the set

intersect transversally (i.e., at any common point of these lines their tangentsare different) and not at the turning points and the 0-turning points of thepairs;

(3) on the surface there are no points at which four different fields belonging tothe set of fields are collinear.


(a)

FIGURE 2.8

(b)

DEFINITION. For a typical set of vector fields on a surface, let the cone of a pointz be smaller than the tangent plane, and contain the zero velocity. This point is calleda saddle (or nodal or focal) zero passing point or a a-turning point if it is a point ofthis kind for a pair of fields belonging to the set, a double 0-passing point if z is a0-passing point of two pairs of fields from a triple of fields belonging to the set, and a,9-passing point if z is a 9-passing point for a pair of fields belonging to the set but isnot a double 0-passing point.

DEFINITION. By eigendirections at a zero-passing point are meant the eigendirec-tions of the linearization at this point of the feasible velocity field vanishing at thepoint.

Assume that the surface (the phase space of a system) is oriented and that acontinuous direction from which the angles are counted in the tangent planes is fixedon it. If the cone of a point z is smaller than the tangent plane, then we denote by L' (z )and L2 (z) the limiting directions at this point determining, respectively, the supremumand infimum of the velocity directions belonging to the cone of the point and by C' (z)the number of fields belonging to the set such that the value at this point of each ofthem belongs to L' (z). An integral curve of the field L' will be called an i-limiting line.

REMARK. Here and later, unless the contrary is stipulated, the number i takesvalues either 1 or 2.

DEFINITION. For a typical set of vector fields on an oriented surface, let the cone ofa point z be smaller than the tangent plane and do not contain the zero velocity. Thepoint z is called an i-regular point if C' (z) = 1, a double i-passing point if C' (z) = 3, ani-turning point if C' (z) = 2 and z is a turning point for a pair of fields whose values atthe point belong to L' (z), and an i passing point if C' (z) = 2 and z is not an i-turningpoint.

EXAMPLE 1. The triple of vector fields (1, 2), (-x, y), (1, 2x - 1) in the planeOxy is typical. For this triple, (0, 0) is a saddle zero-passing point, (1/3, 1/9) is a0-turning point, and (3/2, -3) is a double passing point; the points of the parabolay = x (1 - 2x) and of the line y = -x that lie in the right half plane are a-turningpoints (Figure 2.8a).


EXAMPLE 2. The triple of vector fields (1, 2) (x, -y), (1, 2x - 1) in the planeOxy is typical. For this triple, (0, 0) is a saddle zero-passing point, the points ofthe lines y = -2x > 0 and y = x(1 - 2x) < 0 are a-passing points, (3/2, -3) is adouble i-passing point (i = 1, 2), (1/3, 1/9) is a 1-turning point (the angles in thetangent planes are counted in the standard manner, i.e., counterclockwise), the pointsof the lines -3 < y = -2x < 0, y = x(1 - 2x) < -3, {x = 1} n {y > -1} are2-passing points, and the points of the lines y = -2x < -3, {x = 3/2} fl {y < -3},{y = x(1 - 2x)} fl {x 54 1/3} n {0 < x < 3/2} are 1-passing points (Figure 2.8b).

In Figure 2.8, in addition to the notation from Figure 2.6, the interior of the restzone (the local transitivity zone) is shaded and the double i-passing points and doublea-passing points are encircled.

THEOREM 3.3 (on the boundary of the steep domain of a tridynamical system). Fora control system determined on a surface by a typical triple of vector fields the followingfive assertions hold:

(1) the complement of the steep domain coincides with the rest zone;(2) the interior of this zone coincides with that of the local transitivity zone and

consists of all points such that the cone of each of them coincides with the tangent plane;(3) the boundary of the steep domain coincides with the set of all 19-passing points,

a-turning points, double a passing points, and zero passing points, and each of the pointsof the boundary belongs to only one of the enumerated types of points;

(4) the intersection of this boundary with the local transitivity zone consists of alla-passing points, double a -passing points, and all a-turning points such that no singlelimiting line passes through all of them;

(5) the germ of the steep domain at each of its boundary points is C°°-diffeomorphicto the germ at zero of one of the following three sets: (a) y > 0, (b) lyl > Ix I, and (c)y < Ix I if this point is (a) a 0 -passing point or a a-turning point, (b) a double 0 -passingpoint, and (c) a zero passing point, respectively.

THEOREM 3.4 (on the family of limiting lines of a tridynamical system). For acontrol system determined on an oriented surface by a typical triple of vector fields thefollowing three assertions hold:

(1) the germ of the family of limiting lines at each point z of the boundary of thesteep region is Ck-diffeomorphic to the germ at zero of

(a) the family of semiparabolas y = Ix + C1112 if z is a 0-passing point; in this casek = oo; or

(b) the family of curves y + (3xy - x3) = c, where either y - x2 > 0 (a focal a-turn)or y - x2 < 0 (a saddle 0-turn) if z is a a-turning point; in this case k = oo;

(c) the family of integral curves of either the equation (y')2 = sgn(jyj - IxI)/4 orthe equation (y')2 = 4sgn(lyI - Ix 1) if z is a double a-passing point; in this casek = 0;

(d) the family of curves (i) (x + y'/2)2(x/2 y'12) = c (a folded saddle point),(ii) y = Ix + c1112 (a family of semiparabolas), and (iii) (2x3 - 3xy - c)2 =y(2y - 3x2) (a folded monkey saddle) if z is a saddle zero passing point andthe velocities belonging to its cone can determine (i) only one, (ii) both, and (iii)neither of the two eigendirections, respectively; in this case k = 0;

(e) the family of curves (i) (x + y'i2)2(x/2 + y'12) = c (a folded saddle point), (ii)(c(x + y'/2)2 = (x/2 + y'/2)) U ((x + y'i2) = 0) (a folded saddle-node), (iii)y = Ix + C1112 (a family of semiparabolas), and (iv) the family of limiting lines


of the system determined by the triple of vector fields (-x, -2y), (1, 1), (3, 2) (afolded saddle-node) if z is a nodal zero passing point and the velocities belongingto its cone can determine (i) the eigendirection at this point belonging to theeigenvalue with only the smaller modulus, (ii) the eigendirection belonging to theeigenvalue with only the greater modulus, (iii) both eigendirections belonging tothe elgenvalues with the smaller and greater moduli, and (iv) neither of the twoeigendirections; respectively; in this case k = 0;

(f) the family of semiparabolas y = IX + C1112 if z is a focal zero passing point; inthis case k = 0;

(2) each point in the steep domain belongs to one of the following four types: ani-regular point, an i passingpoint, an i-turning point, and a double i passingpoint, wherei is equal to either 1 or 2;

(3) the germ at zero of the family of i-limiting lines is C°°-diffeomorphic to thegerm at zero of (a) the family of curves y = c, (b) the family of curves y = x Ix I = c,(c) the family of integral curves of the equation y' = ly - x21, and (d) the family ofintegral curves of the equation y' = max{-x, x, Y(x, y)}, where Y is a smooth function,Y(0, 0) = 0 Yy (0, 0) (Y, (0, 0) - 1) if z is (a) an i -regularpoint, (b) an i passing point,(c) an i-turning point, and (d) a double i passing point, respectively; i = 1, 2; c E R.

REMARKS. 1. In the above theorem the names of some singularities are indicatedin the parentheses; when necessary, they will be used in the sequel. It was shownin Chapter 1 that (topologically) the families of trajectories of a folded saddle, afolded node, and a folded focus were the families of integral curves of the equationy = (y' + kx)2 for k = -1, 1/10, and 1, respectively. Topologically, a folded saddleand a folded focus are identical with a saddle 0-turn and a focal ,9-turn, respectively;therefore the names of the last two singularities include the words "saddle" and "focal".A folded saddle-node has some common features with both a folded saddle and a foldednode, which accounts for its name. A folded monkey saddle is the image of the familyof curves g (v, w) = c, where g (v, w) = (v - w) (2v + w) (ti + 2w), under the folding(v, w) H (x = v, y = w2); the monkey can place its tail comfortably on the saddlet =g(v,w).

2. The orientability of the surface in Theorem 3.4 is important only for assertions(2) and (3) of the theorem in which (global) indexing of the limiting directions is used.On a nonorientable surface these assertions are true locally in a neighborhood of eachpoint on the surface after indexing of the limiting directions is introduced. Assertions(1d), (le), and (If) of the theorem in which k = 0 become false even for k = 1 becauseat zero-passing point there is a modulus relative to the diffeomorphisms, which is equalto the exponent of the singular point.

3. For a control system determined by a typical set of more than three vector fieldsthe assertions of the above two theorems remain true if ly I is replaced simply by y inassertion (5) of the first theorem and in assertion (lc) of the other theorem.

The theorems on a tridynamical system will be proved in three stages. We firststudy the interior points of the steep domain in Section 3.3 and establish the validityof assertions (2) and (3) of the second theorem; then we shall show in Section 3.4 thatassertions (1)-(3) of the first theorem are true; and, finally, in Section 3.5 the otherassertions of these theorems will be proved.


3.3. The points of the steep domain. At a point z in the steep domain one of thefollowing three cases is possible: the number C' (z) is equal to 1, or 2, or 3. We shallconsider these cases in turn.

If C' (z) = 1, then z is an i-regular point, and the field L' is determined in aneighborhood of this point by a field v belonging to the triple, v (z) E L' (z), v (z) # 0.Consequently, at this point the family of i-limiting lines has a singularity (3a) becausea smooth vector field is rectifiable in a neighborhood of its nonsingular point.

If C' (z) = 2, then the field L' in a neighborhood of the point z results fromcompetition between the two fields in a pair belonging to the triple, whose values liein L' (z ). These are nonzero values because the point z belongs to the steep domain.Consequently, by definition, z is an i-turning point or an i-passing point if it is a turningpoint or a passing point, respectively, for this pair of fields. This fact and Theorem 2.2about a pair of fields imply that the family of i-limiting lines has a singularity (3b) or(3c) at the point z.

If C' (z) = 3, then, by definition, z is a double i-passing point. By condition (2)in the definition of a typical set of fields, z is a passing point for every pair of fieldsbelonging to the triple. According to the theorem on a pair of fields, we can select asystem of smooth coordinates in a neighborhood of this point with origin at the pointso that the family of phase trajectories of one of the pairs of fields becomes the familyof phase trajectories of the fields (1, x) and (1, -x). In this coordinate system the thirdfield takes the form (1, Y(x, y)), where Y is a smooth function, after multiplication bya positive function. We have Y(0, 0) = 0 because the values of the fields at the pointz are collinear; and Yy (0, 0) 0 Y? (0, 0) - 1 because, by the definition of a typicalset of fields, the collinearity lines of the pairs of fields intersect transversally and not atthe turning points of these pairs. Therefore, in the case under consideration the familyof i-limiting lines has a singularity (3d) at the point z.

Assertions (2) and (3) of the second theorem are proved.

3.4. The boundary of the steep domain and the interiors of the zones. By Theo-rem 1.4 on the boundaries, the convex hull of the velocity indicatrix of each boundarypoint of the steep domain (of the rest zone and the local transitivity zone) contains azero velocity. Consequently, for a certain pair of feasible velocity fields this point iseither a0-passing point or a a-turning point or a zero-passing point. In particular, bycondition (1) in the definition of a typical set of fields and assertion (1) of Theorem 3.1on a bidynamical system, each of these points lies in the rest zone, and hence the entireboundary of the steep domain belongs to the rest zone of the system. A zero-passingpoint of a pair of fields belonging to a typical triple of fields cannot be a double a-passing point of the latter (or in fact, of any typical set of fields). Indeed, by definition,a double 9-passing point is a a-passing point of two pairs of fields belonging to thetriple. Therefore, the values of all the three fields at a double a-passing point mustbe nonzero. A a-turning point of a pair of fields belonging to the triple cannot be adouble a-passing point either because, by the definition of a typical set of fields, thecollinearity lines of the pairs of fields do not intersect either at the turning points or atthe a-turning points of these pairs.

Consequently, for a typical triple of fields a double a-passing point must necessarilybe a a-passing point for two of the three pairs of fields. Hence, each point on theboundary of the steep domain (of the local transitivity zone and the rest zone) is eithera a-passing point or a a-turning point or a double a-passing point or a zero passingpoint of the triple of fields. The set of a-passing points is dense in the set of points of


X = 0

FIGURE 2.9

these four types. In a neighborhood of a0-passing point of the triple we reduce two ofthe three fields to the form (1, x) and (1, -x) by selecting a smooth coordinate systemwith origin at the point and multiplying by positive functions. The value of the thirdfield at this point is noncollinear to the values of the other two fields (by Theorem 2.2on a pair of fields). Consequently, the convex hulls of the velocity indicatrices ondifferent sides of the axis of ordinates contain the zero velocity in their interior andexterior, respectively (Figure 2.9). Therefore, the points lying on one side of the axisof ordinates belong to the steep domain and the local transitivity zone and those onthe other side belong to the rest zone. The axis of ordinates itself is the boundaryof both the steep domain and each of these zones. Thus, all 0-passing points and,consequently, all a-turning points, double 0-passing points, and zero-passing pointsbelong to this boundary, and hence assertion (3) of the first theorem is true. It followsthat at any interior point of each of these zones the convex hull of the velocity indicatrixcontains the zero velocity. Consequently, the interiors of these zones are identical, andthus assertion (2) of the first theorem is true.

The above arguments also imply that assertion (1 a) of the second theorem is truefor a a-passing point and that at such a point the steep region has a singularity (5a)indicated in the first theorem.

As we have proved above, the boundary of the steep domain belongs to the restzone of the system. Therefore, assertion (1) of the first theorem is true. Assertions(1)-(3) of the first theorem are proved.

3.5. Singularities on the boundary of the steep domain. By definition, a0-passingpoint of a typical triple of vector fields is a a-passing point for a certain pair of fieldsbelonging to the triple. By Theorem 3.1 on a bidynamical system, this point belongsto the local transitivity zone of the control system determined by this pair of fields.Hence, this point also belongs to the local transitivity zone of the control systemdetermined by the triple of fields. Assertion (4a) of the first theorem is proved fora-passing points.

By what was proved above, each point of the boundary of the steep domain of thesystem is either a a-passing point, or a0-turning point, or a double a-passing point, ora zero-passing point. All the assertions of the two theorems are proved for a-passingpoints. The singularities of the steep domain and the family of limiting lines at thepoints of the last three types will be studied in turn.


y=x2

(a) (b) (c) (d)

FIGURE 2.10.

A a-turning point. By definition, such a point is a 0-turning point for a pair offields belonging to the triple. According to Theorem 2.2 on a pair of fields, aftermultiplication by a smooth positive functions and selection of a smooth coordinatesystem with origin at that point, this pair of fields can be reduced in a neighborhoodof the point to either the form (1, y - x2), (-1, y - x2) or (-1, x2 - y), (1, x2 .- y).The value of the third field at this point is noncollinear to the values of these two fields.Consequently, the convex hulls of the velocity indicatrices on different sides of theparabola y = x2 contain the zero velocity in their interior and exterior, respectively.Thus, the points on one side of the parabola belong to the steep domain and the localtransitivity zone and those on the other side belong to the steep domain and the restzone, respectively. In particular, the steep domain has a singularity (5a) at a a-turningpoint. Depending on the position of the steep domain (above or below the parabola)we obtain four patterns. of behavior of the family of limiting lines in a neighborhoodof a a-turning point (Figures 2.10a, b, c, and d; the notation in the figure is the sameas in Figure 2.8). The first and third (second and fourth) patterns can be reduced toeach other by changing the direction of motion along the limiting lines so that theydetermine the same singularity (not counting the direction of motion), which we calleda saddle (focal) a-turn. The family of curves y + (3xy - x3) = c, c E IR, is determinedby the typical pair of fields (1 + 3x, 3(x2 - y)), (-1 + 3x, 3(x2 - y)) for which theparabola y = x2 is the collinearity line and (0, 0) is a a-passing point. This factand the theorem on a pair of fields imply that for a control system determined by atypical triple of fields the germ at a a-turning point of the family of limiting lines isC°°-diffeomorphic to the germ at zero of the family of curves y ± (3xy - x3) = c,c c R, where y - x2 < 0 for a saddle a-turn (Figures 2.10a and c) and y - x2 > 0 fora focal a-turn (Figures 2.1Ob and d). Assertion (I b) of the second theorem is proved.

We now prove assertion (4) of the first theorem for a a-turning point. A saddlea-turn point does not belong to the local transitivity zone. This is clearly seen inFigures 2.1 Oa and c: either a point on the negative half-axis of ordinates is unattainablefrom zero (Figure 2.10a) or zero is unattainable from that point (Figure 2.1Oc). Weshall show that a point of local a-turn belongs to the local transitivity zone of thesystem. For definiteness, consider the case in Figure 2. l Ob (the argument is similar forFigure 2.10d).

In a neighborhood of zero the third field has a positive component along the axisof ordinates because (1) the first two fields are directed as the fields (1, y - x2) and(-1, y - x2), respectively, (2) the value of the third field at zero is noncollinear to thevalues of the other two fields, and (3) the steep domain lies above the parabola y = x2.For a sufficiently small e > 0 there is a cycle composed of sections of trajectories ofthe first two fields passing through the point (-e, 0) and the trajectory of the third


FIGURE 2.11

field passing through the point (e, 0) (Figure 2.11). Denote by T(e) the time neededto move over this cycle with feasible velocity fields and by R (e) the maximum time ofmotion along the sections of phase trajectories of the third field cut by the cycle. Anytwo points in the region bounded by the cycle are attainable from each other in timenot exceeding T (e) + 2R (e) (the motion begins and terminates with velocities of thethird field, and in the intermediate part the motion over the cycle is used). It is clearthat this time tends to zero as e - 0. Consequently, a focal 0-turn point belongs to thelocal transitivity zone. Assertion (4) of the first theorem is now proved for e-turningpoints too.

A double a -passing point of a typical triple of fields becomes a double i-passingpoint if one of the three fields (namely the one "looking" in the opposite direction)is multiplied by -1. Hence, by assertion (3) of the second theorem (already proved),in a neighborhood of a double e-passing point a typical triple of fields is reducibleto the triple of fields (1, x), (1, -x), (-1, Y(x,y)) where Y is a smooth function,Y (0, 0) = 0, Y, (0, 0) ( Y? (0, 0) - 1) ; 0. In the vicinity of zero each of the collinearitylines x = - Y(x, y) and x = Y(x, y) lies in one of the two regions xy > 0 and xy < 0.The two lines lie in the same region and in different regions when I Y, (x, y) > 1 andY, (x, y) < 1, respectively, because the tangents at zero to these lines are determined

by the equations Yy (0, O) y + (Y, (0, 0) + 1)x = 0 and Yy (0, O) y + (Y, (0, 0) - 1)x = 0,respectively. In a neighborhood of zero each of the points on these lines belongsto the local transitivity zone as a e-passing point of a pair of fields (formed of thefirst and third or the second and third fields, respectively), which proves assertion(4) of the first theorem for double 8-passing points. The local transitivity zone liesbetween these lines and does not contain the third collinearity line, namely, the axisof ordinates. Consequently, the steep domain of the system under study has thesingularity (5b) indicated in the first theorem at a double e-passing point because thefirst two collinearity lines intersect at zero transversally.

Now we prove that at a double 0-passing point the family of limiting lines has oneof the singularities listed in assertion (1c) of the second theorem. In a neighborhoodof this point one of the limiting directions in the steep domains is determined by thefield (-1, Y) and the other results from competition between the other two fields (1,x) and (i, -x) and is determined by the field (1, - IxI sgn(Y(x, y) - x)). Near zero weuse a smooth change of variables to transform the field (-1, Y) to the form (-1, 0);in this case the coordinates are chosen so that the boundary of the rest zone becomesthe union of the diagonals of the quadrants. There are two possible cases here: therest zone either contains the axis of abscissas or does not when this zone is the regionI x I > IyI or I x I lyis uniquely determined by the points of attainment of the boundary of the region by


(a)

FIGURE 2.12

(b)

the limiting lines passing through the point A. A similar indexing in this region is alsodetermined by the integral curves of the equation (y' )2 = 4 sgn (Ix I - l y I) . As can easilybe seen, the mapping of this region transforming the points with the same indexes intoeach other is a homeomorphism in a neighborhood of zero. We extend in in a suitablemanner this homeomorphism to a homeomorphism of some neighborhoods of zero.Thus, in the second case assertion (1c) of the second theorem is true.

The arguments are based on the same idea in both cases. In a neighborhoodof zero each point in the region I x I < L

Iis uniquely determined by the ordinates u

and v of the points of attainment of the diagonal y = x by the limiting lines passingthrough this point. A similar indexing in this region is also defined by the integralcurves of the equation (y')2 = sgn(jyj - IxI)/4. However, generally speaking, themapping transforming the points with the same indexes into each other cannot beimmediately determined in this case because the first and second indexing systems maydefine different regions in the plane Ouv (see Figures 2.12a and b, respectively; A andB are increasing functions). We must first apply a change of variables of the form ofu = uh(u), v = vh(v) to transform these regions into each other and then use thistransformation and the first and second indexing systems to determine a mapping ofthe region Iy I ? IxI onto itself in a neighborhood of zero. This mapping is obviouslya homeomorphism. We extend this homeomorphism to a homeomorphism of someneighborhoods of zero in any suitable way. Thus, assertion (1c) of the second theoremis also true in this case. Assertion (1c) is proved completely.

REMARK. The above two cases differ at a double a-passing point in that in one ofthem the straight line along the limiting direction at this point lies in the tangent set tothe rest zone at the point, while in the other case it does not.

A zero-passing point. Let v denote a vector field in the triple for which z is a singularpoint. By the definition of a typical set of fields, this singular point is nondegenerate.The values of two other fields at the point z are noncollinear because the lines ofcollinearity of these fields and the field v intersect transversally. In a neighborhood ofthe point z the rest zone coincides with the closure of one of the four sectors into whichthese lines break up a sufficiently small neighborhood of the point. Consequently, at azero-passing point the steep domain has singularity (5c) indicated in the first theorem.Assertion (5) of this theorem is completely proved.

Now we complete the proof of the second theorem and then prove assertion (4)of the first theorem. Here the following three subcases are possible: z is a saddle ornodal or focal zero-passing point. We begin with the third subcase.


(a)

FIGURE 2.13

(b)

A focal zero passing point. Near the point z we reduce the other two fields to thepair of fields (1, 0) and (0, 1) by multiplying by smooth positive functions and selectinga smooth coordinate system with origin at that point. In a neighborhood of zero thecollinearity lines break up the surface into four sectors, the closure of one of thembeing the rest zone. The field of limiting directions is determined by the fields (1, 0)and (0, 1) in the sector.which is vertical relative to the rest zone and by the pairs offields y, (0, 1) and v, (1, 0) in the two remaining sectors, respectively. In the vicinityof zero the points belonging to the closure of the steep domain are indexed using thepoints of intersection of the limiting lines passing through them with the boundary ofthe rest zone (Figure 2.13a; in addition to the notation of Figure 2.8, in Figure 2.13and in the subsequent figures the numbers mark the lines whose intersections withlimiting lines near zero determine the indexing of the points in the steep domain,which is used to construct a normalizing homeomorphism). Similarly, the points inthe region y > 0 are indexed using the points of intersection of the lines of the familyof semiparabolas y = Ix + c1I/2 with the axis of abscissas (Figure 2.13b). We firstuse a homeomorphism of some neighborhoods of zero to transform the set of indexedlines in Figure 2.13a (the boundary of the rest zone) into the set of indexed lines inFigure 2.13b (the axis of abscissas) with preservation of the indexing; in particular,the image of zero is zero. After this we use these indexing systems for the points toredefine this homeomorphism in the steep domain so that the germ at zero of the newmapping transforms the germ at zero of the family of limiting lines of the system inquestion into the germ at zero of the family of semiparabolas. In a neighborhood ofzero the new mapping is a homeomorphism. Assertion (If) of the second theorem isproved.

To investigate the cases of a saddle and a node let us select a coordinate systemof class C2 in a neighborhood of the point z with origin at this point so that (1) thecoordinate semiaxes are phase trajectories of the field v and (2) the axis of ordinatescorresponds to the greatest (in modulus, in the case of a node) eigenvalue of thelinearization of the field v at the point z, and one of the other two fields becomes thefield (1, 1). The value at zero of the third field w is noncollinear to the vector (1, 1)because, by the definition of a typical set of fields, the collinearity lines of the pairsof fields v, (1, 1), and v, w intersect at zero transversally. Consequently, there areonly six essential types of location of the vector w (O, 0) (see Figure 2.14). SubcaseF is reduced to subcase A by rectifying the field w to obtain the field (1, 1) withthe preservation of coordinate axes. Subcase D is reduced to subcase C by central


FIGURE 2.14

FIGURE 2.15

symmetry and rectification of the field -w(-x, -y) transforming it into the field (1,1) with the preservation of coordinate axes. Thus, four subcases remain, namely, A, B,C, and E. We consider them in turn beginning with the case of a saddle zero-passingpoint and then investigate the case of a nodal zero-passing point.

A saddle zero passing point; subcase A. The collinearity lines of the pairs of fieldsv, (1, 1) and v, w lie in the region xy < 0. The rest zone is located in the closure ofthe second quadrant and is bounded by these lines. The field of limiting directionsis determined by the pairs of fields v, w and v, (1, 1) in the regions on the left andon the right of the two lines, respectively, and by the fields (1, 1) and w in the regionbetween the lines in the fourth quadrant. Figure 2.15a illustrates the behavior of thefamily of limiting lines in a neighborhood of zero. The family of trajectories of afolded monkey saddle is illustrated in Figure 2.15b in normal form. The transforminghomeomorphism is constructed as in the case of a focal zero-passing point. Thus,assertion (1 d) is true for this subcase because none of the velocities in the cone of thepoint z can determine any of eigendirections at this point.

A saddle zero passing point; subcase B. The collinearity lines of the pairs of fieldsv, (1, 1) and v, w lie in the regions xy < 0 and xy > 0, respectively. The rest zone isto the right of the two lines and is bounded by them. The field of limiting directions isdetermined by the pairs of fields (1, 1), w; v, w; and v, (1, 1) in the regions to the left,above, and below these lines, respectively. Figures 2.16a,b illustrate the behavior, in aneighborhood of zero, of the family of limiting lines of the system and of the family oftrajectories of a folded saddle in normal form. The transforming homeomorphism isconstructed as in the case of a focal zero-passing point.


(a)

FIGURE 2.16

FIGURE 2.17

(b)

Thus, assertion (1 d) of the second theorem is true in this subcase because thevelocities of the cone of the point z can only determine one eigendirection at thispoint.

A saddle zero passing point; subcase C. The collinearity lines of the pairs of fieldsv, (1) 1) and v, w are located in the region xy < 0. The rest zone lies on the right of thetwo lines and is bounded by them. The field of limiting directions is determined by thefields (1, 1) and w in the region to the left of these lines and by the pairs of fields v, wand v, (1, 1) in the regions between the two lines in the second and fourth quadrants,respectively. Figures 2.17 and 2.13b illustrate the behavior in a neighborhood of zeroof the limiting lines of the system in question and of the family of semiparabolas,respectively. The transforming homeomorphism is constructed as in the case of a focalzero-passing point. Hence, assertion (1 d) is true in this case too because here thevelocities belonging to the cone of the point z can determine two eigendirections atthis point.

A saddle zero passing point; subcase E. The investigation of this subcase is practi-cally the same as the study of subcase B. We leave it to the reader to draw illustrativefigures and verify that in this subcase the germ at zero of the family of limiting linesis homeomorphic to the germ at zero of the family of trajectories of a folded saddlepoint. This is exactly what would be expected in view of assertion (1 d) because herethe velocities belonging to the cone of the point z can determine only one of theeigendirections at this point.

Assertion (1 d) is proved completely.


FIGURE 2.18

A nodal zero passing point; subcase A. The collinearity lines of the pairs of fieldslie in the region xy > 0, and the rest zone is located in the closure of the first quadrantand is bounded by these lines. The field of limiting directions is determined by thepairs of fields v, (1, 1) and v, w in the regions above and below these lines, respectively,and by the fields (1, 1), w in the regions between these lines in the third quadrant.Figure 2.18 illustrates the behavior of the family of limiting lines in a neighborhood ofzero.

We index the phase trajectories of the field v in a neighborhood of zero by thepoints where they intersect the circle x2 + y2 = e for a sufficiently small e > 0; theindex of the phase trajectory of this field is also assigned to the limiting line having apoint of contact with the trajectory near zero. We thus obtain an indexing system forthe limiting lines in a neighborhood of zero. The control system determined by thetriple of fields (-x, -2y), (1, 1), (3, 2) has an analogous indexing system. Considerthe mapping transforming the family of limiting lines of this system and of the systemabove into each other with preservation of indexing. It can readily be seen that in aneighborhood of zero this mapping is a homeomorphism of the steep domains of thesystems. We can extend it to a homeomorphism of some neighborhoods of zero insome suitable manner. Assertion (le) is proved in this subcase because the velocitiesbelonging to the cone of the point z cannot determine either of the two eigendirectionsat this point.

A nodal zero-passing point; subcase B. The collinearity lines of the pairs of fieldsv, (1, 1) and v, w belong to the regions xy > 0 and xy < 0, respectively. The rest zoneis located on the right of the two lines and is bounded by them. The field of limitingdirections is determined by the pairs of fields (1, 1), w; v, (1, 1); and v, w in the regionslocated on the left, above, and below these lines. Figures 2.19 and 2.16b illustrate thebehavior in a neighborhood of zero of the family of limiting lines of this system andof the family of trajectories of a folded saddle in normal form. The transforminghomeomorphism is constructed as in the case of a focal zero-passing point. Thus,assertion (le) is also true in this subcase because the velocities belonging to the coneof the point z can only determine the eigendirection belonging to the eigenvalue withthe smaller modulus.

A nodal zero passing point; subcase C. The collinearity lines of the pairs of fieldsv, (1, 1) and v, w lie in the region xy > 0. The rest zone is located on the right of thetwo lines and is bounded by them. The field of limiting directions is determined bythe pairs of fields v, w and v, w in the regions between these lines in the first and thirdquadrants, respectively, and by the fields (1, 1), w in the region located above these two


FIGURE 2.19

FIGURE 2.20

lines. Figures 2.20 and 2.13b illustrate the behavior in the vicinity of zero of the familyof limiting lines of the system and of the family of semiparabolas. The transforminghomeomorphism is constructed as in the case of a focal zero-passing point. Therefore,assertion (le) is true in this case because the velocities belonging to the cone of thepoint z can determine both eigendirections at the point.

A nodal zero-passing point; subcase D. The collinearity lines of the pairs of fields v(1, 1) and v, w lie in the regions xy > 0 and xy < 0, respectively. The rest zone is abovethe two lines and is bounded by them. The field of limiting directions is determinedby the pairs of fields v, (1, 1); v, w; and (1, 1), w in the regions on the right of, onthe left of, and below these two lines, respectively. Figures 2.21a and 2.21b illustratethe behavior in a neighborhood of zero of the limiting lines of this system and of thefamily of trajectories of a folded node in normal form, respectively. The constructionof the transforming homeomorphism differs only slightly from the previous subcases.In a neighborhood of zero the indexing of the trajectories of the family is determinedby the points where its trajectories passing through a point A close to zero intersectwith the axis of ordinates. The index of the trajectory is preserved under reflection inthe boundary of the rest zone and transition to the other branch of the field of limitingdirections. Trajectory indexing determines in a neighborhood of zero the indexing ofthe points of the closure of the steep domain. Consider the closed regions boundedby the arcs AC and AB of the trajectories in the family and by the thick line BOC.We first apply a homeomorphism of neighborhoods of zero containing these regionsto transform the regions into each other so that similar sections of their boundariesgo into each other, by introducing these indexing systems for the points. Then weuse the introduced indexing systems for the points to modify this homeomorphism ina neighborhood of zero so that the germ at zero of the new mapping transforms the


(b)

FIGURE 2.21

germ at zero of the family of limiting lines of this system into the germ at zero of thefamily of trajectories of a folded node. The new mapping is clearly a homeomorphismin a neighborhood of zero. Consequently, assertion (le) of the second theorem is alsotrue in this subcase because here the velocities in the cone of the point z can onlydetermine at this point the eigendirection corresponding to the eigenvalue with thegreater modulus.

The proof of the second theorem is completed.To complete the proof of the first theorem it remains to show that none of the

zero-passing points of a typical triple of fields belongs to the local transitivity zone. Insix of the subcases we have considered (subcases A, B, and C for a saddle zero-passingpoint and subcases A, B, and D for a nodal zero-passing point) it is easy to show thatsome points in a neighborhood of zero that are unattainable from zero in a short time.For instance, for the pairs in subcases A and B these are the points on the negativehalf-axis of abscissas, and in subcases E and D for a saddle zero-passing point anda nodal zero-passing point the points of this kind are on the negative half-axis ofordinates. In the remaining three subcases (subcase C for a saddle zero-passing pointand a nodal zero-passing point and the subcase of a focal zero-passing point) there arealso such points in a neighborhood of zero, but it is somewhat more difficult to provethis. In contrast to the previous six subcases, any neighborhood of zero contains asmaller neighborhood of zero any two points of which are attainable from each otherwithout leaving the greater neighborhood (i.e., as can easily be seen, a zero-passingpoint possesses the property of local transitivity in arbitrary time). However, thereare also pairs of points in the smaller neighborhood lying arbitrarily close to zero suchthat any motion between them requires a long time (exceeding a positive constant) atthe velocity field v. For example, in the above three subcases this occurs for motionfrom zero to a point close to zero on the negative half-axis of abscissas. Thus, in thesethree subcases the zero-point does not belong to the local transitivity zone of a typicaltriple of fields.

The proof of the first theorem is completed.Note that we have also incidentally proved

LEMMA 3.5. For a typical triple of fields its saddle or nodal zero passing point insubcase C and focal zero passingpoint belong to the interiors of their positive and negativeorbits, respectively.


§4. Classification of singularities

In this section we present classification results for singularities of a control systemin general position and also consider the stability of these singularities with respect tosmall perturbations of the system.

4.1. Systems in general position. By the definition of the class of systems understudy, a control system is defined by a smooth mapping F of a bundle space P over Mwith fiber U into the tangent bundle space TM such that the diagram

P TM

Mis commutative, where z is the bundle projection and 7r is the canonical projection. Weidentify the space of control systems with the set of such mappings and endow it withthe fine Whitney C4-topology. In the case of a compact phase space the proximity oftwo systems in this topology means the proximity, at all points in the space P, of the4 -jets of the mappings defining the systems. For a noncompact phase space the degreeof this proximity is controlled arbitrarily well at "infinity". By a typical (generic)control system or a system in general position is meant a system belonging to an openeverywhere dense set in the space of systems with topology.

THEOREM 4.1. In the space of polydynamic systems, systems determined by typicalsets of vector fields form an open everywhere dense set.

Hence, a control system determined by a typical set of vector fields is a system ingeneral position. Theorem 4.1 is proved in Section 5.

4.2. Singular controls. In a sufficiently small neighborhood of a point of the phasespace, a control system F can be written in the form F : (z, u) H (z, f (z, u)), wherez is a point in the phase space of the system, u is a control parameter, f is a smoothmapping, and f (z, u) is a feasible velocity at the point z corresponding to the valueu of the control parameter. Here and later z and u are local coordinates normalizingthe bundle z : (z, u) -* z, unless otherwise stated.

Limiting lines are important in the investigation of positive and negative orbits ofpoints. A feasible motion along these lines has limiting velocities and corresponds tolimiting controls. We now give the precise definition.

DEFINITION. A velocity belonging to the velocity indicatrix of a point is called alimiting velocity at the point if it belongs to a limiting direction at this point. The valueof the control parameter is called a limiting control at a point if it determines a limitingvelocity at the point.

For a limiting control u at a point z we have L(z, u) T-M, where L(z, u) is thesubspace R f (z, u) + u) T U in the tangent plane. In particular, if f (z, u) 0,then the variations in the velocity direction produced by small variations of the controlhave a high order of smallness.

DEFINITION. A value u of the control parameter is called a singular control ata point z c M if L(z, u) T_ M. A singular control at a point determines asingular velocity at the point. The set of all points (z, u) such that u is a singu-lar control at the point z is called a singular surface of the control system. The

§4. CLASSIFICATION OF SINGULARITIES 61

restriction of a control system to its singular surface determines a field of singularvelocities on the phase space of the system.

Thus, a limiting control (velocity) at a point is always a singular control (velocity)at this point. In the general case the converse is not true.

Generally, the field of singular velocities of a system is multivalued, and the numberof its branches is different for different points of the phase space.

EXAMPLE 1. Consider a control system in the plane Oxy with the set of values ofthe control parameter consisting of a circle (let u be the angle in the circle, 0 < u < 27r)and the number 7. If f (x, y, u) = (/ + cos u, sin u) and f (x, y, 7) = (0, 1), then ateach point in the plane we have three singular controls: it ± m/4 and 7, three singularvelocities: (1, ±1) and (0, 1), two limiting controls: it + 7r/4 and 7, and two limitingvelocities: (1, 1) and (0, 1). Hence, the singular surface of the system is the union oftwo disjoint surfaces u = 5mr/4 and u = 7 smoothly embedded into the space of thevariables z, u.

ExAMPLE 2. For a polydynamic system each value of the control parameter is asingular control at each point in the phase space, and the singular surface coincideswith the bundle space P.

DEFINITION. By the folding of a system we mean the restriction of the projection zto the singular surface of the system.

DEFINITION. A point in the preimage of a smooth mapping between smooth two-dimensional manifolds is called a regular point (a singular point of the type of a Whitneyfold, or a Whitney double cusp) of the mapping if locally in a neighborhood of thatpoint it has the form r = x and s = y (accordingly, s = y2, or s = y3 - xy, ors = y3 - x2y) for some suitably chosen local coordinates x, y in the source spaceand local coordinates r, s in the target with origins at the given point and at its image,respectively.

THEOREM 4.2. The singular surface of a control system in general position is a smoothclosed two-dimensional submanifold in P (possibly empty), and each of the critical pointsof the folding of the system is either a Whitney fold, or a Whitney cusp, or a Whitneydouble cusp.

Theorem 4.2 is proved in Section 6.2. The types of critical points enumerated inthe theorem are illustrated in Figures 2.22a-c. The arrow in the figures symbolizes themapping z, the thin double line represents the sets of critical points and values of thesystem folding, and the heavy (dashed and solid) lines denote the singular surface. AWhitney fold and a Whitney cusp are well-known singularities; a Whitney double cuspwas first obtained when studying singularities of the projection of surfaces [A6, Go, P].

4.3. The critical set of a system.

DEFINITION. A singular control u I at a point z is called a passing control at z ifthere is another singular control u2 at this point such that L(z, u I) + L(z, u2) $ T:M.The set of all points in the phase space such that for each of them there is a passingcontrol, is called the passing set of the system. The union of the passing set and theset of critical values of the folding of a control system in general position is calledthe critical set of the system.


I i

/lz::: . I/.

(a) (b) (c)

FIGURE 2.22

THEOREM 4.3. The critical set of a control system in general position is closed, andthe germ of this set at each of its points is C°°-diffeomorphic to the germ at zero of oneof the nine sets in Table 1. In this case the passing set can be located in the critical set inonly one of the ways indicated in the third column of the table, and this location is onlypossible for the number of different values of the control parameter indicated in the fourthcolumn.

The symbol oo in the table means that dim U 0; the versions of location arecoded with Latin letters; the passing set and the set of critical values of the folding arerepresented by the thin double line and by the heavy line, respectively; r > 1; cok aresmooth functions satisfying the condition 0 < W'(0)(0) < cp2 (0) < . . . < cpr(0) < 1. Incontrast with the sets 1-3 and 5-9, the set 4,. is not encountered in the case of generalposition in the absence of isolated values of the control parameter. This theorem isproved in Section 7.4.

REMARK. As will be seen in Chapter 3, when investigating the attainability sets,two subsets of the critical set are important, namely, the boundary of the steep domainand the limiting-passing set, that is, the set of all points of the phase space at each ofwhich there is a limiting-passing control.

4.4. The boundary of the steep domain. Denote by C (z) the number of pairwisedistinct limiting values of the control parameter at a point z.

DEFINITION. Let the indicatrix of a point z in the phase space contain the zerovelocity and let the cone of the point be smaller than the tangent plane. This pointis called a zero passing point if there exists an isolated value of the control parameterdetermining the zero velocity at the point, a zero point if C(z) = 1, and a nonzero-passing point if C(z) = 2 and one of the limiting values of the control parameter atthe point determines a nonzero velocity there, whereas the other value determines thezero velocity and is a nonisolated value of the control parameter.


TABLE 1

I y = 0 -- ------ r- (a) 00

(a) (b)(b) > 1

2 xy = 0 1 + (a), (b) oo

(a) (b) (c) (c) > 2

3 x(x2 - y2) = 0 > 2

I ... r

4,r

x(x2-y2)rl(x-(Pk(Y))=0 > 3 + rk=1

5 Y(y-x2)=0

6 Y(y - x") = 0

7 X(X t y32) = 000

8y =x2J!

9 y2 = xs oo

EXAMPLE 3. Let M be the plane Oxy, U the disjoint union of a circle and thenumber 7, u the angle in the circle (0 < u < 2mc), f (z, u) = (x/ (1 + x2)1/2 + cos u, (1 +


Y)/(l + x2)'/2 + sin u), and f (x, y, 7) _ (-2x, 2 - y). For this system (+1, 0) arenonzero-passing points, the points in the interval (-1, 1) of the axis of abscissas arezero-points, and (0, 2) is a zero-passing point.

DEFINITION. Assume that the indicatrix of a point z does not contain the zerovelocity and let the cone of the point contain this velocity and be smaller than thetangent plane. Then the point z is called a double 0 -passing point if C(z) = 3 and a0-passing point (a 0-turning point) if C (z) = 2 and each of the limiting directions atthe point does not lie (lies) in the tangent set to the passing set at this point.

REMARK. The previouisly defined 0-passing points, a-turning points, double 19-passing points, and zero-passing points of a typical set of vector fields are also points ofthese kinds in the sense of the above two definitions for the control system determinedby this set of fields. Thus, Example 1 in Section 3.2 demonstrates the notions involvedin the last definition as well.

THEOREM 4.4 (on the boundary of the steep domain). The following four assertionshold for a control system in general position:

(1) the complement of the steep domain coincides with the rest zone;(2) the interior of the rest zone coincides with that of the local transitivity zone and

consists of all points such that the cone of each of them coincides with the tangent plane;(3) the boundary of the steep domain belongs to the critical set of the system and is

the union of all a passing points, 0-turning points, double 0 -passing points, zero passingpoints, nonzero-passing points, and zero points. Moreover; each of the points of theboundary belongs to only one of the there six types of points;

(4) the germ of the steep domain at each point z of its boundary is C°°-diffeomorphicto the germ at zero of one of the seven sets in Table 2, and in this case (a) the boundaryof the steep domain can be located in the critical set in only one of the ways indicated inthe third column of the table, (b) such a singularity is possible for the number of differentvalues of the control parameter indicated in the fourth column, and (c) the point z belongsto the type indicated in the fifth column of the table.

Contrary to the notation in Table 1, the double line in Table 2 denotes the limiting-passing set, the double dashed line denotes the rest of the passing set, and the steepdomain is shaded; r _> 0. Some of the assertions in Theorem 4.4 are proved inSection 7.4 and the rest in Section 9.2.

REMARK. In contrast with Theorem 3.3, Theorem 4.4 contains no indication ofwhich points on the boundary of the steep domain belong to the local transitivity zoneof the system. Here this is stated as a separate assertion (Theorem 4.11).

4.5. Singularities of the family of limiting lines on the boundary of the steep do-main. By the third assertion of Theorem 4.4, each point of the boundary of the steepdomain of a control system in general position belongs to one of the following six types:a a-passing point, a double a-passing point, a 0-turning point, a zero-passing point, anonzero-passing point, and a zero-point. We shall first classify the singularities of thefamily of limiting lines of a system in general position at points of the first four typesand then proceed to points of the last two types.

THEOREM 4.5. For a control system in general position, (z, u) is a regular point ofthe folding of the system if (1) z is a point belonging to one of the following four types:a a -passing point, a double 0 -passing point, a a-turning point, and a zero passing point,and (2) u is a limiting control at this point.


TABLE 2

I (Y 0) u (x < 0)

MUMMMEMMUM= 2 zero-passing point

2 y * 0 = 2 a-passing point or

a-turning point

3 lyl > Ixl = 3 double a-passingpoint

4 y > 1XI >3 double a-passingpoint

// II

5, y < Ixl > 2 +r zero-passing point

/ `fiffffffff ii (a)-(c) zero-po nt

((b)

c)

(a)-(c) (e), (f) a-passing6 y > 0 (d) > 2 point

(e) °° (d) a-passing point

11 (f) > 3 or a-turning point(d) (e) M

7 (x < y'12) n (y > 0) nonzero-passingpoint

This theorem is proved in Section 7.4. Hence, a system in general position in aneighborhood of each of the a-passing, double (9-passing, ,9-turning, and zero-passingpoints has a set of smooth branches of the field of singular velocities such that thevalue of each of them at the point is a limiting velocity at that point. This set will becalled the determining set of this point for the following reason.

THEOREM 4.6. For a control system in general position and any 0-passing point, dou-ble apassing point, a-turning point, and zero passing point of this system the followingtwo assertions hold in a sufficiently small neighborhood of the point:

(1) the determining set of the point is a typical set of 2, or 3, or 2, or min{3, #U}vector fields for which this point is a a -passing point, or a double 0 -passing point, or a0-turning point, or a zero passing point, respectively;

(2) in the closure of the steep domain of the system its field of limiting velocitiescoincides with the field of limiting velocities of the control system determined by thedetermining set of the point.


The second assertion of this theorem and the part of the first assertion concerningthe number of vector fields are proved in Section 7.4. The proof is completed inSection 9.1. Theorem 4.6 immediately implies

COROLLARY 4.7. For a control system in general position the classification of singu-larities of its family of limiting lines, of the rest zone, and of the local transitivity zoneat 0 -passing points, double 0 -passing points, a-turning points, and zero passing pointscoincides with the same classification for a control system determined by a typical set of2, or 3, or 4 vector fields if the number of different values of the control parameter is equalto 2, or 3, or is greater than 3, respectively.

REMARKS. 1. As in the case of a polydynamical system, the zero-passing pointsare subdivided into saddle, nodal, and focal points of these kinds, and the 19-turningpoints are subdivided for # U > 2 into saddle and focal ones.

2. The singularities of the family of limiting lines of a system in general positionat the a-passing, double 0-passing, a-turning, and zero-passing points will be calledsimilarly to the corresponding singularities of a polydynamical system.

The next theorem gives a complete classification of singularities of the family oflimiting lines of a control system in general position at the a-passing, double a-passing,a-turning, and zero-passing points.

THEOREM 4.8. For a control system in general position the germ of the family oflimiting lines at a point z on the boundary of the steep region is Ck-diffeomorphic to thegerm at zero of the one of the following families:

(1) the family of curves y + x I x I = c for # U = 2 and of the family of semiparabolasy= IX +

c I 1 /2 for # U> 2 if z is a a -passing point and k = oo;(2) the family of integral curves of the equation (y') = a sgn(Iy I - I x 1) for # U = 3

and of the equation (y')2 = a sgn(y - Ix1) for #U > 3, where a is equal to 4 or 1/4 inboth the cases if z is a double 0 -passing point and k = 0;

(3) the family of integral curves of the equation y' = ±Iy - x21 for #U = 2 andthe family of curves y + (3xy - x3) = c, where either y - x2 > 0 (a focal a-turn) ory - x2 < 0 (a saddle a- turn) for #U > 2 if z is a a-turning point and k = oo;

(4) the family of limiting lines of the control system determined by one of the followingfive pairs of vectorfields: (a) (1, 1) and (-x, y), (b) either (1, 1) and (-x, -2y) or (1, 1)and (x, 2y), and (c) either (1, 0) and (x - y, x) or (1, 0) and (-x - y, x) if #U = 2and z is (a) a saddle zero passing point, (b) a nodal zero passing point, and (c) a focalzero passing point, respectively, and k = 0;

(5) the family of curves (a) (x + y1/2)2(x T y112) = c (a folded saddle), (b)y = Ix + c1'/2 (a family of semiparabolas),

and

(c) (2x3 - 3xy - c)2 = y(2y - 3x2)2(a folded monkey saddle) if # U > 2, z is a saddle zero passing point, and the velocitiesbelonging to its cone can determine (a) only one of the two eigendirections, (b) botheigendirections, and (c) neither eigendirection at this point, respectively; in this case k = 0;

(6) the family of curves (a) (x + y'12)2(X T y112) = c (a folded saddle), (b)(c(x + y'/2)2 = (x + .y'/2)) U ((x + y'/2) = c) (a folded node), (c) y = IX + C1112

(afamily of semiparabolas), and (d) of the family of limiting lines of the system determinedby the typical triple of fields (-x, -2y), (1, 1), (2, 3) (a folded saddle-node) if #U > 2,z is a nodal zero passing point, and the velocities belonging to its cone can determine (a)the eigendirection corresponding only to the eigenvalue having the smaller modulus, (b)the eigendirection corresponding only to the eigenvalue with only the greater modulus,(c) the eigendirections corresponding to the eigenvalues with both the smaller and greater


moduli, and (d) neither of the two eigendirections at the point z, respectively; in thiscase k = 0;

(7) the family of semiparabolas y = x + c 11/2 if # U > 2, z is a focal zero passingpoint, and k = 0.

Everywhere in the theorem c E R.

This theorem directly follows from Corollary 4.7, Theorems 3.3 and 3.4, and alsoRemark 3 after Theorem 3.4. The proof of Theorem 4.8 is not presented here.

Now we consider a nonzero-passing point z. By definition, at this point thereare two limiting values of the control parameter: one of them determines a nonzerovelocity at this point and the other determines the zero velocity and is a nonisolatedvalue of the control parameter. We denote these values as u 1(z) and u° (z ), respectively.

THEOREM 4.9. For a control system in general position and any of its nonzero-passingpoints the following three assertions hold:

(1) the limiting directions at this point do not lie in the tangent space to the boundaryof the steep domain at the point;

(2) the points (z, u°(z)) and (z, u' (z)) are, respectively, a critical point which is aWhitney fold, and a regular point of the folding of this system;

(3) the germ at this point of the family of limiting lines is homeomorphic to the germat zero of the family of semiparabolas y = I x + c 1/2, c c R.

Theorem 4.9 is proved in Section 9.3.Finally, we investigate the singularities of the family of limiting lines of a control

system at its zero-point.

DEFINITION. A zero-point of a control system is said to be black if the limitingdirections at the point lie in the tangent set to the boundary of the steep domain at thispoint.

REMARK. We will show in Chapter 3 that in the case of general position blackpoints on the attainability boundary are unattainable, and, consequently, if attainabil-ity is interpreted as illumination, then the black points are in the shadow region (thephase space is compact, and we start from a fixed point), which motivates the name"black".

EXAMPLE 4. For the control system describing a swimmer drifting in a stream withfield (x2 + y2, 0), the zero-points form the circle x2 + y2 = 1, and in this case thepoints (+1, 0) are black.

The next theorem completes the classification of the typical singularities of thefamily of limiting lines on the boundary of the steep domain that was started inTheorems 4.8 and 4.9.

THEOREM 4.10. For a control system in general position and any of its zero-points zthe following two assertions hold:

(1) (z, u(z)) is a critical point which is a Whitney fold of the folding of the system,if u(z) is a limiting control at the point;

(2) the germ of the family of limiting lines of the system at this point is C°°-diffeomorphic to the germ at zero of the family of integral curves of the equation (y')2 = xif z is not a black point, and of equation (y' + a(x, y)) 2 = yb(x, y), where a and b aresmooth functions, b (0, 0) = 1, a (0, 0) = 0 ax (0, 0) 1/8, if z is a black point.


This theorem is proved in Section 9.4. The family of limiting lines of a system ingeneral position has a folded saddle, a folded node, or a folded focus at a black pointfor a, (0, 0) < 0, 0 < a, (0, 0) < 1/8, or 1/8 < a, (0, 0), respectively (these singularitieswere studied in detail in Chapter 1). Topologically, all folded saddles (accordingly, allfolded nodes and foci) are identical (see Theorem 2.8 in Chapter 1).

THEOREM 4.11. For a control system in general position, the difference between theboundary of the steep domain and the local transitivity zone is exactly the set of allzero passing points, black points, and also 0-turning points such that not a single limitingline enters each of them (i.e., focal 0-turn points).

This theorem is proved in Section 9.5.

4.6. Singularities in the steep domain. As in the investigation of a tridynamicalsystem, we assume that the phase space of the system is oriented and that a continuousreference direction for measuring angles in the tangent plane is fixed (all the results inthis subsection are automatically extended to the case of a nonoriented phase space).If the cone of the point is smaller than the tangent plane, then, as before, we use L I (z)and L2(z) to denote the limiting directions at this point, which determine, respectively,the minimal and maximal directions of the velocities belonging to the cone point, andn' (z) to denote the angle in the tangent plane corresponding to the direction L' (z).An integral curve of the field L' will be called an i-limiting (or, simply, limiting) line.A limiting control and a passing control at a point z will be called an i-limiting controland an i-passing control at this point, respectively, if the velocity determined by them atthe point lies on the straight line RL' (z ). The number of i-limiting controls at a pointz will be denoted C' (z). Here and henceforth, unless otherwise stated, i is equal to1 or 2.

DEFINITION. By the i-passing set we mean the set of all points at each of whichthere is an i-passing control.

DEFINITION. A point z in the steep domain is said to be i-regular (a double i-passingpoint or an i-cutoff point) if C' (z) = 1 (accordingly, C' (z) = 2 or C' (z) = 1) and ifthe order of contact of the indicatrix at this point with the ray L' (z) at each of theircommon points is less than two (accordingly, less than two or equal to three).

DEFINITION. A point z in the steep domain is called an i-passing point (or ani-turning point) if (1) C' (z) = 2, (2) the direction L' (z) lies (or does not lie) inthe tangent set to the i-passing set at this point, and (3) the order of contact of theindicatrix at this point with the ray L' (z) at each of their common points is less thantwo.

REMARKS. 1. We define the order of contact of a straight line and a set at theircommon point as the number equal to the order. of the zero at this point of the functionthe "distance from a point on the line to the set" minus one. For instance, the ordersof contact at zero of the axis of abscissas with the point (0, 0), the axis of ordinates,and the parabola y = x2 are equal to 0, 0, and 1, respectively.

2. The i-passing, i-turning, and double i-passing points defined in the previoussection (in Section 3.2) for a typical set of vector fields are also points of the samekind in the sense of the above definitions for a control system determined by that setof fields.


DEFINITION. The germ (gl,z1) of a function gl at a point zl is said to be R+-equivalent to the germ (g2, z2) of a function $2 at a point z2 if there is a C°O-diffeomorphism 0 of a neighborhood of the point z1 into a neighborhood of thepoint z2 and a smooth function g such that O(zl) = Z2 and (92,Z2) = (91 00-1 +g, z2).

THEOREM 4.12. For a control system in general position each of the points in itssteep domain belongs to one of the following five types: (1) an i-regular point, (2) ani-passing point or an i-turning point, (3) a double i-passing point, and (4) an i-cutoffpoint. In this case the germ of the function n' at this point is R+-equivalent to thegerm at zero of the function (1) 0, (2) (-1)' Ixl, (3) (-1)'(Ixl + ly + jx1j), and (4)(-1)' max{-w4 + yw2 + xw I w c R}, respectively; i = 1, 2.

Theorem 4.12 is proved in Section 8.1. The singularities of the functions at zeroindicated in the theorem are well-known singularities of the maximum and minimumfunctions of a typical family of functions depending on a two-dimensional parameter[Bzl, Bz2]. This theorem immediately implies

COROLLARY 4.13. For a control system in general position the following two asser-tions hold:

(1) the intersection of the set of i-passing points and the steep domain is exactly theset of points at each of which the function n' is nondifferentiable;

(2) the germ of the family of i-limiting lines at each of the points in the steep domainis homeomorphic to the germ at zero of the family of curves y = c, c E R; i = 1, 2.

By the defining set of a system in general position we mean the closure of the unionof the boundary of the steep domain and the limiting-passing set.

THEOREM 4.14. For a system in general position the following five assertions are true:(1) the defining set belongs to the critical set; (2) in the steep domain the germ of thedefining set at each point of the set is C°°-diffeomorphic to the germ at zero of one of thefive sets in Table 3; (3) the defining set can be located in the critical set in only one of theways indicated in the third column of the table; (4) such a singularity is possible only forthe number of different values of the control parameter indicated in the fourth column;and (5) the point z belongs to the type indicated in the fifth column of the table.

The notation in Table 3 is the same as in Table 2. Theorem 4.14 is provedin Section 8.2. Together with Theorem 4.4 it gives a complete list of the typicalsingularities of the defining set.

The next theorem provides a smooth classification of the singularities of the familyof i-limiting lines of a typical control system at the points in the steep domain.

THEOREM 4.15. For a control system in general position the following two assertionshold:

(1) the germ of the family of i -limiting lines at each of the points z in the steep domainis C°°-diffeomorphic to the germ at zero of the the following family (a) the family of thecurves y = c or y - x I x = c if z is an i-regular point or an i passing point, respectively;(b) the family of the integral curves of the equation y' = iy - x21 if z is an i-turningpoint; (c) the family of the integral curves of the equation y' = max{-x, x, Y(x, y)},where Y is a smooth function, Y (0, 0) = 0 $ Yy, (0, 0) (Y2 (0, 0) - 1), if z is a doublei-passing point; and (d) of the integral curves of the equation y' = Y(x, y), where Y isa continuous function that is nondifferentiable only on the positive semiaxis of ordinatesand is smooth outside its closure, Y(0, 0) = 0, if z is an i-cutoff point;


TABLE 3

I I- (a) z 21 y 0 II (b) 2 3 !-passing point

n

(a) (b) (c)

(c)

2 xy = 0 4 1- and 2-passingpoint

3 x(x2 - y2) = 0 = 3 double 1- and2-passing point

4 (y = IWI) V 2 4 double i-passing((x=0)n(y<_0)) point

5 (x = 0) n (y > 0) - i-cutoff point

(2) for #U > 2 no i-turning point belongs to the closure of the j -passing set; i,j=1,2;iLj


REMARKS. 1. In the analytic case the normal form of the germ of the familyof i-limiting lines at an i-turning point contains the same functional modulus as thenormal form of the germ of the family of phase trajectories of a typical pair of vectorfields at a turning point. We have mentioned that this modulus is related to the molulidescribed by Ecalle [Ec] and Voronin [Vo].

2. For # U = 2 the singularities of the family of limiting lines of a typical controlsystem were found in [B!, B2]. Topological classification of singularities of the familyof limiting lines of analytic semisystems was obtained in [F2].

4.7. Stability of singularities.

THEOREM 4.16. The singular surface of a control system in general position is trans-formed into the singular surface of any control system that is sufficiently close to theformer by a C°°-diffeomorphism, which C°-close to the identical one and isfibered overM, of the bundle space P. In this case the corresponding diffeomorphism of the manifoldM carries the steep domain, the i-passing set, the set of i-turning and 8-turning points,and the set of black points of the former system into the corresponding sets of the latter.

Theorem 4.16 is proved in Section 8.5.

4.8. Generalization. Let Y be the disjoint union of a finite number of compactsmooth manifolds of dimension higher than 1 and let G : Y H TM be a smooth

§5. TYPICALITY OF SYSTEMS DETERMINED BY TYPICAL FIELDS 71

mapping such that 7I o G is a proper submersion "onto" that has more than onepreimage for each point z E M (recall that 7r: TM -p M is the canonical projection).The mapping G determines a control system on M. At a point z in the phase space thevelocity indicatrix is G ((m o G) -'(z)). All the results in this section are automaticallyextended to this class of control systems.

4.9. Remark. The results on the classification of the singularities of the fields oflimiting direction of a typical control system are proved in the next sections of thischapter for the case M = R2. The general case is obtained using the globalizationtheorem in [Hi]. For M = 1(82 we identify P with R2 X U, TM with R2 x R2, andthe set of control systems with the set of mappings f E C°°(R2 X U,R2), namelyF = (id, f) so that F : (z, u) H (z, f (z, u)).

§5. The typicality of systems determined by typical sets of vector fields

In this section we prove Theorem 4.1. The assertion of the theorem is first es-tablished for typical pairs of vector fields and then for typical sets of more than twofields.

5.1. The proof of Theorem 4.1; #U = 2. We must show that the pairs of fieldssatisfying conditions 1-3 of typicality in the definition in Section 2.1 form an openeverywhere dense set in the space of pairs of fields in the finite C4-topology of Whitney.

Smooth vector fields with nondegenerate singular points form a subset in thespace of smooth vector fields which is open in the fine C 1-topology and dense in thefine C°°-topology. Consequently, the first typicality condition is satisfied on an openeverywhere dense subset C1 in the space of pairs of vector fields in the fine C4-topology.

Furthermore, in the space of 1 jets of pairs of vector fields, the system of equationsVI W2 - v2w, = 0, d (vl w2 - v2w,) = 0 determines a closed algebraic subset Q ofcodimension 3. There exists a strictly analytic stratification of this subset in the senseof Whitney [Ma]. (A stratified smooth (analytic) manifold is said to be Whitney-stratified if any embedding transversal to a stratum of lower dimension is transversalto all adjacent strata of higher dimension in a neighborhood of that stratum of lowerdimension.) By Thom's transversality theorem, in the space of pairs of vector fields,the subset of all pairs satisfying the condition that the image of its 1 jet extension doesnot intersect Q, is open in the fine C2-topology and dense in the fine C°°-topology.Consequently, the second typicality condition is satisfied on an open everywhere densesubset C2 in this space in the fine C4-topology.

Finally, the system of equations p = 0 = L p = Lw p = L p (p = 0 = Lv p =Lw p = L2J p), where LS. denotes the derivative along a field s, determines a closedalgebraic subset S of codimension 3 in the space of 2-jets of pairs of vector fields.Similarly, there exists a strictly analytic stratification of the field. By Thom's transver-sality theorem, in the space of pairs of vector fields, the subset of pairs, satisfyingthe condition that the image of the 2-jet extension of the pair does not intersect S,is open in the fine C3-topology and dense in the fine C°°-topology. Hence, the thirdtypicality condition is satisfied on an open everywhere dense set C3 in this space in thefine C4-topology of Whitney.

Thus, the typical pairs of vector fields form an open everywhere dense set Cl flC2 n C3 in the space of pairs of vector fields in the fine C4-topology. Theorem 4.1 isproved for typical pairs of vector fields.


5.2. The proof of Theorem 4.1; # U > 2. In the space of sets of vector fields, thesets satisfying the first typicality condition (in the definition in Section 3.2) form aset CI, which is everywhere dense in the fine C4-topology of Whitney. This obviouslyfollows from the assertion of Theorem 4.1 for # U = 2, which we have just proved.

As before, the violation of the second and third typicality conditions (in thedefinition in Section 3.2) means that the 1 -jet of the set of vector fields can belongto an algebraic subset Q in the space of 1 -jets of sets of vector fields of codimensionhigher than two. There exists a strictly analytic stratification of this subset. By Thom'stransversality theorem, in the space of sets of vector fields, the sets of vector fieldssuch that for each of them the image of its 1 -jet extension does not intersect Q, formof set, which is open in the fine C2-topology and is dense in the fine C°°-topology.Consequently, the second and third typicality conditions are satisfied on an openeverywhere dense subset C23 in this space in the fine C4-topology.

Thus, the typical sets of vector fields form an open, everywhere dense subsetCI n C23 in the space of sets of vector fields in the fine C4-topology.


§6. The singular surface of a control system

Here we prove Theorem 4.2.

6.1. Stratification of jets of control systems. Now we define some sets that will beused below. They are subsets in the space j3 (P, R2) of 3 -jets of mappings f. Denoteby Pr the union of s-dimensional connected components of the manifold P. It is clearthat P2 belongs to the singular surface of any control system. We set

Y2 = J3 (P2,1R2), Y° = { E Y2 I j 3 f (P) _ f (P) = 01,

where j 3 f is the 3-jet extension of the mapping f. Here and below, unless otherwisestated, p is the projection of the jet on the preimage. For -s > 2, at a point p E P.sof the singular surface of the system f we have rank (f, ft,)(p) < 2. We break the setof jets Z = { E J3(P,,]R2) 1 j 3 f (p) = 4 = rank (f, f,,) (p) < 2} into the subsetsconstructed below using rules (6.1) and (6.2) below. In (6.1) and (6.2) the arrowmeans "see"; "t.t.c." means "there are two cases", and the symbol "N" means "isreduced to the following form under a suitable choice of smooth local coordinates z,u with origin at the point p and fibered over M (possibly with the aid of the Morselemma with a parameter [AS, AGV])"; z = (x, y); fI and f2 are the components ofthe velocity f along the axes x and y, respectively; A = f I / f 2; and A is a column ofheight (s -- 1) (s - 2)/2 composed of all 2 x 2-minors of the matrix (f, f t, ).

(6.1) s = 3; the definition of f0, YIl, y2 l, Yi , Y2 and X3.

1. FEZ, j3.f(p)=t.t.c.:

rank (f, f,,) (p) = 1 H 2=0H 13

2. t.t.c..

f(p)54 0F--+ 3

=0H9

§6. THE SINGULAR SURFACE OF A CONTROL SYSTEM 73

3. f ,., f,t.t.c.:

f l (p) 54 0

Au,, (P) 54 O F---> 4

=0t-+5

4.E Y°°5. t.t.c..

A14,11, (P) 0 1--* 6

6. A E Y11

=OF-*7

7. t.t.c..

f:,(P) 0H8=0 13

8. E Y21

9.

J N

t.t.c.:

fI(z,u) = ulf2 (Z, u) = g(Z, ul ), ul = u, g(P) = gu, (P) = 0

g",,, (P) L O H 10

=0H 11

10. E Yl11. t.t.c.:

gu, u, u, (P) 54 0 i 12=0- 13

12. E Y23

13. 5E X3

(6.2) s > 3; the definition of f00, Y11, Y2 ,

y12, Y2, and X, .

1. FEZ, J3.f(P)=t.t.c.:

rank (f, fu)(p) = 1 H 2

=0-162. t.t.c..

rank A,, (p) > s - 3 H3<s-3F--+ 16

3. t.t.c..

f(p)540 4

=0H 10


4. f,.,f, fi(P) 0t.t.c.:

rank t,u (p) = s - 2 --> 5

5. E Y°°6. t. c.:

fu(p) 54 OF-* 7

= 0 F-> 16

7. A - A(z, u) = A(z, ul) ± u2 f ... + us-2,t.t.c.:

Au1 (p) = A,,,u, (p) = 0

Au, u, u, (P) 54 0 H 8

=0H98. E Y11

9. 4 E Yz10.

t.t.c.:

fI(z,u) = u1f2(Z, u) = h(z, u1, u2) ± u3 f ... f us-2, h(p) = 0, hu(P) = 0

hu2,r2 (P) 54 0 ,-i 11

=0H 15

11.

NJ 2 2

t.t.c.:

fI(Z,u) = UIZ, U) = gf2( )

Z, ut) ± u2 ± ... f us-2,

gu,u,(P) 54 0' 12=0H 13

12. ' E Y1213. t.t.c.:

gu, u, u, (P) 54 0 14

=0- 16

g(P) = gu, (P) = 0

14. E Y215. E Y216. SEX.We denote by YO O, Y, X, Yji, and YS the union f00 U (Y2\ Y°), the union of all

YJ.

, the union of all XS, the closure of YJ.

in Y, and the intersection Y fl j3 (PS, IR2),respectively.

§6. THE SINGULAR SURFACE OF A CONTROL SYSTEM 75

REMARKS. 1. It is clear that the above sets XS and Y.i are pairwise disjoint, andthe fact that the jet belongs to one of them does not depend either on the choiceof the representative f, j 3 f (p) or on the local coordinate systems used in thedefinitions.

2. The intersection Yji n j3 (PS,1R2) is empty if and only if (i, j) = (0, 2) and s > 2or i > 1, j assumes an arbitrary possible value, and s = 2 or i = j = 2 and s = 3.

Later the following results will be useful.

LEMMA 6.1. Y, is a smooth closed submanifold in J3(PS, R2)\XS of codimensions-2.

LEMMA 6.2. The intersection Y n YS is either empty or is a smooth closed subman-ifold in YS of codimension j.

These lemmas and definitions (6.1) and (6.2) imply

COROLLARY 6.3. The intersection Y n j3 (PS,1R2) either is empty or is a smoothclosed submanifold in J3(PS,R2)\XS of codimension s + j - 2.

COROLLARY 6.4 Y is a smooth closed Whitney-stratfed submanifold in J3 (P, 1182)\Xwith the following adjacency diagram of strata Yf :

Y2 y1' Yo Y°

Y3 y12 Y22 2

We now state the following important result.

LEMMA 6.5 (the first basic lemma). For a control system in general position, (1) theintersection j3 f (p) n X is empty and (2) the mapping j3f is transversal to the stratifiedsubmanifold Y.

We do not present the proofs of Lemmas 6.1 and 6.2. They can be reduced to theverification of their assertions in local coordinates. Lemma 6.5 is proved in Section 6.3.

6.2. Proof of Theorem 4.2. The theorem follows from Lemma 6.5 and the lemmabelow.

LEMMA 6.6. Let the first basic lemma hold for a system f . Then Theorem 4.2 holdsfor this system. Moreover; for each point p on the singular surface of the system we havej3 f (p) E Y, and p is a regular point or is a critical point of the type of a Whitney foldor a Whitney cusp or a Whitney double cusp for the folding of the system if and only ifj3 f (p) belongs to Yo n Y°, or to V11 n Y12, or to Y2 n Y2, or to Y2, respectively.

This lemma is proved in Section 6.4.

6.3. Proof of Lemma 6.5. The second assertion of this lemma follows directlyfrom the first assertion, Corollary 6.4, and Thom's strong transversality theorem. Letus prove the first assertion. We introduce the set X' = X n ((f, 0) for s > 3,the sets X3 = (X3\X3') n (f = 0) and X3 = X3\(X3' n X3) for s = 3, and the setsXs = (XS \X') n (rank A < s - 3) and Xs = (XS \X') n (rank 0 = s - 3) fors > 3. It can easily be shown that X' is a smooth closed submanifold in J3(PS,R2) ofcodimension higher than s, XS is a Whitney-smoothly-stratified (simply smooth for


s = 3,4) closed submanifold in J3(PS, R2)\Xt of codimension higher than s, X, is asmooth closed submanifold in J3(PS, l 2)\(Xl UX?) ofcodimension higher than s, andX, = X, U X, U X,3. Applying Thom's strong transversality theorem consecutively toX, , X,2, and Xr we conclude that for a system f in general position the image j3 f (p)does not intersect either Xr , or K,2, or X, for any s and hence does not intersect X.The lemma is proved.

6.4. Lemma 6.6 is proved using the notation in definitions (6.1) and (6.2). Assumethat the first basic lemma holds for a system f. Since j 3 f (p) n X = 0, the singularsurface of the system is determined by the equation

(6.3) rank(f, 1.

It is clear that this surface is a closed subset of P. For a point p on the surface thejet j3 f (p) can belong to only one of the following six sets: Y2, Y2, Yi , Y2, Y1I, orYo0 U Y20 .

We consider these six cases one by one.If j3 f (p) E

Y21 n Y then for s > 3 we have f - f : f i (z, u) = u1, f 2(z, u) _h(z, uI, u2) f u3 f f u2s _2, where h is a smooth function. Equation (6.3) yields

ulh,,, -h=u3=...=Us_2=0.Case 1. j 3 f (p) E y22. We have (p) = 0 and h,21(p) 0 because

rank 0 (p) = s - 3 and f (p) = 0. The condition j 3 f rh Y2 (the symbol rh means"transversal to") implies

rank

h,12142 Xto- h,,hu,x

h14214211 1

1 0

0 0

h142,11 0

h111-112142

(to simplify the notation in this and subsequent matrices we replace quantities whichmust be zero at the point in question by zeros). Consequently,

(6.4) hill 1nu2(P) 54 0 0 (hxhu2)' - hyh112X)(P)

By the implicit function theorem, we find from the equation h,,, = 0 that uI = r(z, u2),where r is a smooth function, and (p) = 0 54 ",421i (p) because h142142 (P) = 0

Substituting ul = r(z,u2) into the equation uIhill - h = 0, we obtainthe equation H(z, u2) = 0, where H is a smooth function, and H,12 (p) = H121 = 0because (u1 h,,, - h),,, (p) = 0 = h11, (P) = (P) = (p). It follows from (6.4) thatH,12141- (p) $ 0 54 (H, y, - H3 , H,,,,) (p). Consequently, p is a Whitney fold criticalpoint of the folding of the system.

REMARK. We do not dwell on the choice of normalizing coordinate systems. It issimple and can be done using the Mather division theorem.

If.3 f(p) 2

2E(-y2 \ Y2) n Y then for s > 3 the coordinate u2 can be chosen so

that h (z, u 1, u2) = g (Z' U I) f U2 2, where g is a smooth function because h1,,, (p) 54 O.Equation (6.3) yields uig,,, - g = 0 = u2 = U3 = = Us-

2-Case 2. j3 f (p) E YY , gill,,, (p) = 0 g1,,,41141 (p). The condition j3f rh YY implies

914111lx gUI UI l' g111U1141

rank 0 0 1 (p) = 3gx g,' 0

§7. THE CRITICAL SET OF A CONTROL SYSTEM 77

Consequently, (g,,,,,,,g), -g,,,,,,),gx)(p) 0. Furthermore, (ulg,,, -g),,,(z,0) = 0.Hence, p is a double Whitney cusp critical point of the folding of the system.

Case 3. j3 f (p) E Y12, g,,,,,, (p) # 0. The condition j3f rh YY implies

rank C0 0 1 (p) =

2,\ gx g), 0

whence g2 (p) 0. Therefore p is a Whitney fold critical point of the folding of thesystem.

If j3f(P) f ^'f, fl(P) 34 0, A(z, u) =A(z,u1) fu2 f...+us-2Equation (6.3) yields A,,, = 0 = u2 = = Us-2-

Case 4. j3f E Y2, A,,,,,, (p) = 0 = A,,,,,,,,, (p). The condition j3f rh Y2 implies

rank A,,,,,,,tAU, U,UIX AU, U,U,)' AII,U,u,ll,

0Afl, l/,)'

All,. All,)' 0

whence All, ,,,,,,,,, (p) 0 (Al,,,,, ,A,4,), -A,l,,,,),,,,,) (p). Consequently, p is a Whitneycusp critical point of the folding of the system.

Case 5. j3 f (p) E Y11, A,,,,,, (p) = 0 < A111141141 (p). The condition j3f rh Y11

implies

rank (A4X All, , )Al,, (p) = 2,All,, A141), 0

whence All, z (p) 0. Consequently, p is a Whitney fold critical point of the folding ofthe system.

Case 6. j 3 f (p) E Yo U Y°. Obviously, p is a regular point of the folding of thesystem.

Thus, p is a regular point or is a Whitney fold or a Whitney cusp or a doubleWhitney cusp of the folding of the system f if the jet j 3 f (p) belongs to Yo u Y20, orto YIl U Y2 , or to Y2 U Y2 , or to Y2 , respectively. Lemma 6.6 is proved.

We note that in Cases 1 and 2 the interior of the convex hull of the indicatrix atthe point r(p) contains the zero velocity, and in Case 5 the velocity f (p) cannot be alimiting velocity at the point r(p). Therefore, we have

COROLLARY 6.7. If the first basic lemma holds for a system f, then for each point zof the closure of its steep domain (of the boundary of the rest zone and of the boundaryof the zone of local transitivity) (1) the intersection j 3 f (z, U) n (Y2 U Y2) is empty and(2) j3 f (z, u) V Y1' if u is a limiting control at the point z.

§7. The critical set of a control system

In this section Theorems 4.3 and 4.5 and some of the assertions of Theorems 4.4and 4.11 are proved.

7.1. Stratification of multijets. Here and below, unless otherwise stated, we shalluse the following notation: the sets X, Xr, Y, and Y were introduced in §6; p" _(q", u") is the projection of the 3 jet ,, onto the preimage; 1" = lR f (p") +f,, (p") U;s = yb = W E R2); y = s1 + s2 + + Ski i = 02 ... ik,0 < ii 3, j = jlj2. A, 0 < Jl < 2, IjI = jI + j2 + + A, i,. and j,. are integers;and1<r<k.


Let us define the following sets in the space Jk (p, R3) of k-fold 3 jets of controlsystems, k > 1. We put

Xk ={ E Jk(P,R2)IE3r, 1 < r < k, c, E X};yk ={ E J3(P, E Yj;, qr = q", r, n E {1, 2, ... , k}

Ci ={gE Yklj, >0, 1<r<k};0 ={1; E Yllr +l" # R2, r,n E {1,2,... ,k}};

1 k ili2...iY I( 1, 2,.EOJIJ2...jr( rxx

xx+l b,+2, k) E

1<n<r;I< r <k - 1;

x 'x E 1 < r< k- 1;(br+1, br+2, , bk)

O!(r,n) ={ E Yk ,br) E 01 ' ,i2...ir

xx xx xx it+Iir+2...in(b,'+1, br+2, , b») E

rr xx x E Oin+I 1 < r < n- 1<k-2 ;(bn+1, bn+2, , bk) Jn+I

CO'(r,n)={g E Yklj., >0, 1 <s <r;xx xx rr it+I it+2...i

(br+1, br+2, , bn) E(x xx in+I in+2...Jkbn+l, b,I+2, ,k) E 1<r<n-1<k-2.

Denote by Ck (Ok, Ok (r), COk (r), Ok (r, n), and COk (r, n)) the union, over allpossible pairs of multi-indices (i, j), of the sets C (O, 1 < r < k - 1; CO(r),1 <r<k-1;0'(r,n),1 <r<n-1 <k-2;andCO; (r,n),1 <r<n-1 <k-2,respectively), and by A the closure of a set A in Yk. It can easily be shown that theabove definitions of the sets and Lemmas 6.1 and 6.2 imply

LEMMA 7.1. Locally, in a neighborhood of one of its points (, the set yk is a smoothclosed submanifold in Jk (P, ][82)\Xk of codimension E - 2.

LEMMA 7.2. The set C,' (O,'; -6,i (r), 1 < r < k - 1; CO! (r), 1 < r < k - 1; Oj'(r, n),

1 < r < n - 1 < k - 2; or COQ (r, n), 1 < r < n - 1 < k - 2, respectively), is aclosed smooth submanifold in yk of codimension IJ I (Ij I + k - m - 1, I j I + k - m - 2,IJI +k-in -r-1, IjI+k-m-3,or ljI+k-m-r- 2, respectively,where misthe number of different values of s such that (i,,, j.,) = (0, 2) and 1 < s < k, 1, < s < k,or rk < s < k, respectively).

LEMMA 7.3. The set Ck (Ok, Ok (r), COk (r), Ok (r, n), Or COk (r, n), respectively)is a smooth closed Whitney-stratified submanifold in Yk with strata Cj' (0' Oa(r), 1 <r < k - 1; C01(r), 1 <r<k-1;0'(r,n), 1 <r<n-1 <k-2;oi- C0'(r,n),1 < r < n - 1 < k - 2, respectively).

7.2. Transversality. Denote by jk f a k-fold 3 jet extension of a system f, andlet p(k) = {(p', p2, ... pk) I pi E P; pi p" for 1 n}. We state the followingimportant result.

LEMMA 7.4 (the second basic lemma). For a control system in general position, (1)the intersection jk f (P(k)) n Xk is empty and (2) the mapping jk f is transversal to each


of the stratified submanifolds Ck; Ok; Ok (r), 1 < r < k - 1; Ok (r, n), 1 < r < n - 1 <k-2; andCOk(r,n), I< r < n -1 <k-2.

This lemma is proved in Section 10.1. Together with Lemmas 7.1 and 7.2, itimplies

COROLLARY 7.5. The images of k -fold 3 -jet extensions of a control system f ingeneral position do not intersect any of the sets of the form of Ok (r, n) and COk (r, n),and they can have nonempty intersections only with (1) the strata Cl 2'2; (2) the strata 000O1'°°, 0012, 020, 002, and 0000; (3) the stratum 00000 (2); and (4) the stratum COioo°(1)of the sets of the form (1)Ck, (2) 0k, (3) 0k (r), and (4) CO' (r), respectively; iI = 1, 2;i2 = 1, 2.

Together with Lemma 6.6 this corollary implies

COROLLARY 7.6. The points of the critical set of a control system f in general positionare broken up into 16 classes such that for a point q of the n th class the images of the fibreq x U (of the bundle z: P -* M) under a 3 -jet extension and multiple 3 -jet extensionsof the system have nonempty intersections with the nth set among the following 16 sets:

(1) 00000(2), (2) 0000 (3) CO100(1), (4) CO100(1),

(5) 020 U 002, (6) O10 U 001, (7) O10 U 001, (8) 01111

(9) CIIi U CIiI , (10) 011 , (11) 000, (12) Y11,

(13) Y12, (14) Y2, (15) Y2, (16) Y2,

and the index of the class of a point is defined as the smallest of all the indices of theenumerated sets with which these images have nonempty intersections.

7.3. The beginning of the proof of Theorem 4.4. Later the following lemma will beuseful.

LEMMA 7.7. A point on the boundary of the steep domain (of the rest zone or of thelocal transitivity zone) of a control system f in general position belongs to the limiting-passing set if there is no nonisolated value of the control parameter determining the zerovelocity at this point.

COROLLARY 7.8. The boundary of the steep domain (or of the rest zone, or of thelocal transitivity zone) of a control system in general position belongs to the critical setof the system.

PROOF OF THE COROLLARY. Consider a point q on the boundary of the steep do-main (or of the rest zone, or of the local transitivity zone). Let u° be a nonisolatedvalue of the control parameter determining the zero velocity at this point. We haverank(f, u°) < 1 because, by Theorem 1.4 (on boundaries) the boundary of theconvex hull of the velocity indicatrix at the point q contains the zero velocity. Accord-ing to Definitions (6.1) and (6.2), the jet j3 f (q, u°) belongs to one of the sets Y,2, Y22,or Y2 because, by the first basic lemma, we have j 3 f (P) n X = 0 for a typical system.Whence, by Lemma 6.6, the point z is a critical value of the folding of the system, and,consequently, it belongs to the critical set of the system.

By Lemma 7.7, a point q belongs to the limiting-passing set if there is no suchvalue u° of the control parameter. However, by definition, the limiting-passing set iscontained in the critical set as part of the passing set. Hence, Corollary 7.8 is true. Inparticular, it implies the first part of assertion 3 in Theorem 4.4.


PROOF OF LEMMA 7.7. By Theorem 1.4 (on boundaries), the boundary of the.convex hull of the velocity indicatrix at every point q on the boundary of the steepdomain (of the rest zone and of the local transitivity zone) contains the zero velocity.It is clear that this point belongs to the limiting-passing set if the velocity indicatrix atq itself does not contain the zero velocity.

Let the velocity indicatrix of a point q contain the zero velocity and let u bean isolated value of the control parameter determining this velocity. There exists alimiting control u1 at the point q distinct from u. Indeed, if the zero velocity belongsto the set f (q, U\{u}), then the value of u which belongs to U\{u} and determinesthe zero velocity at q can be taken as u 1. Otherwise any value of u belonging to N\{u }and determining a limiting direction velocity at the point q can be taken as u'. Thus,the point q belongs to the limiting-passing set. Lemma 7.7 is proved.

7.4. The classes of points will be investigated for a system f, for which the firstand second basic lemmas are true, and, consequently, Lemma 6.6 and Corollaries 7.5and 7.6 hold. As a result of the investigation, we shall prove Theorems 4.3 and 4.5 andmake substantial progress in proving Theorems 4.4 and 4.6.

LEMMA 7.9. The intersection j 3 f (x x U) n Y2 is nonempty if the first basic lemmais true for the system f and the point z does not belong to the passing set but belongs toits closure.

This lemma is proved in Section 7.5. Together with Lemma 6.6, it implies that thelimiting points of the passing set not belonging to it lie in the set of critical points ofthe folding of the system. The latter set is obviously closed. Consequently, the criticalset of the system is closed.

We denote the critical set, the steep domain, the rest zone (stationary zone), andthe local transitivity zone of a control system and the interior and the closure of aset A by CS, SD, SZ, LTZ, Int A, and A, respectively. Consider a point q c CS.For j 3 f (q, u") E Y0 U Y° we denote by v" = (vi', v2) the smooth field of singularvelocities determined locally, in a neighborhood of the point q, by the restriction off to a sufficiently small neighborhood of the point (q, u") on the singular surface (wenote that v" (q) = 0 j 3 f (q, u") E Y20; vl and areare the velocities along theaxes x and y, respectively). For j3f (q, u") E Yj, i > 0, we take a system of localcoordinates in a neighborhood of the point (q, u") with origin at the point and fiberedover M such that (1) locally, in a neighborhood of the point, the singular surface ofthe system is determined by the system of equations A (z, u1) = 0 = u2 = = US _2(in this connection, the coordinates u2, u3, ... , Us,, _2 are regarded as missing, and weset ul = u) and (2) for j = 1 we also have A(z, u) = a"(z) - (u)2. The expression(CS, q) - (B, 0) (or (SC, q) , (B, 0)) means that the germ of the set CS (SD) at thepoint q is C°°-diffeomorphic to the germ at zero of the set B in Table 1 (or Table 2).For v' and v" we put V'"" = v2v1'. The arguments are local in character andrelate to a small neighborhood of the point q.

00000(2). We have V1'2(q) = V3'4(q) = 0. Furthermore, V2'3(q) 0 because,by Corollary 7.5, j43f (P(4)) n 04 = 0. The set CS is determined by the equationV 1,2 V3'4 = 0. Since j43 f rh 00000(2), we have

V,1'2 1'2 3'4 1,2 3 4rank V3,4 (q) = 2, that is, (V, Vy, - Vi, V; )(q) 0.


Thus, the values of the functions V',' and V3,4 in a neighborhood of the point q canbe taken as coordinates in the phase space. Consequently, (CS, q) - (2c, 0). It isobvious that # U > 3.

If the point q belongs to the boundary of the steep domain (of the rest zone orof the local transitivity zone), then by Theorem 1.4 the boundary of the convex hullof the velocity indicatrix at this point contains the zero velocity. The indicatrix itselfdoes not contain this velocity because j3 f (P(5) n CO5 (l, 3) = 0. Hence, the limitingdirections at the point q constitute a straight line, and q is a limiting-passing point.The controls u I and u2 or u3 and u4 are the only limiting controls at the point becausethe images of the 3 -jet extensions of the system f do not intersect any of the sets 04,05(2), and 06(2,4).

For definiteness, assume that u I and u2 are limiting controls at the point q. Thevelocities v3(q) and v4(q) have the same direction and are not limiting at the point q.However, we have V I,2 (q) = 0 and d V I ,2 (q) 54 0. Consequently, in a neighborhoodof the point q, the steep domain lies on one side of the line V1,2 = 0 and the interiorof the rest zone (of the local transitivity zone) lies on the other side, where the cone ofeach of the points contains the zero velocity.

Here the behavior of the cone of a point is seen to be the same as that in Figure 2.9.Hence, (Int SZ, q) = (Int LTZ, q) and (SD, q) - (6f, q), and in this case q is a a-passing point if v I (q) does not belong to Tq ( V 1'2 V3'4 = 0) and is a a-turning pointotherwise. The defining set of this point consists of the two fields v I and v2. The points(q, u') and (q, u2) are regular points of the folding of the system.

0000 .We have V 1,2 (q) = V 1,3 (q) = V2,3 (q) = 0. The set CS is determined by the

equation V1,2 V1,3 V2'3 = 0. By the condition j3 f fh 00000, we have

rank V'1,3 (q) = 2, that is, (V;'2 .,1,3 - ,'1,2V;'3)(q) 54 0.

In a neighborhood of the point q, in the coordinates x" = V 1,2 (x, y), y = V 1,3 (x, y), wehave v2 = i (y)v I + zv2 and v3 = ,u(z)v I + yv3, where A and u are smooth functions,A(0) 0 ,u(0), and v2 and v3 are smooth vector fields. Consequently,

V2,3(x,Y) = X'U (X)(vivi - viv2)(x,Y) +YA(Y)(v,'i - vivi)(z,+ v2v3)(X, y)

_ -,U(x)VI°2(x,y) +A(y)V1"3(z,.v) +x5 52,5)(2,5)

Therefore, (CS, q) ' (3, 0). It is clear that # U > 2.If the point q belongs to the boundary of the steep domain (of the rest zone or

of the local transitivity zone), then by Theorem 1.4 the boundary of the convex hullof the velocity indicatrix at this point contains the zero velocity. The indicatrix itselfdoes not contain this velocity because j4 3f (P(4) n (C04 (1) U 04) = 0. Hence, thelimiting directions at the point q form a straight line, and q is a limiting-passing point.The controls u1, u2, and u3 are the only limiting controls at this point because, bythe relations j3 f (P(S)) n05(2) = 0 = j43 f (P(4)) n04' there are no other passingcontrols at the point. Thus, q is a double a-passing point, and (q, u'), j = 1, 2, 3,are regular points of the system folding. Two of the three velocities v 1(q), v2(q),and v3(q) have the same direction whereas the third one has the opposite directionrelative to these two velocities. For definiteness, let this third velocity be v3(q). In a


neighborhood of the point q, when leaving the open region, we intersect the collinearitylines V',3 = 0 and V2'3 = 0 and enter the interior of the rest zone (of the localtransitivity zone) where the cone of each point coincides with the tangent plane.It follows that (Int SZ, q) = (Int LTZ, q), and (SD, q) N (3, 0) if # U = 3 and(SD, q) - (4, 0) if # U > 3. The defining set of the point q consists of the threevelocity fields v1, v2, and v3.

'00 We have a' (q) = 0 = V2'3(q). The set CS is determined by theC0100equation a' V2,3 = 0. By the condition j3 3f rh CO'oo (1), we have

1

rank V2,3 (q) = 2, that is, V2,3) (q) 54 0.

Thus, the values of the functions V2,3 and a' in a neighborhood of the point q can beused as coordinates in the phase space. Consequently, (CS, q) - (2b, 0). Obviously,#U=oo.

If the point q belongs to the boundary of the steep domain (of the rest zone orof the local transitivity zone), then by Theorem 1.4 the boundary of the convex hullof the velocity indicatrix at this point contains the zero velocity whereas the indicatrixitself does not because j4 f (P(4)) n C04 (j) _ 0 for j = 1, 2. Thus, the limitingdirections at the point q is a straight line, and q is a limiting-passing point. The controlu' is not a limiting control at the point because, by the condition j 3 f (q, u') E Y,',the velocity f (q, u') lies inside the convex hull of the velocity indicatrix at the pointq. The controls u2 and u3 are the only limiting controls at this point because, byj5 f (P(1)) n CO5(1, 3) = 0 = j4 f (P(4)) n C04(1), there are no other passing controlsat the point. Hence, the defining set of the point q consists of the fields v2 and v3whose values at this point have opposite directions. However, we have V2'3 (q) = 0 andd V2.3 (q) 0. Consequently, in a neighborhood of the point q, the steep domain lies onone side of the line V2'3 = 0 whereas the interior of the rest zone (of the local transitivityzone) lies on the other side where the cone of every point coincides with the tangentplane. Here in the behavior of the cone of a point the same phenomenon is observedas in Figure 2.9. Consequently, (Int SZ, q) = (Int LTZ, q) and (SD, q) , (6e, 0); inthis case q is a 0-passing point if v2(q) Ty(V2'3 = 0) and is a (9-turning point ifv2(q) E Tq(V2'3 = 0). The points (q, u2) and (q, u3) are regular points of the systemfolding.

C0200 (l ). Similarly to the class CO "0 (1), we obtain (CS, q) N (2b, 0) and also100 100

#U=oo.Furthermore,1' +1' = R2, s = 2, 3, because j3 f (p(3)) n 0100 = 0. Consequently,

the vectors v2(q) and v3(q) have the same direction if the point q belongs to theboundary of the steep domain (of the rest zone or of the local transitivity zone). Thecontrol u' is the only limiting control at the point q because j43f (P(4)) n 0ijo (2) _ 0for all possible i and j. Therefore, q is a zero-point. In a neighborhood of this pointthe steep domain and the interior of the rest zone (of the local transitivity zone), wherethe cone of each point coincides with the tangent plane, are separated by the line ofcritical values of the system folding. Consequently, (Int SZ, q) = (Int LTZ, q) and(SD, q) - (6c, 0). The point (q, u') is a critical point, which is a Whitney fold of thesystem.

02°° U 00°. Let v' (q) = 0 54 v2(q) (the case v' (q) 54 0 = v2(q) is considered ina similar way). Here u ' , U , u are singular controls at the point q, j 3 f (q, u") E


Yo because j2 3f (P(2)) n 00 = 0, j = 1, 2, and V"," (q) 54 0 because j3 f (P(3)) n0000 = 0, n, r E {2, 3,... , k}, n # r. The set CS is determined by the equation

V',2 V' ,3 ... VIA = 0. By j 3 f rh Y°, the matrix of the linearization of the field v'at the point q is nonsingular. Consequently, (dV',r)(O) 54 0, 2 < r < k. Thus, thegerm (Cs, q) is C°°-diffeomorphic to the germs (lb, 0), (2c, 0), (3, 0), and (4k_4, 0)if k = 2 (in this case # U = 2), k = 3, k = 4, and k > 4, respectively.

If the point q belongs to the boundary of the steep domain (of the rest zone or ofthe local transitivity zone), then q is a zero-passing point because there is an isolatedvalue of the control parameter determining the zero velocity at this point, and byTheorem 1.4 the cone at that point is smaller than the tangent plane.

The convex hull of the vectors v2(q), v3 (q), ... , vk (q) does not contain the zerovelocity because j3 f (P(3)) n 0200° = 0. Consequently, the field v'', 2 < r < k, belongsto the defining set of the point q if v"(q) is a limiting velocity at the point. Thus,the defining set of the point q consists of the two fields v' and v2 if k = 2 (and inthis case # U = 2) and of the three fields v' , vr, and vs (where yr (q) and vs (q) arelimiting velocities at the point q, 2 < r < s < k) if k > 2 (and then # U > k). In thelatter case the cone of each of the points in the interior of the rest zone (of the localtransitivity zone) coincides with the tangent plane. Accordingly, (SD, q) N (1, 0) and(SD, 0) - (5k_3, 0). In both cases (Int SZ, q) = (Int LTZ, q), and (q, u) is a regularpoint of the system folding if u is a limiting control at the point q.

O10° U 001. Let j 3 f (q, u I) E Y11 (the case j 3 f (q, u2) E Y1' is considered ina similar way). We put y = a' (z). The set CS is determined by the equationyH(x,y, + f) = 0, where H = f;/fj - v?/v and (i, j) = (2, 1) if vi (q) 0 and(i, j) = (1, 2) if vj (q) = 0. By the condition j2 f rh 010, we have

Huiu1

rank H 0H- 0

(q, ul) = 3,

that is, (H,,H,4, -H,H14 )H,,,,t,(q, u') 54 0. However, we (q, u') = H,,,,Y(q, u')= 0 because, by the choice of the local coordinates, H (z, u) = (y - (u)2)R(z, u), whereR is a smooth function. Therefore, (H, H,,), H1,4,4) (q, u I) # 0. By the implicit functiontheorem, in a neighborhood of the point (q, u') the equation H = 0 is equivalent toan equation of the form x = h (y, u), where h is a smooth function. We represent thefunction h as a sum of an even function and an odd function with respect to u: h (y, u) =A(y, (u)2) + uB(y, (u)2), where A and B are some smooth functions. In this case theset CS is determined by the equation y(yB2(y, y) - (x - A(y, y))2) = 0. However,we have B(0, 0) = 0 B(y, y),,(0) because u) = 0 54 u').Consequently, in the smooth coordinate system x = x - A (y, y), y = (yB2 (y, y)) 1/3

the set CS is determined by the equation (we drop the tilde) y (y - x2/3) = 0, i.e.,(CS, q) - (6, 0). Obviously, # U = oo.

The investigation of this subcase is completed by the following lemma.

LEMMA 7.10. A point belonging to any of the classes 6, 8, 10, 12, 14, 15, and 16cannot belong to the boundary of the steep domain (of the rest zone and of the localtransitivity zone) of a control system for which the first and second basic lemmas hold.

This lemma is proved in Section 7.6. It implies that the point q cannot belong tothe boundary of the steep domain (of the rest zone and of the local transitivity zone).


010 U O01. Let j 3 f (q, u') E Yi (the case j 3 f (q, u2) E Y, is considered in a similarway). Let y = a' W. The set CS is determined by the equation yH(x, y, fVFy-) = 0,where H = v,? /v? and (i, j) = (2, 1) if vI(q) 54 0 and (i, j) = (1,2) ifVI (q) = 0. We set S = f ,, f 2 - f i f 2 ,, . Since j2 f rh O10°, we have

Sux Stiy, Suitrank S, Sy, 0 (q,u') = 3, that is, (S,u,(S,H3, - H,S,,))(q,u') # 0.

H, H), 0

However, we have S, (q, u') = 0 = (q, u') because, in view of the choice of the localcoordinates, S(z, u) = (y - (u)2)R(z, u), where R is a smooth function. Consequently,(St,,, S3, H,) (q, u') 0 0. Furthermore, H (q, u') 54 0 since S,,,, (q, u') # 0. By theimplicit function theorem, in a neighborhood of the point (q, u') the equation H = 0is equivalent to the equation u = h (x, y), where h is a smooth function, h (0, 0) = 0 54h, (0, 0), because H (0, 0, 0) = 0 H, (0, 0, 0). The critical set is now determined bythe equation y(y - h2(x, y)) = 0. Making change of variable x" = h(x, y) (the tilde isomitted) we bring the equation to the form y(y - x2) = 0, i.e., (CS, q) - (5, 0). It isclear that # U = oo.

If the point q belongs to the boundary of the steep domain (of the rest zone orof the local transitivity zone), then u' and u2 are limiting controls at this point. Thisfollows from the fact that the cone of the point q is obviously no smaller than the openhalf plane bounded by the straight line L(q, u'), and by Theorem 1.4 it is no greaterthan its closure. At the point q there are no limiting controls distinct from u I andu2 because by Corollary 7.5 j3f (P(3)) n 020 = 0 for any possible i and j. Hence,

3 10j

C(q) = 2, and q is a nonzero-passing point.The restriction of f to a sufficiently small neighborhood of the point (q, u I) on

the singular surface determines two branches of the field of singular velocities inthe region y > 0. The two branches vanish on the abscissa axis. One of them is.collinear to the field v2 on one of the branches of the parabola y = x2 and the other iscollinear to v2 on the other branch of the parabola. The collinear fields have the samedirection on one of the two branches, and opposite directions on the other branch(Figure 2.23). By the condition j2 If rh 02, the transition from the steep domain tothe local transitivity zone is at the intersection of the latter branch from the side of theregion y > x2. In a neighborhood of the point q the boundary of the steep domainis a C' -submanifold obtained by pasting together this semiparabolic branch and thesemiaxis of the abscissa axis. On the other side of this boundary lies the interior of therest zone (of the local transitivity zone), where, as can easily be seen, the cone of each ofthe points coincides with the tangent plane. Consequently, (Int SZ, q) = (Int LTZ, q)and (SD, q)) , (7, 0).

CIII' , CI'I U CI I' , and C H. The set SD is determined by the equation a' a2 = 0.The condition j23f rh Ci ltz, il, j2 E 11, 2} implies

ranka2

(q) = 2, that is, 0.

Thus, the values of the functions a' and a2 in the vicinity of the point q can be takenas coordinates in the phase space. Therefore, (CS, q) - (2a, 0).

According to Lemma 7.10, for classes 8 (C1111) and 10 (C2i) the point q cannotbelong to the boundary of the steep domain (of the rest zone and of the local transitivityzone).


FIGURE 2.23

CS

FIGURE 2.24

Now let a point q of class 9 (CI11 U C2 II) belong to the boundary of SD (SZ orLTZ). For definiteness, assume that j 3 f (q, u 1) E YII. We have 11 + 12 = R2 becausej2 3f (P(2)) n (Oi 1 O ) = 0, which follows from Corollary 7.5. Clearly, the union ofthe limiting directions at the point q coincides with the straight line 12. Hence, u2 is alimiting control at the point q whereas u 1 is not. There are no other limiting directionsat the point q because, according to Corollary 7.5, we have j33f (P(3)) n COi 2i (1) = 0for any possible i and j. Consequently, q is a zero-point.

In a neighborhood of this point, the zero velocity intersects the boundary of theconvex hull of the velocity indicatrix at the intersection of the boundary of SD (SZ orL TZ). The phenomenon determining the boundary of SD (SZ or L TZ) in a neighbor-hood of the point q is observed in "pure" form in class 13 (Y,2; see Figure 2.24) investi-gated below. Consequently, we have (SD, q) - (6b, 0) and (Int SZ, q) = (Int LTZ, q),and the cone of each of the interior points of SZ (LTZ) coincides with the tangentplane.

000. The set CS is determined by the equation V 1,2 = 0. In view of the conditionj2 3f rh 000, we have V1'2 (q) 0. Consequently, (CS, q) - (lb, 0). Obviously,#U> 1.

If the point q belongs to the boundary of the steep domain (of the rest zone or ofthe local transitivity zone), then u 1 and u2 are limiting controls at this point. Indeed,otherwise the point would belong to one of the previous classes. For the same reason,there are no other limiting controls at the point. Therefore, q is a 8-passing point ifv 1(q) Ty (V 1.2 = 0) and is a 0-turning point if v 1(q) E Tq (V 1'2 = 0). The points


(q, u') and (q, u2) are regular points of the system folding. The defining set of thepoint q consists of the two fields v' and v2. The germ (SD, q) is C°°-diffeomorphicto the germ (2, 0) if #U = 2 and to the germ (6d, 0) if #U > 2. In the latter casethe cone of each of the points of the complement of the closure of the steep domaincoincides with the tangent plane, and we have (Int SZ, q) = (Int LTZ, q).

Y11. Lemma 7.9 and Corollary 7.5 imply that the point q does not belong to theclosure of the passing set. Hence, by Lemma 6.6, we have (CS, q) , (la, 0). It isclear that #U = oo. According to Lemma 7.10, the point q does not belong to theboundary of the steep domain (of the rest zone and of the local transitivity zone).

Y,2. Lemma 7.9 and Corollary 7.5 imply that the point q does not lie in theclosure of the passing set. Consequently, by Lemma 6.6, we have (CS, q) - (la, 0).Obviously, in this case # U = oo.

If the point q belongs to the boundary of the steep domain (of the rest zone orof the local transitivity zone), then, as can easily be seen, u' is a limiting control atthis point. There are no other limiting controls at the point q because otherwise qwould belong to class 7. Hence, q is a zero-point. In a neighborhood of this point, thezero velocity intersects the boundary of the convex hull of the velocity indicatrix at theintersection of the critical set (Figure 2.24). Consequently, we have (SD, q) , (6a, 0)and (Int SZ, q) = (Int LTZ, q), and the cone of each of the points in the interior ofthe rest zone (of the local transitivity zone) coincides with the tangent plane.

Y21. We assume, without loss of generality, that f I (q, u') # 0. Set A = f2 If I .

We have A(z, u) = a(z) + ub(z) + u2c(z) + u3d (z) + u4R(z, u), where a, b, c, d, andR are smooth functions, and b (q) = c (q) = d (q) = 0 because j 3 f (q, u') E Y21. Sincej3.f fi I'2,

rankAuuZ 0 (q, u') = 3,

AUZ 0 )that is,

(7.1) (A,4u:ru(AuuxAUJ, - 21114),A,,x))(q,uI) 0.

Consequently, the mapping (z, u) F--* (z, 2(z, u)) has a Whitney singularity 1111 (aswallow tail) at the point (q, u') [AGV]. A change of variables of the form of i = Z(z),A = A(z, A) reduces the set of critical values of this mapping to the set of critical valuesof the mapping (z, u) --> (z, uz + u2y + u4) [AGV]. The Maxwell stratum correspondsto the passing set and the cuspidal edge of the swallow tail corresponds to the set ofcritical values of the system folding. Consequently, (CS, q) - (7, O). By Lemma 7.10,the point q does not belong to the boundary of the steep domain (of the rest zone andof the local transitivity zone) of the system f .

Y2 and Y21. From Lemma 7.9 and Corollary 7.5 it follows that the point q doesnot lie in the closure of the passing set. Consequently, by Lemma 6.6, the germ (CS, q)is C°°-diffeomorphic to the germ at zero of sets 8 and 9 in Table 1 for the cases of Y2and Y2 , respectively. By Lemma 7.10, the point q does not belong to the boundary ofthe steep domain (of the rest zone and of the local transitivity zone) of the system f.

The investigation of the classes of points can be completed modulo Lemmas 7.9and 7.10. To sum up the investigation, Theorems 4.3 and 4.5 have been provedcompletely. In Theorem 4.4 it remains to prove assertions (1) and (4), which will bedone in Section 9.2. In Theorem 4.6 we have proved the second assertion and the part


of the first assertion relating to the number of fields in the defining set. The other twoparts of this theorem will be proved in Section 9.1.

7.5. Proof of Lemma 7.9. Consider a sequence of points of the passing set: zk qfork -* oo and sequences uk, I and uk,2 of distinct passing controls at the point zk whichdetermine the collinear velocities at this point. We select converging subsequences{ukr"I } and {uk"'2} from these sequences (which is clearly possible). We can assume,without loss of generality, that ki = 1. These subsequences converge to the same pointu° because otherwise the point q would belong to the passing set. We shall show thatj3 f (p) E Y2, where p = (q, u°). We have j3 f (p) C Y because p is obviously a pointof the singular surface.

We first show that f (p) 54 0. If f (p) = 0, then locally, in a neighborhood of thepoint p, we have f I (z, u) = u1, f2 (Z, u) = h(z, u1i u2) + u3 f f us_2 in a suitablesystem of smooth local coordinates with origin at p and fibered over M (see Part 10 inDefinition (6.2)). If h (z, u1, u2) = h(z, uI) + u2 for a suitable choiceof the coordinate u2; in this case we have u2 = U3 Us-2 on the singular surface.The vectors (the tilde is omitted)

(ul'I, h(z', ul'1)), (ul'2, h(z', ul'2)),

(1, ht,, (Z', ui'1)), (1, hill (z', u1'2))

are collinear by the choice of the subsequences. By applying Lagrange's mean-valuetheorem, the collinearity of the first two vectors implies that the vector

1,hW, u'11 - h(112z1, u12)

_ (1, hill (z', ui'2 + 0(u 1'1 - ui'2)))u1 ill

1

where 0 < 0 < 1, is collinear to those two vectors. Since (z', u'"") - p as 1 - oo,n = 1, 2, it follows that h,,,,,, (p) = h,,,,,,,,, (p) = 0. Consequently, j3 f (p) E X, which isimpossible because of the choice of the system. Thus, there must be (p) = 0. Forhu2u2 (p) = 0 we obtain j3 f (p) E Y2 by the choice of the system, and, in particular,hu,,,2 (p) ; 0. On the singular surface we have 0 = u3 = = us_2. By theimplicit function theorem, the equation 0 implies that locally, in a neighborhoodof the point p, we have u1 = r(z, u2), where r is a smooth function and (p) = 0because 0. According to the choice of the subsequences, the vectors(1, hill (z', r(z', u2"), u2")), n = 1, 2, are collinear. Therefore, applying Lagrange'smean-value theorem, we find

1,

)0 = hill (z', r(z', u,'2), u2'2) - h,,, (z', r (z', u2u2

= (hu,u, (z', r (z', v), v)r112 (z', v) + huiu2 (z', r(z', v), V))(U12,2 - u21),

where v = u21 + 0(u22 - u2'1), 0 < 0 < 1. Since (p) = 0 and (z',ui,") -* pas1 - 0, n = 1, 2, it follows that h,,,,,2 (p) = 0, and thus we arrive at a contradiction.Consequently, f (p) 54 0.

For f (p) 54 0, locally in a neighborhood of the point p we have 2(z, u) = A(z, u1)±

u2 ± ... + us_2 in a suitable system of smooth local coordinates with origin at p andfibered over M (see Part 7 in Definition (6.2)). Furthermore, A(z', A(z', ui'2)and All, (z', U1,11) = 0, n = 1, 2. Since (z', u'"") -p p as 1 --+ oo, n = 1, 2, it follows thatA,,, (p) = Au,,,, (p) = A,,,,,,,,, (p) = 0. Therefore, j3 f (p) E Y2, which is what we hadto prove. Lemma 7.9 is proved.


It is easy to see that we have also proved

LEMMA 7.11. If the first basic lemma holds for a system f and the image of themapping j23f intersects 02 in an arbitrarily small neighborhood of the point ( , ) on thediagonal in j3 (P, R2), then j3 f (p) E Y2.

7.6. Proof of Lemma 7.10. For classes 15 (Y2) and 16 (Y2) the assertion of thelemma follows directly from Corollary 6.7. For classes 6, 8, 10, 12, and 14 we can provethe lemma by contradiction. Assume that a point q of one of these classes belongs tothe boundary of SD (SZ or LTZ) for a system f. We shall consider the five subcasesin question in the reverse order.

Class 14 (Y2). By Lemma 7.7, the image of a 3 -jet extension of the system f hasa nonempty intersection with at least one of the sets of the form of 0"i12

Cii: 2, and

CO2 This contradicts Corollary 7.5. Therefore, the assertion of Lemma 7.10 istrue for class 14.

Class 12 (Y11). According to Corollary 6.7, the control u' cannot be limiting at thepoint q. This fact and Lemma 7.7 imply that the image of a multiple 3 -jet extensionof the system f has a nonempty intersection with at least one of the sets CIIj andCOi J23 (1). Hence, the point q does not belong to class 12, which is a contradiction.Consequently, the assertion of Lemma 7.10 is true for class 12.

Class 10 (C). We have 11 + 12 = I[82 because, by Corollary 7.5, j2 f (P(2)) n022-11 i

0. Together with the definition of the set Y,2, this implies that the cone of the pointq coincides with the tangent plane, and this contradicts Theorem 1.4 on boundaries.Consequently, the assertion of Lemma 7.10 is true for class 10.

Class 8 (C1111). By Corollary 6.7, neither of the controls u' and u2 is limiting at thepoint q. This fact and Lemma 7.7 imply that the image of a multiple 3 -jet extension ofthe system f has a nonempty intersection with one of the sets of the form of C i and

C1I11614 (2), which contradict Corollary 7.5. Therefore the assertion of Lemma 7.10 istrue for class 8.

Class 6 (O10° U Oo ). Neither of the controls u' and u2 is limiting at the pointq because, by Corollary 6.7, at least one of these controls is not. Together withLemma 7.7, this fact implies that the image of a multiple 3 -jet extension of the systemf has a nonempty intersection with one of the sets io (1) and Oi03 J4 (2). But thiscontradicts Corollary 7.5. Therefore the assertion of Lemma 7.10 is true for class 6.

Lemma 7.10 is proved.

§& Singularities of the defining set and their stability

In this section we first prove Theorems 4.12 and 4.14 and then show that thecritical set, the defining set, and the set of black points and turning points are stableunder small perturbations of a control system in general position. After that we proveTheorem 4.16.

U. Proof of Theorem 4.12. It suffices to show that the assertions of the theoremhold for a system for which the first and second basic lemmas are true and, conse-quently, Lemma 6.6 and Corollaries 6.7, 7.5, and 7.6 hold. The following lemma willbe useful later.

LEMMA 8.1. If for a system f the first basic lemma holds and u° is a limiting controlat a point q in the steep domain, then the jet j3 f (q, u°) belongs to either Y0 or Y2. In

§8. SINGULARITIES OF THE DEFINING SET AND THEIR STABILITY 89

the first case (j 3 f (q, u°) E Y00), either we have (q, u°) E P2 or (f1,,, f 2 - f I f 2,,,) (q, uo)is a definite matrix.

PROOF. The second assertion of the lemma is obvious. The first assertion followsfrom Lemma 6.6 and Corollary 6.7. Indeed, first of all we have f (q, u°) 54 0 because thepoint q belongs to the steep domain. Consequently, j3 f (q, u°) E Yo if (q, u°) E P2. If(q, u°) V P2, then L(q, u°) 54 R2 because u° is a limiting control at the point q. Hence,(q, u°) is a point on the singular surface of the system f. By Corollary 6.7, we havej3 f (q, u°) V Y11 U Y2 U Y23. Furthermore, j3 f (q, u°) V Yi because f (q, u°) : 0.Therefore, the jet j 3 f (q, u°) belongs either to Yo or to Y2. In the first case, the matrix(f 1,,, f 2 -f I f2,1, ) (q, u°) is definite because it is nonsingular and u° is a limiting controlat the point q (and, consequently, at this point the velocity indicatrix lies on one sideof the straight line L(q, u°)). Lemma 8.1 is proved.

We now prove the theorem. According to Corollary 7.5, the number m; of i-limiting controls at the point q is less than 4. If m; = 1 and u is an i-limiting controlat the point q, then, by Lemma 8.1, either (1) j3 f (q, u°) E Yo or (2) j3 f (q, u°) E Y2.In the first case q is an i-regular point, and the germ (n', q) is R+-equivalent to thegerm (0, 0). In the second case, by condition (7.1), q is an i-cutoff point, and thegerm (n', q) is R+-equivalent to the germ ((-1)' max{-w4 + yw2 + xwJw E R}, 0).

If m; = 2, then q is not a point of class 14. This fact and Lemma 8.1 implythat j2 f (q, u I, u2) E 000, where u 1 and u2 are i-limiting controls at the point q, andthe order of contact of the indicatrix of the point q and the straight line R f (q, u') ateach of their common points is less than two. Consequently, q is either an i-passingpoint or an i-turning point. The condition j2 If rh 000 implies that the germ (n', q) is00R+-equivalent to the germ ((-1)x1, 0).

For m, = 3 we similarly conclude that q is a double i-passing point, and the germ(n', q) is R+-equivalent to the germ ((-1)'(Iy + IxI I + Ix1), 0).


82. Proof of Theorem 4.14. By definition, the limiting-passing set belongs to thepassing set and, consequently, to the critical set of the system. By the third assertionof Theorem 4.4 (proved in the previous section), the boundary of the steep domain ofa system in general position is also contained in the critical set of the system. Thus,the first assertion of Theorem 4.14 is true.

To prove the remaining assertions of the theorem we take a point q lying in thesteep domain and belonging to the defining set. By Corollary 7.6, this point belongsto one of the sixteen classes. It cannot be a point of any of the eight classes 4, 5, 7,9, 10, 13, 15, and 16 because the velocity indicatrix of a point in any of these classescontains the zero velocity (and, consequently, this point does not belong to the steepdomain). The point q cannot be a point in classes 8 and 12 either because singularities2a and la in Table 1 correspond to these classes, and in a neighborhood of thesesingularities there are no points belonging to the passing set. Finally, the point qcan neither be of class 6 because singularity 6 in Table 1 corresponds to this class, inwhose neighborhood, by Corollary 6.7, the passing must be in a direction which is notlimiting.

Therefore, the point q may only belong to one of the five classes 1, 2, 3, 11, and 14in Corollary 7.6. Taking into account which singularities of the critical set correspondto the points of these classes, it is easy to show that these points lead to the followingsingularities of the defining set (according to Table 3).


TABLE 4

Class ofthe point

Critical set The boundaryof the steepdomain

The definingset in the steepdomain

Remarks

1 2c 6f lb,2

2 3 3,4 3,4

3 2b 6e lc

4 2b 6c -5 lb,2c,4,4, 1,5,. -6 6 - -7 5 7 -8 2a - -

9 2a 6b -

10 2a - - E Int LTZ

11 lb 2,6d la12 la - -

13 la 6a -

14 7 - -

15 8 - -

16 9 - - EIntLTZ

Singularity 2 for class 1 if the number C (q) of different limiting controls at thispoint is equal to 4, i.e., q is a 1- and 2-passing point; singularity lb if C(q) = 3, i.e., q

I

is an i-passing point and a j-regular point.Singularity 3 for class 2 if #U = 3; in this case q is a double 1- and 2-passing

point; singularity 4 if #U > 3; in this case q is a double i-passing point and a j-regularpoint.

Singularity lc for class 3; obviously, #U = oo; in this case q is an i-passing pointand a j-regular point.

Singularity 1 a for class 11; in this case q is a 1- and 2-passing point if # U = 2 andan i-passing point and a j-regular point if # U > 2.

Singularity 5 for class 14; the point q itself is an i-cutoff point and a j-regularpoint. Here everywhere i, j E{ 1, 2}, i 54 j.


U. The correspondence between classes and singularities. Table 4 summarizes,for a system in general position, the results of our investigations of singularities ofthe critical set, the boundary of the steep domain, and the defining set in the steepdomain. The first column of the table enumerates the classes of points in Corollary 7.6;the second, third, and fourth columns give the singularities of these three sets in thenotation corresponding to Tables 1, 2, and 3 respectively, if the points of the sets belongto the corresponding class. A dash in a column means that for a system in generalposition a point in the given class cannot belong to the corresponding set.

§8. SINGULARITIES OF THE DEFINING SET AND THEIR STABILITY 91

94. The stability of the sets and their singularities.

LEMMA 8.2. For a control system in general position and any other close system thecritical set and the steep domain of the latter system are transformed into the criticalset and the steep domain of the former by a nearly identical homeomorphism of thephase space. The transforming homeomorphism can be chosen so that it is a C°°-diffeomorphism everywhere except possibly at singular points of type 4,. of these criticalsets.

PROOF. For a control system in general position, to each class in Corollary 7.6there corresponds a specific singularity of the critical set and, if a point of this classappears on the boundary of the steep domain, a specific singularity of the steep domainon its boundary. Consequently, the stability of the critical set and the steep domainwith respect to small perturbations of the system follows from the transversality ofthe 3-jet extension and multiple 3 -jet extensions of the system to the correspondingsubmanifolds. Outside type 4,. at these singular points the transforming mapping canbe chosen so that it belongs to class C°°, which follows from the expressions for thenormal forms in Table 1. Generally speaking, at these singular points of the criticalset the mapping is only continuous. Indeed, at a type 4,. singular point, the cross-ratioof four different straight lines lying in the tangent set to the critical set of the systemat that point is a modulus relative to the C '-diffeomorphisms. Generally, this ratiovaries under perturbation of the system. Lemma 8.2 is proved.

REMARKS. 1. In Lemma 8.2 a C°°-diffeomorphism can be used as the transform-ing mapping provided that the critical set of the first system (and, consequently, of anysystem sufficiently close to it) has no type 4,. singular points.

2. For a control system in general position and any system sufficiently closeto it, the defining sets of the systems are transformed into each other by a C°°-diffeomorphism of the phase space, which is C1-close to the identity diffeomor-phism. Indeed, as can easily be shown, the singularities of the defining set dependCI-continuously on the control system in general position. Moreover, it can be seenfrom Tables 2 and 3, smooth normal forms of the typical singularities in this set containno moduli. Thus, for the defining sets the transforming C°°-diffeomorphisms can bechosen so that it is C'-close to the identity diffeomorphism.

LEMMA 8.3. For a control system in general position, the field L' can only touch theboundary of the steep domain and of the i passing set at nonsingular points of the criticalset and with the first order of contact. The set of these points of tangency (i.e., the set ofblack points, 0-turning points, and i-turning points) depends continuously on this system.

PROOF. The condition of tangency of the field L' with the defining set imposes anadditional independent constraint on the jets of a control system and, consequently,cuts out of the corresponding stratum in the space of 3 -jets or in the space of multiple3 -jets of the control systems a submanifold of codimension 1 (in this stratum). By thetransversality theorems, the 3-jet extension and multiple 3 -jet extensions of a system ingeneral position are transversal to these submanifolds. It is this property that impliesthe assertions of Lemma 8.3.

8,5. Proof of Theorem 4.16. According to Lemmas 8.3 and 8.2, for a systemin general position and any system sufficiently close to it, the critical set, the steepdomain, and the set of black points and (a- and i-)turning points of the latter system


are transformed into the same objects of the former by a C°°-diffeomorphism of thephase space, which is C°-close to the identity diffeomorphism. The correspondingdiffeomorphism of the bundle space P, which is fibre-wise identical, is also C°-closeto the identity diffeomorphism. It arranges the sets of critical points of the foldings ofthese systems one above the other (more precisely, in the same fibres). By Lemmas 6.5and 6.6, these sets (and the singular surfaces spanning them) are close. By Theorem 4.2,the smooth normal forms of the germs of the folding of a system in general positioncontains no moduli. Consequently, these sets and the singular surfaces spanning themare transformed into each other by a C°°-diffeomorphism of the bundle space P,which is C°-close to the identity diffeomorphism and preserves the bundle fibres (i.e.,is identical on M).


§9. Singularities in the family of limiting lines in the steep domain

In this section the justification of the results in §4 is completed. We first completethe proofs of Theorem 4.6 (namely the part of its first assertion concerning the typicalityof a set of vector fields and the types of points) and Theorem 4.4 (assertions (1) and(4c)). Then we consecutevely prove Theorems 4.9, 4.10, 4.11, and 4.15. Below.we consider only typical systems and will not mention this specifically in differentassertions.

9.1. Completion of the proof of Theorem 4.6. So far, we have only proved the partof the first assertion about the number of vector fields in the defining set of fields duringthe investigation of the classes of points.

The statement that defining sets of the vector fields are typical will first be provedfor double a-passing and zero-passing points and then for 0-turning and a-passingpoints.

By Corollary 7.6, for a system f in general position, double a-passing points andzero-passing points belong to classes 2 and 5, respectively. According to Table 4, thesepoints are singular points of the boundary of the steep domain. By Lemma 8.3, thelimiting directions at such points do not lie in the tangent set to the defining set. Thisfact and the condition j2 3f rh O000 imply that for a typical system the defining set ofeach of the double a-passing points in a neighborhood of this point is a typical tripleof fields.

For a system in general position, to each value of the control parameter therecorresponds a (feasible) velocity field with nondegenerate singular points. (This is asomewhat stronger requirement than the condition j 3 f rh Y° because it additionallyrequires that the eigenvalues of the linearization of the field must be distinct at eachnode.) This property and Lemma 8.3 imply that for a system in general position thedefining set of each of the zero-passing points (consisting of min{3, # U} vector fields)is a typical set of vector fields in a neighborhood of this point.

By Corollary 8.4, a a-passing point of a typical system is a point of class 11.According to Lemma 8.1, the field of limiting directions touches the boundary of thesteep domain with top first order of contact. This result and the condition j23f rh 000imply that in a neighborhood of the point its defining set is a typical pair of fields.

Finally, according to Lemma 8.1 and Corollary 7.6, a a-passing point can belongto one of the three classes 1, 3, and 11 in this corollary. Because of the conditionj2 f rh 000, the defining set of this point is a typical pair of fields.

§9. SINGULARITIES IN FAMILY OF LIMITING LINES IN STEEP DOMAIN 93

It is clear that a-passing points, double a-passing points, 0-turning points, andzero-passing points of a typical system are points of the same type for their definingsets.

Theorem 4.6 is proved completely.

9.2. Completion of the proof of Theorem 4.4. Assertion 4c of the theorem directlyfollows from Lemma 8.4 and the results of the analysis of the classes of points (Table 4).

By the second assertion of this theorem (proved during the investigation of classesin Section 7.4), to prove the first assertion of the theorem we only need to show thatfor a typical system the boundary of the steep domain belongs to the rest zone. Thethird assertion of the theorem (also proved in the investigation of classes) implies thatthis boundary consists of a-passing points, a-turning points, double a-passing points,zero-passing points, nonzero-passing points, and zero-points.

A point of any of the last three types belongs to the rest zone since, by definition,the zero velocity is feasible at such a point.

According to the first assertions of Theorems 3.3 and 4.6, a-passing, a-turning,and double 0-passing points of a typical system also belong to the rest zone.


9.3. Proof of Theorem 4.9. By Theorem 4.4 about the boundary of the steepdomain, for a typical system f each of its nonzero-passing point belongs to theboundary of the steep domain. By the same theorem and Table 4, this point is alsoa point of class 7 in Corollary 7.6. Whence, if u I and u° are limiting controls at apoint z and f (z, u') 0 = f (z, u°), then j3 f (z, u') E Y0 and j3 f (z, u°) E Y1'. ByLemma 6.6, the points (z, u°) and (z, u') are, respectively, a critical point of the typeof a Whitney fold and a regular point of the system folding. The second assertion ofthe theorem is proved.

The limiting directions at the point z form a straight line L(z, u°). According toLemma 8.3, this line does not lie in the tangent space to the boundary of the steepregion at that point. This proves the first assertion of the theorem.

In a neighborhood of the point z the field of limiting directions is continuous inthe closure of the steep domain and is locally Lipschitzian (by Theorem 4.12) in thesteep domain. Its values are noncollinear in the steep domain and form a straightline transversal to the boundary of the steep domain at the points on this boundary.Consequently, the third assertion of Theorem 4.9 is true.


9.4. Proof of Theorem 4.10. Let z be a zero-point of a typical control system fand let u(z) be the only limiting control at the point. According to Theorem 4.4 andTable 4, the point z belongs to one of the three classes 4, 9, and 13 in Corollary 7.6. Thisfact and Corollary 6.7 imply that j 3 f (z, U (Z)) E y,1. Consequently, by Lemma 6.6,(z, u(z)) is a critical point, which is a Whitney fold of the system folding. The firstassertion of the theorem is proved.

We select a system of local coordinates in a neighborhood of this point, with originat the point and fibered over M, such that in these coordinates the singular surfaceis determined by the system of equations x - (u1)2 = 0 = u2 = = u, if the pointlies in PS+2. Therefore, we can regard the coordinates u2, u3, ... , u., as being absent,and put ul = u. The germ of the field of limiting directions at the point z is lifted tothe germ at the point (z, u(z)) of a smooth direction field on the singular surface. Thelatter field is cut out on the singular surface by the field of planes f2,,, dx - f I ,,, dy = 0.


In a sufficiently small neighborhood of the point (z, u(z)) this field of planes is acontact structure because for a{system in general position we have j3f rh Y,, and,consequently, (f2,u f I,uu - ,/ I,u,/ 2,uu) (Z, a (z)) 54 0.

Hence, in a neighborhood of the point (z, u (z)) we obtain a pattern that exactlycorresponds to the pattern in a neighborhood of a point of the criminant of a typicalequation F(x, y, y') = 0 for which the discriminant curve is smooth. The singularitiesof this equation were studied in detail in Chapter 1. This implies the second assertionof Theorem 4.10.


9.5. Proof of Theorem 4.11. For a-passing points, 0-turning points, and zero-points the assertion of Theorem 4.11 is a direct consequence of Corollary 4.7, Theo-rem 3.3 (on the boundary of the steep domain of a tridynamical system), and Theo-rem 4.1. Hence, to complete the proof of Theorem 4.11 it suffices to find for a systemin general position which of the points of the following three types belong to the localtransitivity zone: the nonzero-passing points, the zero-points that are not black, andthe black zero-points. We first show that this zone contains each point of the first twotypes and then prove that no black zero-point belongs to the zone.

Let q be a nonzero-passing point or a zero point that is not black for a typicalsystem. Denote by u° the limiting control at the point q that determines the zerovelocity there. By Theorems 4.9 and 4.10, (q, u°) is a critical point of a Whitney foldfor the system folding. In a neighborhood of the point (q, 0°), we choose a systemof local coordinates with origin at this point and fibered over M so that the singularsurface is determined by the system of equations y - (u1 )2 = 0 = u2 = u3 = . . . =Us -2 (if (q, u°) E P.s) and the field of limiting directions f (x, y, +y'/2, 0, 0, ... , 0) isrepresented as (y, ±y 1/2)h (x, y), where h is a smooth function, h (0, 0) > 0. Accordingto Theorems 4.9 and 4.10, such a coordinate system exists. Therefore, we regard thecoordinates u2, u3, ... , Us-2 as being absent, and set ul = u.

Consider the velocity field v± = f (x, y, ±y2/3). We represent it in the form

f (x, y, +y2/3) = a (x, y) + y2/3b(x, y) + .Y4/3C(S, y),

where a, b, and c are smooth vector fields (generally, different for v+ and v_). Wehave a (x, 0) - 0 since f (x, 0, 0) = 0, b, (x, 0) = 0 since the field of limiting directionsis vertical on the axis of abscissas, and a I,,, (0, 0) > 0 and b2 (0, 0) > 0 since h (0, 0) > 0.Hence, on the axis of ordinates we have

f (0, y, +y2i3) = (at.v (0, 0) y +... + y2/3b2(0,0) + ... ),

where the dots symbolize the terms of higher degree with respect to y. It is easy tosee that at the points on this axis that are close to zero, the fields v± point in thedirection toward the left half plane if y < 0 and toward the right half plane if y > 0.Consequently, for a sufficiently small e > 0 the phase trajectories of the fields v+ andv_ extended across the axis of abscissas and passing through the point (0, e) intersectin the vicinity of zero in the lower half plane. Therefore, we obtain a cycle enclosinga neighborhood of zero (Figure 2.25). It is not difficult to show by direct calculationthat the time taken to move over the cycle with the chosen velocity fields is 0(63/8),and hence it tends to zero as e - 0. Using these fields we can reach the cycle from anypoint in this neighborhood or reach any point in the neighborhood from the cycle alsoin time 0(e3/8). It is clear that motions with the chosen velocity fields are feasible.

§9. SINGULARITIES IN FAMILY OF LIMITING LINES IN STEEP DOMAIN .95

FIGURE 2.25

(a)

FIGURE 2.26.

(b)

Consequently, for a typical system, the nonzero-passing points and the zero-pointsthat are not black belong to the local transitivity zone.

Now we assume that q is a black point of a typical system. According to Theo-rem 4.10, in a neighborhood of this point we can choose a system of local coordinateswith the origin at the point such that in this coordinate system the family of limitinglines of the system is the family of integral curves of equation (y'+a (x, y) )2 = yb (x, y),where a and b are smooth functions, b (0, 0) = 1, a (0, 0) = 0 54 ax (0, 0) 1/8. Tocomplete the proof it suffices to show that there exists T > 0 such that for any suffi-ciently small e > 0 one of the two points q or (0, e) is unattainable from the other intime less than T. We first consider the cases of a saddle and a node (0 54 ax (0, 0) < 1/8)and then the case of a focus (1/8 < ax (0, 0)).

For a saddle and a node we take the limiting lines passing through the point (0, e).The set A, (0, e) of points attainable from the point (0, e) in a time t lies in the regionabove these lines for either positive or negative t, I t I < T, where T is some value of timenot depending on E. Consequently, one of the two points q or (0, e) is unattainablefrom the other in a time less than T (Figure 2.26a for a saddle point with incomingseparatrixes above the outgoing ones and Figure 2.26b for an unstable node).

In the case of a focus, any two points in a neighborhood of q are attainable fromeach other, and therefore the proof of the desired assertion is more complicated. For


definiteness, consider the case when the cone of the point q contains the velocity (0, 1)(the subcase with velocity (0, - 1) reduces to it by the time inversion). Let a denotethe exponent of the folded singularity q. The next lemma will be useful.

LEMMA 9.1. For a control system f, f' (x, u) = (cos u, 1 + y -kx2 +sin u), wherek > 1/8 and u is the angle in a circle, -7t < u < it, none of the points of the parabolay = kx2 is attainable in a time not exceeding (2k)-' from any of the points on thepositive semiaxis of ordinates.

We use the lemma to complete the proof of Theorem 4.11. As was shown inChapter 1, the only invariant of a singular point of the folded focus type under C°°-diffeomorphisms is its exponent. For the system f 1 in Lemma 9.1, zero is a singularpoint of the field of limiting directions of the type of a folded focus with exponent(8k - 1)1/2. In a neighborhood of the point q we select a system of local coordinatesx, y with origin at q, in which the field of limiting directions of the system f coincideswith the field of limiting directions of the systen f 1 in Lemma 9.1 for k = (a2 + 1)/8.ForcERwesetf`=cf1.

LEMMA 9.2. There is a constant c > 0 such that in a neighborhood of zero everyfeasible velocity of the system f in the region y > kx2 is less than the maximum velocityof the system f ` in the same direction.

Lemmas 9.1 and 9.2 imply

COROLLARY 9.3. For a sufficiently small e > 0 the time needed to move from point(0, e) on the axis of ordinates to any point on the parabola y = kx2 along any feasibletrajectory of the system f is greater than a positive constant T not depending on e.

The assertion in Theorem 4.11 for focal zero-points now follows from Corol-lary 9.3. Indeed, assume that zero belongs to the local transitivity zone of a system f.By definition, any two points lying sufficiently close to zero are attainable from eachother in a small time. Take this time t > 0 and a number 6 > 0 so small that in movingfrom any point in the 6-neighborhood of zero along any feasible trajectory one cannotleave the neighborhood of zero where the field of limiting directions of the systemf coincides with that of the system f t . For a sufficiently small e > 0 the limitinglines issuing from the point (0, e) intersect the parabola y = kx2 at some points B±(Figure 2.27). Consider an arbitrary feasible motion starting from this point. If eis sufficiently small, then, by Corollary 9.3 and the choice of t, the trajectory of thismotion does not intersect the parabola y = kx2 in a time not exceeding both T and t,and, consequently, it does not attain zero (i.e., the point q). We thus get a contradic-tion, and, hence, for a typical system the focal black points do not belong to its localtransitivity zone.

To complete the proof of Theorem 4.11 it remains to prove Lemmas 9.1 and 9.2.

9.6. Proof of Lemma 9.1. We consider the system f' and take the parabolay = kx2 as an objective set for time-optimal motion from the region above theparabola. According to the maximum principle [PBMG], we introduce the function

7-1(x,y,u,y/I,y/2) = yrlcosu+yr2(1+y-kx2+sinu),

where (yr1, yr2) is an auxiliary vector. By the maximum principle, the following equa-tions are exact on the time-optimal extremal:(9.1).z = cos u, ,y = 1 + y - kx2 + sin u, y11 = -7I, = 2kxyr2, y12 = -7-1y = -yr2,

§9. SINGULARITIES IN FAMILY OF LIMITING LINES IN STEEP DOMAIN 97

= kxz

FIGURE 2.27

where the time-optimal control u(t) can be found from(9.2)

H(x(t), y(t), u(t), VI (t), W2(t)) = max{7-l(x(t), y(t), u, V/I (t), yr2(t))I u E U}.

According to the transversality condition, at the time of arrival at the parabola theauxiliary vector is perpendicular to the parabola and, by condition (9.2), it is directedtowards the region below the parabola. This vector is defined up to multiplication bya positive constant. For definiteness, assume that its value at the time of arrival at thepoint (xo, kxo) is (2kxo, -1) . Without loss of generality we can also assume that thetime of arrival at the parabola is zero because the system in question is autonomous.Thus, the relations

(9.3) x(0) = xo, y(0) = kxo, yiI (0) = 2kxo, X2(0) _ -1

can be taken as the conditions at the end of the time-optimal trajectory. Further-more, it suffices to consider the time-optimal extremal going from a point on thepositive semiaxis of ordinates to the right branch of the parabola because the invo-lution (x, y) -f (-x, y) preserves both the parabola and the indicatrix field of thesystem.

From condition (9.2) it follows that

-y/I (t) sin u(t) + V/2 (t) cos u(t) = 0, that is, cot u(t) = y/I (t)/y/2(t).

This relation and the last two equations in (9.1) imply (d/dt) cot u(t) = 2kx+cot u(t),whence, on the time-optimal extremal, we have

cot u(t) = (_f°2kx(t)exP(_t)dt+cotu(o)) exp(t).

It is clear that, on the extremal in question, x(t) is an increasing function of time. Thisfact and the last of the above relations imply

0

x(0) exp(-t)dt + cot u(0) exp(t) = -2kxocot u(t) > (-I 2k

because cot u (0) = -2kx (0) = -2kxo. Consequently, on this extremal the veloc-ity component along the x axis does not exceed cos u (0), i.e., the magnitude ofthis component at the time of arrival at the parabola. This magnitude is equal to2kxo/(1 + 4k2x02 ) and hence is less than 2kxo. Thus, the time of motion along thetime-optimal extremal exceeds xo/(2kxo) = (2k)-I, which proves Lemma 9.1.


9.7. Proof of Lemma 9.2. Assume the converse, namely that there is no such c.This means, in particular, that for sufficiently large natural values n of the constantthere is a point z belonging to the intersection of the region y > kx2 with the (1 In)-neighborhood of zero and a direction d belonging to the cone of that point such thatthe maximum velocity in the direction d at the point z for the system f is no lessthan the maximum velocity in this direction for the system f " . Select a convergentsubsequence from the sequence {d }. It can be assumed without loss of generality thatthe sequence itself converges. Denote by /1 the acute angle between its limit and theaxis of abscissas.

If /3 74 0, then for sufficiently large values of n the angle between the limitingdirection at the point z and the direction d is no less than /3/2. Consequently,at this point the maximum velocity in the direction d,, for the system f is greaterthan 2n sin(/3/2). However, we have 2n sin(/3/2) -p oo for n -p oo. Hence, thevelocity indicatrices of the system f in a neighborhood of zero are unbounded, whichcontradicts the continuity of the system and the compactness of U. Therefore the case/3 74 0 is impossible.

Let /3 = 0. In a neighborhood of zero, when approaching the parabola y = kx2from above, the magnitudes of the limiting velocities of the system f (f I) decrease as(y - kx2)1/2. Consequently, there is a constant r > 0 such that in a neighborhood ofzero the magnitude of the limiting velocities of the system f' at each point in the regiony > kx2 (these velocities have the same magnitude) is greater than the magnitudes ofthe limiting velocities of the system f. Furthermore, u° is the only limiting controlat the point q, and we have j 3 f (q, u°) E Y,2. This fact and the condition j 3 f rh Y1imply that near zero, in the region y > kx2, in the directions sufficiently close tothe limiting ones the curvature of the boundary of the convex hull of the velocityindicatrix at the points of maximum velocities (belonging to convexified indicatrices)in these directions exceeds a constant m > 0. This means that for c > max{1/m, r}the direction d,, cannot belong to such directions, and hence /3 74 0, and we arrive at acontradiction.

Thus, the assumption that there is no required constant c is wrong. Consequently,the assertion of Lemma 9.2 is true.

9.8 Proof of Theorem 4.15. The second assertion of the theorem follows directlyfrom Lemma 8.3. We shall prove the first assertion. We choose a system of localcoordinates x, y in a neighborhood of a point q belonging to the steep domain of atypical system with origin at q so that the i-limiting direction at this point coincideswith the positive direction of the axis x. As usual, the angles in the tangent planeswill be measured counterclockwise from this direction. The family of i-limiting linesin the vicinity of zero (i.e., in a neighborhood of the point q) coincides with the familyof phase trajectories of the system of equations z = 1, y = tan n (x, y), where n (x, y)is the angle in the tangent plane corresponding to the i-limiting direction at the point(x) y), n (0, 0) = 0. By Theorem 4.12, q is a point belonging to one of the following fivetypes: an i-regular point, an i-passing point, an i-turning point, a double i-passingpoint, or an i-cutoff point. We consider these five types separately.

An i-regular point. By Theorem 4.12, n is a smooth function in a neighborhoodof zero. However, a smooth vector field in the neighborhood of its nonsingular pointis smoothly rectifiable [A3]. Consequently, the germ at the point q of the family ofi-limiting lines is C°°-diffeomorphic to the germ at zero of the family of curves y = c.

An i-passing point. According to Theorem 4.12, the germ of the function n atthe

§10. TRANSVERSALITY OF MULTIPLE 3-JET EXTENSIONS 99

point q is R+-equivalent to the germ at zero of the function (-1)'Ixl. The line ofnon differentiability of the function n is transversal at zero to the axis of abscissasbecause q is not an i-turning point. Consequently, in the vicinity of zero the field ofi-limiting directions of the system under consideration coincides with that of a systemdetermined by a typical pair of smooth vector fields, for which zero is a passing point.This property and Theorem 2.2 imply that the germ at zero of the family of i-limitinglines of the system in question is C°°-diffeomorphic to the germ at zero of the familyof curves y - xlxl = c.

An i-turning point. By Theorem 4.12, the germ of the function n at the point qis R+-equivalent to the germ at zero of the function (-1)' Ix I. The line of nondif-ferentiability of the function n touches the axis of abscissas at zero because q is ani-turning point. By Lemma 8.3, the order of the contact is equal to one. Consequently,in a neighborhood of zero the field of i-limiting directions of the system under studycoincides with that of a system determined by a typical pair of smooth vector fields, forwhich zero is an i-turning point. This fact and Theorem 2.2 imply that the germ at thepoint q of the family of i-limiting lines of the system in question is C°°-diffeomorphicto the germ at zero of the family of integral curves of the equation y' = ix - y21.

A double i-passing point. In accordance with Theorem 4.12, the germ of thefunction n at the point q is R+-equivalent to the germ of the function (-1)'(x l +y + I x 1) at zero. By Lemma 8.3, the vector (1, tan n (0, 0)) does not lie in the tangentset at zero to the set of nondifferentiability of the function n. Consequently, in aneighborhood of zero the field of i-limiting directions of the system under considerationcoincides with that of a system determined by a typical triple of smooth vector fields,for which zero is a double passing point, whence, applying assertion 3 of Theorem 3.4,we conclude that the germ at the point q of the family of i-limiting lines is C°°-diffeomorphic to the germ at zero of the family of integral curves of equation y' =max{-x, x, Y(x, y)}, where Y is a smooth function, Y(0,0) = 0 Y(0, 0) (Y? (0, 0) -1).

An i-cutoff point. By Theorem 4.12, the germ of the function n at the point q isR+-equivalent to the germ at zero of the function (-1)' max{-W4 +yw2+xw lw E R}.The latter function is continuous. It is nondifferentiable only on the positive semiaxisof ordinates and is smooth outside its closure. According to Lemma 8.3, the vector(1, tan n (0, 0)) does not lie in the tangent set at zero to the set of non differentiability ofthe function n. Consequently, the germ at the point q of the family of i-limiting linesis C°°-diffeomorphic to the germ at zero of the family of integral curves of equationy' = Y(x, y), where Y is a continuous function that is nondifferentiable only on thepositive semiaxis of ordinates and is smooth outside its closure.

§10. Transversality of multiple 3-jet extensions

In this section the second basic lemma is proved.

10.1. Proof of Lemma 7.4. By the first basic lemma, the image of the 3 -jet extensionof a system f in general position does not interset X, and, by the definition of Xk, thefirst assertion of Lemma 7.4 follows.

The Whitney-stratified submanifolds Ck, Ok, Ok (r), COk (r), Ok (r, n), andCOk (r, n) are closed in Jk (P, R2) \ Xk. Consequently, to prove the second asser-tion of Lemma 7.4 it suffices to show that for a typical system f the mapping jk f istransversal to any of the stratified submanifolds in a neighborhood of the generalizeddiagonal A in j3 (P, R2). After that the assertion is an immediate consequence of the


first assertion of Lemma 7.4 and the multijet transversality theorem [GG]. As can beseen, the desired transversality in a neighborhood of the generalized diagonal followsfrom the lemmas below.

LEMMA 10.1. If the first basic lemma holds for the system f, then the mapping 2 f istransversal to the stratified submanifold 02 in a neighborhood of the generalized diagonalin J3(P,R2)2.

LEMMA 10.2. Let the assertion of the first basic lemma be true for a system f.Then, in a neighborhood of the generalized diagonal A in J3 (P, R2)k, the image of themapping jk f does not intersect (1) the set Ck; (2) the set Ok, k > 2, if the mappingj2 f is transversal to the stratified submanifold 02; (3) the set COk (r), 1 < r < k - 1,if the mapping j23f is transversal to the stratified submanifolds C2 and 02; and (4)the set Ok(r), 1 < r < k - 1 (Ok(r,n), 1 < r < n - 1 < k - 2; and COk(r,n),1 < r < n - 1 < k - 2) f the mappings j2 f and j3 f are transversal to the stratifiedsubmanifolds C2 and 02 (CO2 (1) and 03), respectively.

The second basic lemma is proved modulo Lemmas 10.1 and 10.2.

10.2. Proofs of Lemmas 10.1 and 10.2. The first lemma follows from Lemma 7.11.Indeed, if the image of the mapping j2 3f intersects the submanifold 02 in any neigh-borhood of the point on the generalized diagonal, then, by Lemma 7.11, wehave j 3 f (p) E Y2. The condition j 3 f rh Y2' implies inequality (7.1) . Conse-quently, the mapping (z, u) H (z, (z, u)) has a Whitney singularity 11,1°1 at the pointp. The Maxwell stratum in the "swallow tail" surface corresponds to the passing set.At the points of this stratum this surface has a transversal self-intersection, which,together with formula (7. 1) implies the transversality of the mapping j2 1f to the strat-ified submanifold 02 in a sufficiently small neighborhood of the point Hence,Lemma 10.1 is true.

Let us prove Lemma 10.2. Its first assertion is a direct consequence of Theorem 4.2.We prove the second assertion. If a point ( = (1, '2, ... , 'k) E A is a limiting point ofthe intersection of the image of the mapping jk f with 0k, k > 2, and n 54 m,then, by Lemma 7.11, we have ,, E Y. The condition j3 f r) Y2 implies that cj 54

for 1 < 1 < k, n 54 1 m. Consequently, j2 f (p", pl) E 02i! . However, Lemma 10.1has already been proved, and therefore we have j2 f rh 02 for a typical system f. Inparticular, the intersection of the image of the mapping j23f with the stratum 021"'/ isempty for any possible it and jr, which is a contradiction. Therefore there are no suchpoints 4, and the second assertion of Lemma 10.2 is true.

The third assertion of this lemma fork > 0 follows directly from the same assertionfor k = 3 and its first two assertions. We now prove the third assertion for k = 3, i.e.,for C03 (1). Let a point = (c1, 2, 3) E A be a limiting point of the intersectionof the image of the mapping j3 3f with the submanifold C03 (1). We have either (1)2 = c3 or (2) c1 and c2 3, where n is equal to 2 or 3.

For 2 = 3, by Lemma 7.11, we have 2 E Y2. The condition j3f rh Y2 impliesinequality (7.1). Consequently, the mapping (z, u) -* (z, 1,(z, u)) has a Whitneysingularity 11,1,1 at the point p. However, the Maxwell stratum and the edges of theregression in the "swallow tail" surface correspond, respectively, to the passing setand to the set of critical values, whence 154 2. Consequently, j2 f (p', p2) Ejj, > 0. But the first assertion of Lemma 10.2 has already been proved, and thereforewe have j2 f rh C2 for a typical system f. In particular, the intersection of the image

§10. TRANSVERSALITY OF MULTIPLE 3-JET EXTENSIONS 101

of the mapping j2 If with the stratum CSI I is empty for any possible i1 and jI. We thus12

arrive at a contradiction excluding this case.For I = ' 2 (the case I = 3 is considered in a similar way) the point p2(= p 1)

is a critical point of the system folding, which is either a Whitney cusp or a Whitneydouble cusp, because in an arbitrarily small neighborhood of this point a critical pointexists and a regular point of this folding whose images under the folding coincide.Consequently, we have 2 f (p2, p3) E 021 3 , i2 > 0. Lemma 10.1 has already beenproved, and therefore we have 2 f rh 02 for a typical system. In particular, theintersection of the image of the mapping j2 3f with the stratum 0' is empty for anypossible i2, 6, and j3, which is a contradiction excluding the case under consideration.Thus, the image of the mapping j3 f does not intersect C03 (1) in a neighborhood ofthe generalized diagonal. The third assertion of Lemma 10.2 is proved.

The fourth assertion of Lemma 7.13 for Ok (r, n) and COk (r, n) follows immedi-ately from the same assertion for Ok(r) and the third assertion of this lemma. ForOk (r), k > 4, the fourth assertion follows directly from the fourth assertion for 04(2).The latter assertion is proved by analogy with the third assertion of the lemma. Let

= 4) E 0 be a limiting point of the intersection of the image of themapping j43 f with 04(2). According to Theorem 4.2, when approaching the diagonalover the set 04(2), no more than three jets can merge. It follows that, generally, oneof the four cases below is possible (probably after the jets are reindexed so that thestructure of this set is preserved): (1) c1 = 2 54 3 = 4, (2) 1 = 2 = 3 54 4, (3)4 54 1 = 2 3 54 4 and (4) 4 54 I 2 = 3 4 We consider these four casesone by one.

According to Lemma 7.11, in the first case we have , E Y2 for i = 1 and i = 3.Consequently, j23f (pl, p2) E C22. However, the first assertion of Lemma 10.2 hasalready been proved, hence we have j23f rh C2 for a typical system f. In particular,the intersection of the image of the mapping j23f with the stratum C22 is empty. Thisis a contradiction, and, consequently, the case under consideration is impossible.

According to Lemma 7.11, in the second case we have I E Y2. Consequently,j2 3f (p3, p4) E O. But Lemma 10.1 has already been proved, so that j2 f rh 02 fora typical system f. In particular, the intersection of the image of the mapping 2 fwith the stratum 02 J4 is empty for any possible i4 and j4, and we get a contradictionexcluding this case.

In the third case, according to Lemma 7.11, we have ci E Y. Consequently,j 3 f (p 1, p3, p4) E C02 J3'j4 (1). However, the third assertion of Lemma 10.2 has already

been proved, and therefore we have j3 f rh CO2 (1) for a typical system f. In particular,the intersection of the image of the mapping j3 f with C02 14 (1) is empty for anypossible 6, i4 and j3, j4. The resulting contradiction excludes this case as well.

Finally, in the fourth case we have 2 E Yj' for j2 > 0 because there are pointson the singular surface lying arbitrarily close to the point p2 such that their imagesunder the system folding are identical. Consequently, j3 f (p 1, p2, p4) E O J1J2J4 for

j2 > 0. However, the second assertion of Lemma 10.2 has already been proved, sothat j23f rh 03 for a typical system f. In particular, the intersection of the image ofthe mapping 2 f with Oi1 J2 J4, j2 > 0, is empty for any possible il, i2, i4 and j1, j2, j4The contradiction thus obtained excludes this case. Assertion (4) of Lemma 10.2 isproved.

The proof of Lemma 10.2 is completed.

CHAPTER 3

Structural Stability of Control Systems

The primary aim in this chapter is to show that typical control systems on compactorientable surfaces possess the same structural stability as typical smooth vector fieldsprovided that the trajectory of a point for a control system is defined as the union of thepositive and negative orbit of the point. The main results of this chapter are formulatedin §1, and in §2 illustrative examples are presented. The subsequent sections of thechapter are devoted to the proofs of these results.

§1. Definitions and theorems

We first recall the definitions of certain concepts used in the previous chapters andthen state the main results of this chapter.

1.1. The class of systems. In this chapter we shall continue to study smooth controlsystems on a smooth surface M without boundary, endowed with a Riemannian metric.It is also assumed that the surface is compact and orientable. The set U of the valuesof the control parameter is the disjoint union of a finite number of compact smoothmanifolds containing at least two distinct points. By definition, a control system isdetermined by a smooth mapping F of a bundle space P over M with fibre U into thetangent bundle space TM such that the diagram

P TM

Mis commutative, where r is the bundle projection and it is the canonical projection. Thespace of control systems is identified- with the set of these mappings and is endowedwith the C4-topology of Whitney. By a typical control system or a control system ingeneral position we mean a system belonging to an open everywhere dense set in thespace of the systems in this topology.

A steep domain of a control system is the set of all points in the phase space forwhich the positive linear hull of the set of feasible velocities does not contain the zerovelocity. The sides of the hull at a point belonging to the closure of the steep domainare called the limiting directions at this point. The integral curves of the field of limitingdirections are called limiting lines. As was shown in the previous chapter, the field oflimiting directions and the boundary of the steep domain of a control system in generalposition can have only typical singularities stable with respect to small perturbationsof the system. For example, by Theorem 4.15 in Chapter 2, for a typical system thefamily of limiting lines of each of the two branches of the field of limiting directions ina neighborhood of every point belonging to the steep domain coincides in a suitable

103

104 3. STRUCTURAL STABILITY OF CONTROL SYSTEMS

system of local coordinates x, y with origin at that point, with the family of integralcurves of one of the following five differential equations (everywhere, unless otherwisestated, x, y are coordinates in the plane and zero is the origin): y' = 0 (a regularpoint of the branch); 2y' = x sgn x (a zero-passing point of the branch); y' = l y - X21(a turning point of the branch); y' = max{-x, x, Y(x, y)}, where Y is a smoothfunction, Y (0, 0) = 0 Y), (0, 0) (Y? (0, 0) - 1) (a double passing point of the branch);and y' = Y(x, y), where the function Y is continuous, continuously differentiableoutside the positive semiaxis of ordinates, and, by Theorem 4.12, differentiable withrespect to y, Y(0, 0) = 0 (a cutoff point of the branch). According to Theorem 4.8 inthe previous chapter, if the number of different values of the control parameter is greaterthan three, then the family of limiting lines of a typical system in a neighborhood ofeach of the boundary points of its steep domain coincides in some suitable continuoussystem of local coordinates x, y with origin at this point either with the family ofintegral curves of one of the following four differential equations: (y')2 = x (a familyof semicubical parabolas); y = (y' + kx)2 for k = -1 (a folded saddle), k = 0.1 (afolded node), and k = 1 (a folded focus), or with the family of limiting lines determinedby one of the following two triples of vector fields: (-x, -2y), (1, 1), (2, 3) (a foldedsaddle-node) and (x, -2y), (1, 1) (2, 3) (a folded monkey saddle).

1.2. Stability of orbits. A control system is said to be structurally stable if thepositive and negative orbits of the points of any system sufficiently close to it aretransformed into the orbits of the points of the original system under a nearly identicalhomeomorphism (i.e., a one-to-one mapping continuous together with its inverse) ofthe phase space. This is an analog of the notion of a structurally stable vector fieldintroduced by Andronov and Pontryagin [A2].

THEOREM 1.1. A control system in general position is structurally stable.

In relation to orbital equivalence, a control system in general position behaves likea differential equation in general position [PM]. This theorem is proved in Section 6.5.

A subset in phase space is said to be stable if for any e > 0 there is b > 0 such thatfor to < t < oo every trajectory of the system with initial point z(to) belonging to theb-neighborhood of this subset exists and belongs to the e-neighborhood of the subset.If, in addition, the distance from the point z(t) to this subset tends to zero as t -p 00,then the subset is said to be asymptotically stable [Fl].

THEOREM 1.2. For a control system in general position, the positive orbit of a pointis asymptotically stable provided that this point belongs to the interior of the orbit.

REMARK. Under a change of the direction of motion the negative orbit of a pointbecomes the positive orbit and vice versa. Therefore, Theorem 1.2 applies to negativeorbits as well, which, however, are asymptotically stable for t -00. When appliedto negative orbits, Theorem 1.2 will be referred to as Theorem 1.2'.

Theorem 1.2 is proved in Sections 6.1-6.4.

1.3. Conditions for structural stability. It turns out that the singular limiting linesof a system are important in the determination of the boundaries of the orbits of points.A limiting line of a system in general position (whose field of limiting directions canhave only typical singularities described in the previous chapter) is said to be singularif it is either a closed curve (a cycle) lying entirely in the steep domain or a separatrixof (at least one) folded singular point of the field of limiting directions. A systemin general position has separatrices at folded saddles, nodes, monkey saddles, and

§1. DEFINITIONS AND THEOREMS

FIGURE 3.1

(c)

105

(d)

saddle-nodes. Figures 3.1 a-d illustrate (up to within a homeomorphism) these fourtypes of folded singularities. The dashed and solid lines represent the integral curvesof the two branches of the field of limiting directions; the separatrices are shown inheavy lines and the boundary of the steep domain is the double line. As is seen, atthese points 4, 2, 6, and 4 separatrices respectively are "born".

The family of limiting lines of a branch of the field of limiting directions of acontrol system is said to be structurally stable if the corresponding family of limitinglines of any sufficiently close system can be transformed into the original family undera nearly identical homeomorphism of the phase space. In this case the branch itself isalso said to be structurally stable. The structural stability of the set of singular limitinglines of a system in general position is defined in a similar way. A cycle of a branchof the field of limiting directions is said to be simple if the derivative of the Poincaremapping at a point of the cycle is different from one.

THEOREM 1.3. For #U > 3 the structural stability of a branch of the field of limitingdirections of a control system in general position is equivalent to the following threeconditions:

(A) there are no double separatrices, i.e., the limiting lines of this branch that areseparatrices of folded singular points for both increasing and decreasing time;

(B) each closed limiting line of the branch belonging to the steep domain is a simplecycle;

(C) when extended in each of the directions, any limiting line of this branch distinctfrom a cycle either ends at a point belonging to the boundary of the steep domain orwounds around the cycle of this branch.

We note that conditions (A)-(C) of this theorem are analogs of the correspondingconditions for structural stability of a smooth vector field on a compact orientablesurface [Pel].

THEOREM 1.4. For a control system in general position the structural stability of thisset of singular limiting lines is equivalent to the following four conditions: conditions(A)-(C) of Theorem 1.3 for each of the two branches of the field of limiting directionsand condition:

(D) for each point on the boundary of the steep domain that is not a folded singularpoint, there is at most one singular limiting line that can reach this point.

In Theorems 1.3 and 1.4, by a system in general position we mean a system thesuch that singularities of its family of limiting lines are typical and stable with respectto small perturbations. These theorems are proved in Section 3.2 and Sections 4.1and 4.2, respectively.

THEOREM 1.5. For a control system in general position the following two assertionshold:


(1) the set of singular lines and, for # U > 3, each of the two branches of the familyof limiting directions are structurally stable;

(2) one of the singular limiting lines contains cutoff points, double passing points,and turning points of any of the two branches of the field of limiting directions, and everyintersection point of these lines is a regular point of each of the branches.


1.4. Nonlocal controllability. The nonlocal transitivity zone (NTZ) of a controlsystem is an open region in the phase space of the system such that the intersection ofthe positive and negative orbits of any point belonging to this region coincides withthe whole region. As can easily be seen, (1) any two points belonging to this zoneare attainable from each other and (2) the zone either does not intersect the positive(negative) orbit of a point or entirely belongs to the orbit. The nonlocal transitivityzones of typical bidynamical systems were studied in [B1].

A point on the boundary of a nonlocal transitivity zone will be called a singularpoint of type p, 1 JxJ or y < JxJ for p = 1; y > JxJxp-1 for p = 2 or 3;eax" <y<x,where e=+1,for p=4;y<(e+c(sgnx-1))Ixl",wheree=+1,e 54 2c E R, for p = 5; y < h(x) for p = 6, where the graph of the function y = h(x)coincides near zero with the closure of the union of two nonsingular phase trajectoriesof a folded node y = (y' + a(a + 1)-2x/2)2 that enter zero from opposite directions;here and above a > 1 is not an integer. A singular phase trajectory of a (folded) nodeis a phase trajectory that reaches the singular point or leaves it and is extendable acrossthe point as a smoothly embedded curve; for a nonintegral exponent of the node thereare exactly four such phase trajectories (recall that the exponent of a node is the ratio ofthe eigenvalue with the greatest modulus of the linearization of the field at this singularpoint to the eigenvalue with the smallest modulus, and the exponent of a folded node isequal to that of the node in the preimage).

THEOREM 1.6. For a control system in general position the following three assertionshold:

(1) the boundary of any of its nonlocal transitivity zones is either empty or is asmoothly embedded curve with singular points of type p, 1 < p < 6;

(2) the closures of any two different nonlocal transitivity zones are disjoint;(3) for any system sufficiently close to the former; the nonlocal transitivity zones of

the two systems are transformed into each other under a nearly identical homeomorphismof the phase space. The transforming homeomorphism can be chosen so that it is aC°°-diffeomorphism everywhere except possibly at singular points of the boundaries ofthe zones of the fourth, fifth, and sixth types.

Thus, for any typical control system, not only the nonlocal transitivity zones butalso the singularities of their boundaries are stable with respect to small perturbations.Theorem 1.6 is proved in Sections 7.1-7.4. In its proof we incidentally show thatfor any starting set the attainability boundary of a typical system is also a smoothlyembedded curve with singular points of type p, 1 < p < 6, provided that the closure ofthis set belongs to the interior of its positive orbit. We also note that if this condition is.dropped, the list of typical singularities of the attainability boundary is the same if thestarting set is a smooth compact submanifold in phase space or even the image under atypical mapping of the disjoint union of a finite number of compact smooth manifolds.

§1. DEFINITIONS AND THEOREMS 107

In this case at a point of the starting set itself this boundary can have singular pointsof the first two types only.

1.5. Orbit boundary. Fix a continuous reference direction on phase space formeasuring angles in the tangent planes. For a point z belonging to the closure ofthe steep domain we denote by L1 (z) and L2(z) the minimal and maximal limitingdirections at this point, respectively; let pr (z) (r7; (z) and r7; (z)) be the limiting lineof the branch L; of the field of limiting directions taken together with the point z andpassing through the point z (accordingly, starting at the point z and entering the pointz). Denote by 0+ (S) and O- (S) the positive and negative orbits of a subset S of thephase space, respectively.

As is known, for a typical system the closure of the positive (negative) orbit of anysubset in the phase space coincides with the closure of the interior of this orbit and is amanifold with boundary. This boundary is a C°-hypersurface in the phase space [D5,D6]. This will be proved in the next chapter for a broader class of control systems thanthat considered here. The following two theorems provide a more precise descriptionof the structure of the boundary of the positive orbit.

THEOREM 1.7. If for a control system in general position a closed set S belongs tothe interior of its positive orbit, then the following five assertions hold:

(1) the boundary of S belongs to the union of the steep domain and the set of foldedsingular points of one of the following four types: a saddle point, a stable node, a monkeysaddle, and a stable saddle-node;

(2) the germ of this boundary at any point z lying in the steep domain coincides withone of the following three germs:

(a) (rll (z), z);(b) (q2 W, z); and(c) (ril (z) U rig (z), z); and, if (r7; (z), z) = (aO+(S), z), then(3) the limiting line ri, (z) is not an incoming separatrix of any of the folded singular

points, and one of the following two assertions holds:(4) the limiting line q+(z) is a cycle of the field of limiting directions that belongs to

aO+(S), and therefore ri; (z), z) = (8O+ (z), z) for any point z E ri; (z); or(5) the limiting line rlr (z) is an outgoing separatrix of a folded singular point of one

of the following three types: a saddle point, a monkey saddle, or a stable saddle-node.

THEOREM 1.8. If for a control system in general position a closed set S belongs tothe interior of its positive orbit, then for every point z belonging to the intersection of theboundary of this orbit with the steep domain (the boundary of the local transitivity zone)the following two assertions hold:

(1) the germ of this orbit at a point z contains the germ of the local transitivity zoneat this point;

(2) the germ at a point z of the boundary of this orbit coincides with one of thefollowing five germs:

(a) (q i (z) U rig (z), z) if z is a folded saddle; in this case, locally in a neighborhoodo f the point z, the outgoing separatrices r l i (z) and i (z) separate the incomingseparatrices qi (z) and r72 (z) from the local transitivity zone;

(b) (r7i (z) U r72 (z), z) if z is a folded stable node; in this case the lines rj (z) andri2 (z) are not separatrices of that node;

(c) (r7i (z) U rig (z), z) if z is a folded monkey saddle; in this case the lines q+(z) andrli (z) are outgoing separatrices of both this singular point and the singular point


z (a saddle point) of the velocity field determined by the corresponding isolatedvalue of the control parameter;

(d) either (rji (z) Ur72 (z), z) or (r7; (z) Ur7; (z), z) if z is a folded stable saddle-node;in this case only the line r7; (z) is a separatrix of this saddle-node.

Some explanations of assertion (1) in Theorem 1.7 should be given. If the numberof different values of the control parameters equals to two, the boundary of the orbitO+ (S) for a typical system may also contain folded foci. In this case the boundary ofthe steep domain is a line, and the appearance of folded saddles and foci on it occurssimultaneously (on different sides of this line) at the a-turning points (see Theorems 3.1and 4.1 in Chapter 2). The stability of a folded saddle-node depends on the stability ofthe corresponding node of the smooth velocity field determined by the isolated valuesof the control parameter or on the number of outgoing separatrices of this foldedsingular point: if there is only one such separatrix, then the saddle-node is stable, andif there are three separatrices, then the saddle-node is unstable. A point z belongingto the intersection of the boundary of the orbit O+ (z) with the steep domain will becalled a point of confluence if the germ of this boundary at the point coincides withgerm (c) in the second assertion of Theorem 1.7. Here and below, unless otherwisestated, the indices i and j take the values 1 and 2, and if they occur simultaneously,then i j. Theorems 1.7 and 1.8 are proved in Sections 5.1 and 5.2, respectively.

REMARK. Under a change of the direction of motion the positive orbits becomenegative and vice versa. Therefore, Theorems 1.7 and 1.8 apply to the negative orbitsas well if in their statements the superscripts + and - of the limiting lines are replacedby - and +, respectively, and the words "incoming", "outgoing", and "stable" arereplaced by "outgoing", "incoming", and "unstable", respectively. In relation tonegative orbits, Theorems 1.7 and 1.8 will be referred to as Theorems 1.7' and 1.8',respectively.

Theorems 1.7, 1.7', 1.8, and 1.8' and condition (A) in Theorem 1.3 imply

THEOREM 1.9. For a control system in general position and any of its nonlocaltransitivity zones z the following three assertions hold:

(1) the germ of this zone at each point z on its boundary coincides with the germ atthis point of the orbit 0+ (Z)(0- (Z)) if the point belongs to the orbit O- (Z)(O+(Z));

(2) if z E a0+(Z) n a0-(Z), then(a) either z is a point lying in the steep domain, and then the germ at this point of the

boundary of the zone coincides with the germ (r7; (z) U qt (z), z); in this case wehave (00+(Z), z) = (r7; (z), z) and (a0- (z), z) = (17j (z), z); or

(b) z is a folded monkey saddle, and the germ at this point of the boundary of thezone coincides with the germ (r7; (z) U r7; (z), z);

(3) the germ of this zone at each of the points of intersection of its boundary withthe boundary of the steep domain contains the germ at this point of the local transitivityzone.

A point z E a0+(Z) n a0-(Z) will be called an angular point.This theorem will be proved in Section 5.3.

1.6. Remark. For the sake of brevity, the proofs of the stated results are given forthe case # U > 3 when the closure of the steep domain of a typical system is a manifoldwith boundary (by Theorem 4.4 in the previous chapter). Results concerning structural

§2. EXAMPLES 109

FIGURE 3.2

stability of a typical system and its nonlocal transitivity zones were announced in [D8,D9], and presented in details in [D4].

§2. Examples

In this section the role of singular limiting lines in the construction of the bound-aries of nonlocal transitivity zones is illustrated, realization of some of the singularitiesof the boundaries of these zones is presented, and an example of a structurally stablesystem is given.

2.1. Swimmer drift. A swimmer in a planar sea RX,Y is carried by a smooth watercurrent with velocity field v(x, y), v(x, y) = ((x2 + y2)1/2, 0) for x2 + y2 > 1/4 andlv(x, y)A < 2/3 for x2 + y2 < 1/4. In standing water the swimmer can swim in anydirection with unit velocity. The swimmer's motion is described by the control system(x, y) = v (x, y) + (cos u, sin u), where u is the circular angle. The steep domain of thesystem is determined by the inequality x2 +y2 > 1, and its complement coincides withthe closure of the local transitivity zone. Only two points of this closure do not belongto the zone itself, namely (- 1, 0) and (1, 0). At these points the field of the limitingdirections of the system has folded saddles. The boundary of the nonlocal transitivityzone consists of these points and of parts of separatrices of the folded saddles startingat the point (-1, 0) and entering the point (1, 0) (see Figure 3.2; here the thick lineis the boundary of the nonlocal transitivity zone, the double line is the boundary ofthe local transitivity zone, and the local transitivity zone itself is shaded). On thisboundary there are two angular points to which two singular points of type 1 of theboundary of the nonlocal transitivity zone correspond.

2.2. Ship drift. In a planar sea IR2X,1 a smooth water current with velocity field(-x -1, -2.1y -2.1) and a near-surface wind with velocity field (-x + 1, -2.1 x +2.1)drift to the points (-x, -1) and (1, 1), respectively. An inertialess ship suffers loss ofspeed and control and is alternately driven by the current and the wind. Here we aredealing with a bidynamical system. Its local transitivity zone is the interval joining thesingular points of the wind and water fields, and the steep domain is the complementof the closure of this interval. The field of limiting directions has folded saddle-nodesat these singular points. The boundary of the nonlocal transitivity zone consists of


FIGURE 3.3

parts of the separatrices issuing from these singular points, and (1, 1) and (-1, -1) aretype 4 singular points of this boundary (Figure 3.3).

If, under the same wind, the water current has a "wirlpool" with the velocity field(y - 2x, -3x), then the boundary of the nonlocal transitivity zone is the separatrix ofa folded saddle-node issuing from the singular point of the wind field. This separatrixencircles the singular point of the water field and again enters the singular point of thewind field.

If the wind also has a "wirlpool" with the velocity field (y - 2x + 1, -3(x - 1)),the boundary of the nonlocal transitivity zone is a limit cycle of one of the branches ofthe limiting direction field of the system that encircles the singular points of the windand water fields. The intersection points of the cycle with the straight line y = x aresingular points of type 2 of this boundary.

It can be easily seen that these realizations of the singularities of the boundaryof the nonlocal transitivity zone are stable with respect to small perturbations of thesystem.

REMARKS. 1. A type 5 singular point of the boundary of a nonlocal transitivityzone is obtained in the case in Figure 3.3 if one more velocity field (-2(x + 2), -3(y -2)) is added. A type 3 singular point is realized at zero on the boundary of the nonlocaltransitivity zone of the bidynamical system determined by the vector fields (1, x2 - y)and (-1, x2 - y) (cf. Example 8 in Section 1.3 of the previous chapter).

2. A type 6 singular point occurs on the boundary of the nonlocal transitivity zoneor in the system describing the swimmer's drift by a water current with velocity field(-x, -fly) or in an arbitrarily small suitable perturbation of this system (here /3 > 2 isnot an integer). This was the case in the example of the swimmer's drift in Section 1.3of Chapter 1.

2.3. A structurally stable system. An object on a sphere admits motions withtwo structurally stable vector fields. One of the velocity fields has a nondegeneratesaddle-node at the south pole and a nondegenerate unstable node at the north pole.The remaining phase trajectories of this field go from the north pole to the south poleand coincide with the meridians outside sufficiently small neighborhoods of the poles.

The other structurally stable velocity field has two simple cycles that are parallelsand are located somewhere north and south of the equator. The northern cycle is

§3. A BRANCH OF THE FIELD OF LIMITING DIRECTIONS 111

FIGURE 3.4

unstable, and a phase trajectory unwinding from it either winds around the southerncycle or enters a nondegenerate stable focus near the north pole.

The southern cycle is stable, and a phase trajectory winding around it eitherunwinds from the northern cycle or starts at a nondegenerate unstable focus near thesouth pole (Figure 3.4).

This control system has two nonlocal transitivity zones, the northern and southernones. They lie above and below the northern and southern cycles, respectively. Thenorthern and southern nonlocal transitivity zones are separated by an equatorialannulus where the phase trajectories of one of the velocity fields are meridians whereasthe phase trajectories of the other field unwind from the northern cycle of this fieldand wind around its southern cycle.

We now discuss how the orbits of the points of the sphere look like. For pointsin the northern (southern) nonlocal transitivity zone, the negative (positive) orbitcoincides with that zone whereas the positive (negative) orbit coincides with the entiresphere. For points of the boundary of this zone, one of the orbits coincides with theclosure of the zone and the other is the closure of the complement of the zone.

Consider a point belonging to the equatorial annulus. Let us draw phase trajec-tories of the fields of feasible velocities through this point and take two sections of thepositive and negative semitrajectories for each of them that lie between the point inquestion and the two nearest (along the semitrajectories) points of intersection of thesetrajectories. The positive (negative) orbit of the point under consideration lies below(above) the union of two of these four sections belonging to the positive (negative)semitrajectories.

It is clear that the structure of the point orbits for the control system in this exampleremains the same for any other system sufficiently close to it in the C I -topology and,even more so, in the C4-topology of Whitney. Under a homeomorphism that is C°-close to the identity, the orbits of the points in one of these systems are transformedinto the orbits of the points in the other system. Thus, the control system under studyis structurally stable.

§3. A branch of the field of limiting directions

In this section Theorem 1.3 is proved. Here we consider one of the two branchesof the field of limiting directions and the corresponding family of limiting lines of acontrol system in general position.


3.1. Differentiability of the family of limiting lines. A rectifying chart of class Ckfor a vector field satisfying the Lipschitz condition is defined as the pair (W, g), whereW is an open set in phase space and g is a Ck-diffeomorphism of this set onto thesquare I xI < 1, 1 y I < 1 under which the trajectories of the field W go into the straightlines y = c E (-1, 1) (see [PM]).

LEMMA 3.1. For any point belonging to the steep domain of a control system ingeneral position and each of the two branches of the field of limiting directions there is arectifying chart of class C 1 containing this point.

LEMMA 3.2. For a control system in general position and a compact nonclosed arc yof a limiting line there is a rectifying chart (W, g) of class C' such that y E W.

LEMMA 3.3. For a control system in general position and any cycle of its field oflimiting directions the Poincare mapping of this cycle is continuously differentiable.

Lemma 3.3 follows from Lemma 3.2, which, in turn, follows from Lemma 3.1.We shall show that the assertion of Lemma 3.1 holds. By Theorem 4.15 from theprevious chapter, for a system in general position the germ of the family of limitinglines for a branch of the limiting directions field at each of the points of the steepdomain is C°°-diffeomorphic to the germ at zero of the family of integral curvesfor one of the following five differential equations: (1) y' = 0; (2) y' = x sgn x;(3) y' _ y - x21; (4) y' = max{-x, x, Y(x, y)), where Y is a smooth function,Y(0, 0) = 0 Yy (0, 0) (Y? (0, 0) - 1); and (5) y' = Y(x, y), where the function Yis continuous,continuously differentiable with respect to y, and differentiable outsidethe positive semiaxis of ordinates. For each of these five cases we consider a firstintegral I of the equation in a neighborhood of zero such that I (0, y) = y. It is easyto show that the integral is a continuously differentiable function. Consequently, in aneighborhood of zero the functions x and I can be taken as rectifying coordinates ofclass C I. Lemma 3.1 is proved.

REMARK. For a cycle of a branch of the field of limiting directions the derivativeof a Poincare mapping h at a point z of the cycle is calculated by the known formula

h' div V (t)dtJ

(z) = exp(fT

where V is the field of singular velocities corresponding to this branch and z is theperiod of the cycle, i.e., the time it takes to move around the cycle with this velocityfield. The cycle is unstable for h'(z) > 1 and stable for h'(z) < 1. A cycle is said to besimple if h' (z) 54 1.

3.2. Proof of Theorem 1.3. The necessity of the first two conditions is obvious.When the third condition does not hold, it is possible to obtain, by an infinitesimalperturbation, another closed limiting line or another double separatrix (the requiredperturbation is constructed in a standard manner [PM] outside the typical singularitiesof the field of limiting directions). Consequently, the third condition is also necessary.

We now show that (A), (B), and (C) are sufficient conditions for the structuralstability of a branch of the field of limiting directions of a typical system F. Indeed,by Theorem 4.16 in the previous chapter, for such a system the sinks and sources of itslimiting lines located on the boundary of the steep domain are stable with respect ofsmall perturbations in the system. Furthermore,

§4. THE SET OF SINGULAR LIMITING LINES 113

(1) Outside an arbitrarily narrow neighborhood of this boundary the field of limit-ing directions satisfies the Lipschitz condition with a constant depending continuouslyon the system. This follows from Lemma 8.1, the first and second basic lemmas, andCorollary 7.6 in Chapter 2.

(2) The singularities of a branch of the field of limiting directions depend C'-continuously on the system. This follows from Lemma 8.3 and Remark 2 in Section 8.4of Chapter 2.

(3) Outside an arbitrarily small neighborhood of these singularities the branchitself depends C'-continuously on this system.

These remarks and condition (B) imply the stability of the cycles of the branch ofthe limiting direction field, i.e., the stability of the sinks and the sources of the limitinglines of the branch lying inside the steep domain, under small perturbations of a typicalsystem. Hence, it can be assumed that the system F and any system .P sufficiently closeto it have the same sinks and sources of the limiting lines of the considered branchof the field of limiting directions. Using the theorem on the continuous dependenceof a solution of a differential equation on the initial data and the right-hand sideof the equation [P2] and taking into account that folded singularities of a system ingeneral position are stable with respect to small perturbations of the system, we seethat the stabilization of each separatrix of a branch of the limiting direction field of thesystem F occurs at the same sink or source as the stabilization of the correspondingseparatrix of the system F. Thus, it can be assumed that the system F and any systemP sufficiently close to it have the same separatrices of the investigated branch of thefield of limiting directions.

Finally, the proof of Theorem 1.3 is completed by partitioning the closure ofthe steep domain (which is a manifold with boundary) into canonical regions andconstructing a nearly identical transforming homeomorphism. These procedures arecarried out by analogy with [Pel] and [Pe2]; a detailed description is omitted here.


§4. The set of singular limiting lines

Here we first prove Theorem 1.4 (Sections 4.1 and 4.2) and then Theorem 1.5(Section 4.3).

4.1. The necessity of conditions (A)-(D). Conditions (A)-(C) are necessary for thestructural stability of the set of singular limiting lines for each of the two branches ofthe field of limiting directions of a control system in general position. Condition (D)is necessary because whenever two singular limiting lines approach a point on theboundary of the steep domain which is not a folded singular point, this phenomenonis destroyed under an arbitrarily small perturbation of the system in an arbitrarily smallneighborhood of the point. Figure 3.5 demonstrates two versions of this destruction.Here the singular limiting lines are shown by thick lines; outside the encircled regionsthe system does not change.

Thus, conditions (A)-(D) are necessary for the structural stability of the set ofsingular limiting lines of a system in general position.

4.2. The sufficiency of conditions (A)-(D). According to Theorem 1.3, conditions(A)-(C) for a system F in general position imply the structural stability of each ofthe two branches of the field of limiting directions. Denote by 111(F) and 112(F) thecorresponding two families of limiting lines. In view of the stability of the families


FIGURE 3.5

III (F) anL. 1-12 (F) it can be assumed that a system P sufficiently close to F has a familyof limiting lines III (P) coinciding with III (F), and the family H2 (P) goes into II2(F)under a nearly identical homeomorphism of the phase space. Let us show that undercondition (D) there is a nearly identical homeomorphism of the phase space preservingthe location of the singular lines of the families III (F) = III (F) and transforming thesingular limiting lines of the family II2(F) into the singular limiting lines of the familyII2(F). This homeomorphism is constructed in three stages, namely (I) in a smallneighborhood of each of the points of attainment of the the steep domain boundaryby the singular limiting lines of the family 112 (F), (II) in a small neighborhood of eachof the limit cycles of the family II2(F), and (III) in the remaining part of the phasespace.

Stage I. The point 0, where a singular limiting line of the family II2(F) attainsthe boundary of the steep domain, can be either (1) a regular point, at which the germof the family of limiting lines is homeomorphic to the germ at zero of the family ofsemicubical parabolas (y + c)2 = x3, or a folded singular point of one of the followingfour types: (2) a saddle point, (3) a monkey saddle, (4) a node, or (5) a saddle-node.We consider these five cases one by one. The arguments are local in a neighborhoodof the point 0, and e > 0 is an arbitrarily small number.

(1) We choose a system of coordinates x, y in a neighborhood of the point 0with origin there in such a way that (a) the steep domain coincides with the half planey > 0, (b) the family II1(F) is a family of vertical lines in this region, and (c) thesingular limiting line of the family II2(F) entering the point 0 is the bisector of thefirst quadrant.

For the system P we have H1(F) = fIl (F). The corresponding singular limitingline of the family 112(F) is the graph of a continuous function y = cp(x), x > xo(F),cp(xo(F)) = 0,cp(x) > Oforx > xo(F),xo(F) -40 forF -* F. We seta = Ixo(F)1+e,N = max{max{c (x) I xo(F) < x < 3a + e}, 3a + e}. Let yr be a continuousfunction on IR that is equal to 1 on the closed interval [-1, 1], to zero outside theinterval [-2, 2], and is linear on the intervals [-2, -1] and [1, 2]. The homeomorphismhI : (x, y) H (x -xo(F)yr((x -xo(F))/a) yr(y/N), y) is identical outside the rectangle0 = {(x, y) I I x 1 < 3a, Iy1 < 2N}. It transforms the graph of the function y = cp(x)into the graph of a continuous function y = co (x), x > 0, cp1(0) = 0, cpI (x) > 0 forx > 0. Consider a homeomorphism h2 preserving the axis of abscissas and the verticallines, identical outside the rectangle A i = { (x, y) I I x I < 3a + e, 1 y I < 2N }, and definedin the rectangle 0 itself in the following way. For a fixed x E [0, 3a] it is linear onthe line segment 0 < y < 2N, in each of the intervals [0, cpi (x)] and [WI (x), 2N] andtransforms the point y = W, (x) on this line segment into the point y = x. In the


remaining part of the rectangle O1 the homeomorphism h2 is defined in any suitablemanner.

The homeomorphism h = h2 o h 1 is identical outside the rectangle A1 and pre-

serves the rectangle A and the family 1-11 (F) outside it. In the rectangle A itself thishomeomorphism transforms the singular limiting line of the family r12(F) into thesingular limiting line of the family II2(F).

(2) We choose a system of coordinates x, y in a neighborhood of the point 0 withorigin at 0 such that (a) the steep domain is the region R2\{x, y) I -x < y < 0}, (b) thefamily III (F) is a family of vertical lines of this region, and (c) the parts of the graphof y = x + I x 1 for x > 0 and x < 0 are singular limiting lines of the family H2 (F) (thepositive and negative semiaxes of the axis Oy are the singular limiting lines of the familyIII (F)). For the system F we have H1 (F) = II1(F), and the corresponding singularlimiting lines of the family II2(F) are parts of the graph of a continuous functiony = cp(x), cp(0) = 0, W(x) > 0 for x > 0, that correspond to x < 0 and x > 0. We setN = max{max { I c p (x) J I x < 261, 4,-J. Consider a homeomorphism h preserving thefamily II1(F), identical outside the rectangle Al = {(x, y) I Ix1 < 2e, IyI c 2N}, anddefined in the rectangle Al itself in the following way. For a fixed x E [-e, 0), on theline segment IyI < 2N it is linear on each of the intervals [-2N, cp(x)] and [W (x), 2N]and transforms the point y = W(x) of this line segment into the point y = 0; for afixed x E (0, e], on the line segment 0 < y < 2N, it is linear on each of the intervals[0, cp(x)] and [W (x), 2N] and transforms the points y = 0 and y = cp(x) of this linesegment into the points y = 0 and y = 2x, respectively. In the remaining part of therectangle Al the homeomorphism h is defined in an arbitrary suitable manner.

The homeomorphism h is identical outside the rectangle A1, preserves the family171, (F) and the rectangle A = J (x, y) l Ix I < e, l y I < 2N I, and in the rectangle A itselfit transforms the singular limiting lines of the family 1`12 (P) entering the point 0 intothe singular limiting lines of the family H2(F).

(3) We take a system of coordinates x, y in a neighborhood of the point 0 withorigin there such that (a) the steep domain is the region R2\{(x, y)I - x < y < 0},(b) the family of limiting lines II1(F) coincides in this region with the family ofcurves xy = c for x < 0 and with the family of vertical lines for x > 0, and (c)the graphs of the functions y = ±x for x < 0 and y = 2x for x > 0 are thelimiting singular lines of the family II2(F) (the semiaxes of abscissas for x < 0 andordinates for both y < 0 and y > 0 are the singular limiting lines of the familyH1(F)). For the system F we have III (F) = H1(F), and the corresponding singularlimiting lines of the family H2 (F) are the three trajectories entering zero and lying inthe first, second, and third quadrants. In the first quadrant the singular limiting lineof the family f12(F) is the graph of the continuous function y = cp(x), W(O) = 0,W(x) > 0 for x > 0. In the second (third) quadrant the singular limiting line of thefamily H2(F) is not, generally, a graph of the function y = y(x). However, if thesecond (third) quadrant is mapped onto the half plane z < 0 by the homeomorphismg:(x,y)H(x=xy,y=x+y)(g:(x,y)H(z=-xy) y=x- y), then theimage of the singular limiting line of the family H2(F) lying in this quadrant is partof the graph of the continuous function y = cp (z), W(0) = 0, for z < 0. Under thehomeomorphism g the singular limiting line of the family 112 (F) goes into the negativesemiaxis of abscissas.

As can easily be seen, the homeomorphism h, h = h for x > 0 and h = g-1 hg forx < 0, where h is the homeomorphism considered in the previous case (2), is thus the


one we are looking for. Indeed, the set Al = {AI n {x > 0}} U g-1(01 n {x < 0})and A = {A n {x >_ 0}} U g-I (A n {z < 0}), where, when obtaining the preimageof the mapping g, the rectangles A and Al introduced in case (2) are placed in ][8zxy

by the identification z = x, y = y. The tilde in h, A, and Al is omitted. Thehomeomorphism h preserves the family 111(F) and the neighborhood A, is identicaloutside the neighborhood A1, while in the neighborhood A itself it transforms thesingular limiting lines of the family 112(F) entering the point 0 into the correspondingsingular limiting lines of the family I12(F).

(4) We choose a system of coordinates x, y in a neighborhood of the point 0 withorigin there so that (a) the steep domain is the region y > 0, (b) the semiaxis of Oy fory > 0 is a separatrix of a folded node of the family III (F), (c) for x > 0, this familyitself is the set of vertical lines in this region, and (d) the n singular limiting lines of thefamily 112(F) approaching the point 0 are the rays y = x/k, x > 0, 1 < k < n, wherek is a natural number. In this case y = x, x > 0, is a separatrix of the family II2(F).

For the system F we have 111(F) = 111(F), and the corresponding singular limit-ing lines of the family 112 (F) are the graphs of the continuous functions y = cpk (x),Wk (0) = 0, 1 < k < n, WI (x) > W2 (x) > > w, ,(x) > 0 for x > 0. LetN = max{max{cpl (x)10 < x < 2e}, 2e}. Consider a homeomorphism h iden-tical outside the rectangle A2 = { (x, y)10 < x < 2e, 0 < y < 2N}, preservingthe family 111(F), and defined in this rectangle in the following way. For a fixedx E (0, e], it is linear on the line segment 0 < y < 2N and on each of the inter-vals [0, W, ,(x)], [cp (x), (x)], ... , [cp2(x), WI (x)], [WI (x), 2N], and transforms thepoints cpl (x), W2 (X), ... , W,,(x) of this line segment into the points x, x/2, ... , x/(n-1),x/n. In the remaining part of the rectangle A2 the homeomorphism h is defined in anysuitable way.

The homeomorphism h is identical outside the rectangle Al = {(x, y) I I xj <2e, ly 1 < 2N}, preserves the family II1(F) and the rectangle A = {(x, y) I 1x j < e, y <2N}, and in the rectangle A it transforms the singular limiting lines of the family 112 (F)entering the point 0 into the corresponding limiting lines of the family 112(F).

(5) The behavior of two branches of the field of limiting directions in a neighbor-hood of a folded saddle-node is quite different, and therefore the construction of thehomeomorphism depends on which of two branches determines 111(F).

If the family 111 (F) possesses one separatrix of a folded saddle-node entering thepoint 0 and one separatrix of a folded saddle-node starting at 0, then we take localcoordinates x, y in a neighborhood of the point 0 with origin there such that (a) thesteep domain coincides with the region I[8 2\{(x, y) I x < y _< 0}, (b) the family oflimiting lines 111 (F) in the half plane x > 0 is a set of vertical lines (the semiaxes ofthe axis Oy are separatrices of a folded saddle-node), and (c) n > 2 singular limitinglines of the family 112(F) are the rays y = x/k, x > 0, 1 < k < n, where k is a naturalnumber. For the system P we have 11, (P) = III (F), and the corresponding n singularlimiting lines of the family II2(F) are, for x > 0, parts of the graphs of continuousfunctions y = Wk W, Wk (0) = 0, 1 < k < n; cpl(x) > cp2(x) > . . . > W,, (x) forx > 0. Let N = max{max{cpl(x) 10 < x < 2e}, I 0 < x < 2e}, 2e}.Consider a homeomorphism h that is identical outside the rectangle A2 = {(x, y) I

0 < x < 2e, jyI < 2N}, preserves the family 111(F), and is defined in the rectangleitself in the following way. For a fixed x E (0, e], on the line segment Iy I < 2N, itis linear on each of the intervals [-2N, cp (x)], [cp,,(x), 1(x)], ... , [cp2(x), WI (x)],[cpl (x), 2N] and carries the points cp (x), (x), ... , WI(x) of this line segment to


the points x/n, x/(n - 1), ... , x, respectively. In the remaining part of the rectangleA2 the homeomorphism h is defined in any suitable manner.

The constructed homeomorphism h is identical outside the rectangle Al ={ (x, y) I x I < 2e, I y } < 2N, preserves the family III (F) and the rectangle A ={ (x, y) I I x I < e , I Y I < 2N I, and in the rectangle A it transforms the singular limitinglines of the family II2(F) entering the point 0 into the corresponding singular limitinglines of the family 112(F).

If at the point 0 the family 111(F) has two incoming or two outgoing separatricesof a folded saddle-node, then local coordinates x, y in a neighborhood of the pointO with origin there are chosen so that (a) the steep domain coincides with the regionR2\{(x, y) I x > 0, y < 0}, (b) the family III (F) is a set of vertical lines in the firstquadrant and a set of horizontal lines in the third quadrant (the negative semiaxis ofabscissas and the positive semiaxis of ordinates are separatrices of a folded saddle-node), and (c) the n + 1 > 2 singular limiting lines of the family 1I2(F) are the raysy = x, x > 0 and x = y/k, y < 0, 1 < k < n where k is a natural number. Forthe system F we have III (P) = III (F), and the n + 1 singular limiting lines of thefamily I12(F) are graphs of the continuous functions y = cp(x), x > 0, W(O) = 0,cp(x) > 0 for x > 0, and x = Wk (A y < 0, Wk (0) = 0, 1 < k < n; cp1(y) <W2 (Y) < . . . < 0 for y < 0. We set N = max{max{cp(x)10 < x < 2e}, 2e}and N1 = max{max{-cp1(y)I - 2e < y < 0}, 2e}. Consider a homeomorphismh preserving the family 111(F), identical outside the union of the rectangles A2 ={(x, y) 0 < x < 2e, 0 < y < 2N} and A3 = {(x, y) 1 -2e < y < 0, -2N1 <x < 0}, and defined in the rectangles themselves in the following way. For a fixedx E (0, e] on the line segment 0 < y < 2N it is linear on each of the intervals[0, cp(x)] and [W (x), 2N] and carries the point cp(x) to the point x; in the remainingpart of the rectangle A2 the homeomorphism h is defined in an arbitrary suitablemanner. For a fixed y c [-e, 0), . on the line segment -2N1 < x < 0, it is linearon each of the intervals [-2N1, WI (y)], [cpl (Y), cp2(Y)], , [cpn-1(Y), cpn (Y)], [cpn (Y), 0]and transforms the points cp1(y),.W2 W, ... , W" (Y) into the points y, y/2,. .. , y/n,respectively; in the remaining part of the rectangle A3 the homeomorphism h is definedin any suitable way. The homeomorphism h is identical outside the rectangle Al ={(x, y) I -2N1 < x < 2e, -2e < y < 2N}, preserves the family III (F) and therectangle A = {(x, y) I -2N1 < x < e, -e < y < 2N}, and in the rectangle A itselfit transforms the singular limiting lines of the family II2(F) entering the point 0 intothe corresponding singular limiting lines of the family II2(F).

The first stage of the construction of the desired homeomorphism is complete.Stage II. Let y be a cycle of the family 1I2(F). For definiteness, assume that the

cycle is stable. We take a system of local coordinates r, cp in a sufficiently narrow tubularneighborhood of the cycle such that (a) on the cycle itself r is equal to 1 and cp variesfrom 0 to 27C and (b) locally, in a neighborhood of this cycle, the family III (F) is the setof radii cp = c E [0, 27r); r and cp will be interpreted as polar coordinates in the plane(Figure 3.6). Take e > 0 so small that in a neighborhood {(r, cp) I Ir - 11 < e} thefamily II I (F) is a set of radii. In this neighborhood we take the points A = (1 - 6, 0)and B = (1 + 6, 0) not lying on the singular limiting lines of the family H2 (F). If6 > 0 is sufficiently small, then the positive semitrajectories of the limiting lines of thefamily II2 (F) passing through the points A and B lie entirely in the neighborhood{ (r, cp) I I r - 11 < e } and wind around y. Denote by A 1, A2 and B1, B2 the points ofintersection of these semitrajectories with the line cp = 0 that follow immediately after


FIGURE 3.6

A and B, and let AI (A) be a closed neighborhood of the cycle bounded by the arcs AA1and BBI (A1 A2 and BI B2) of these semitrajectories and the segments AA1 and BIB(AIA2 and B2B1) of the line cp = 0. According to the choice of the points A and B,each of the singular limiting lines of the family II2(F), which winds around the cycley, enters the neighborhood A at a point belonging to the interval AIA2 or the intervalB1 B2. For a system F (sufficiently close to F) we have III (F) = 1I1(F), and thefollowing assertions hold: (1) the corresponding cycle y lies inside the neighborhoodA; (2) the number of singular limiting lines of the family II2(F) that wind around thecycle y on each of its sides is equal to the number of singular limiting lines of the familyII2(F) that wind around the cycle y on its corresponding sides; (3) each of the singularlimiting lines of the family 1-12 (P) winding around the cycle y enters the neighborhoodA at a point belonging to either the interval AIA2 or the interval B2B1, and, whenfurther extended, it remains inside this neighborhood.

Consider a homeomorphism h preserving the family 1-11 (F) and the neighborhoodA, identical outside the neighborhood AI, and defined in the neighborhood A itself inthe following way. Each arc {cp = c } f1A is divided by the points of its intersection withthe singular limiting lines of the family II2(F) into either two or an infinite number ofsegments. The restriction of h to this arc carries the ends of these segments to the endsof the corresponding segments for the family II2(F). To the segments themselves and tothe difference AI \A the homeomorphism h is extended in an arbitrary suitable manner.The homeomorphism h is identical outside the neighborhood AI, preserves the familyIII (F) and the neighborhood A, and in the neighborhood A itself it transforms thesingular limiting lines of the family 1I2(F) into the corresponding singular limitinglines of the family II2(F).

The second stage of the construction of the homeomorphism is complete.

REMARK. In all cases at the first and second stages we constructed closed neigh-borhood A and Al and a homeomorphism h "normalizing" the singular limiting linesof the family II2(F) in the neighborhood A. The normalized singular limiting linesissue from the neighborhood A across the part of its boundary determined by an arc ofa limiting line of the family 1-11 (F). The smallness of the constructed neighborhoodsA and A2 depends on the magnitude of e and on the degree of closeness of the systemF to the system F, whence, by the stability of the families III (F) and 1I2(F), it followsthat for a system P sufficiently close to F these neighborhoods can be chosen so thatthe following three conditions are satisfied:


(1) any two different neighborhoods AI are disjoint;(2) for each neighborhood A of a folded singular point or of a cycle of the family

112(F) the intersection of this neighborhood with the singular limiting linesof the family 112 (F) consists exactly of the normalized (by an appropriatehomeomorphism h) arcs of these singular limiting lines;

(3) for each of the remaining neighborhoods A the intersection of the neighbor-hood with the singular limiting lines of the system F consists exactly of thenormalized (by means of an appropriate homeomorphism h) arc of a singularlimiting line of the family 112(F).

The first condition makes it possible to define a homeomorphism H that coincideswith the corresponding homeomorphism h in each of the neighborhoods 01 and isidentical outside the union of these neighborhoods. By construction, the homeomor-phism H preserves the family H1 (F) everywhere except, possibly, in the neighborhoodsmentioned in the first condition, whence, by this condition, the homeomorphism Hpreserves the singular limiting lines of the family 111(F) = 11(F). Furthermore, byconstruction and by the second and third conditions, in the union of all neighbor-hoods A the homeomorphism H transforms the singular limiting lines of the familyII2(F) into the singular limiting lines of the family 112(F). Hence, to complete theproof of Theorem 1.4 it suffices to modify the homeomorphism H outside the unionof all neighborhoods A so that the new homeomorphism preserves the family III (F)outside this union and transforms the singular limiting lines of the family 112(F) intothe singular limiting lines of the family H2(F), and this is what will be done at thethird stage.

Stage III. Outside the interior of the union of all neighborhoods A there is a finitenumber of closed arcs of singular limiting lines of the family H2(F). Each of thesearcs y begins at the boundary of a neighborhood A and ends at the boundary of some(different, in the general case) neighborhood A, and, locally, in the neighborhood ofthe beginning (end) of the arc y, the boundary of the neighborhood A is determined byan arc of a line belonging to the family H1(F) (Figure 3.7). Consider a neighborhoodof the arc y such that for different arcs these neighborhoods are disjoint. In a (possiblysmaller) neighborhood V of the arc y we choose a system of local coordinates x, yin such a way that this arc becomes the closed interval [0, 1] of the axis Ox and thefamily 1-11 (F) becomes the family of vertical lines x = c. For a system F (sufficientlyclose to F) we have III (F) = 111(F), and the corresponding arc y lies in V and isthe graph of a continuous function y = cp(x), 0 < x < 1. The curve H(y) is thegraph of a continuous function y = W1(x), 0 < x < 1 , W (0) = cpl (1) = 0. We setN = max{Jcp1(x)J 10 < x < 1}. For a system F sufficiently close to F the rectangleQ = {(x, y) 0 < x < 1, ly1 < 2N} belongs to V. Consider a homeomorphismh1 which is identical outside this rectangle and is defined in the rectangle itself in thefollowing way. For x E [0, 1] it is linear on each of the intervals [-2N, cpl (x)] and[cpl (x), 2N] and carries the point y = cpl (x) to the point y = 0. The homeomorphismh1 preserves the family H(111(F)), and in the rectangle Q it transforms a singularlimiting line of the family H2(F) into a singular limiting line of the family H2(F).

Consider a homeomorphism HI identical outside the union of such neighbor-hoods Q and coinciding in each of them with the corresponding homeomorphismh1. Then HI o H is the desired homeomorphism. The degree of its closeness to theidentical homeomorphism depends on. the magnitude of the number e > 0 chosen at


Ay

I

Y I x I

---JFIGURE 3.7

the beginning of the proof and on the degree of closeness of the system P to the systemF.

The sufficiency of conditions (A)-(D) is established. Theorem 1.4 is proved.

REMARK. Note that the constructed transforming homeomorphism H1 o H pre-serves the family of lines 1-11 (F) everywhere except, possibly, in the neighborhoods Amentioned in the third condition in the remark after the second stage.

4.3. Proof of Theorem 1.5. The second assertion of the theorem follows fromthe first assertion and the stability of the set of cutoff points, double passing points,and turning points of a typical system with respect to its small perturbations (seeTheorem 4.16 in Chapter 2). To prove the first assertion of the theorem it suffices toshow that the set of systems in general position, for which conditions (A)-(C) hold foreach of the branches of the field of limiting directions, is everywhere dense. Indeed,such systems are easily approximated by systems satisfying condition (D) as well (seeSection 4.1 and Figure 3.5). The system is perturbed in the steep domain outsidethe defining set. Under such a perturbation each of the two branches of the field oflimiting directions can be modified independently of the other branch because at eachpoint outside the defining set these branches assume different values (we note that herethe perturbation of a branch of the field of limiting directions is in the class C3 or C4;generally speaking, both classes are possible). As for the approximation of a systemin general position for whose fixed branch of the field of limiting directions conditions(A)-(C) hold, it is carried out using methods similar to those applied to vector fields[PM, Pel]. We do not present this approximation here. Theorem 1.5 is proved.

§5. The structure of orbit boundaries

In this section we prove Theorems 1.7, 1.8, and 1.9. It suffices to prove thesetheorems for a system F in general position with typical singularities of the limitingdirections field, for which the assertions of Theorem 1.5 hold. In particular, conditions(A)-(C) of Theorem 1.3 are satisfied for each of the two branches of the limitingdirections field.

5.1. Proof of Lemma 1.7. We first prove the second assertion of the theorem.Assume that there are points z1 G ,7 (z) not belonging to O+(S). We considersome neighborhoods V (z;) such that V (z;) n O+(S) = 0. The theorem about thecontinuous dependence of a solution of a differential equation on the initial data andthe theorem on the integration of inequalities imply that there is a neighborhood ofthe point z such that when moving backward along any feasible trajectory we attainthe region V (zl) U V (z2)U Int 0-(z) in a finite time (Int W denotes the interior of

§5. THE STRUCTURE OF ORBIT BOUNDARIES 121

the set W). This region does not intersect 0+ (S). Consequently, z V aO+(S), whichis a contradiction. Hence, the above assumption is false.

Furthermore, we have p (z) n Int O+(S) = 0 because otherwise we wouldhave z E Int 0+ (S). Therefore, either (z), z) C (00+ (S), z) and (rj2 (z), z)(aO+(S), z), and then (aO+(S), z) = (i7; (z), z), or (ni (z), z) c (aO+ (S), z), andthen (00+ (S), z) = (,j (z) U q2 (z), z). Thus, the second assertion of the theorem isproved.

The third assertion of Theorem 1.7 immediately follows from condition (A) ofTheorem 1.3 and assertions (4) and (5) of Theorem 1.7.

W e now prove assertions (4) and (5). If (i , (z), z) = (aO+(S), z), then q (z) cOO+(S) because otherwise the second assertion already proved would not hold for apoint on the trajectory qi (z). If, , (z) is a cycle, then assertion (4) is true. If qi (z) is nota cycle, then, by condition (C) of Theorem 1.3, the trajectory 17r (z) either (a) unwindsfrom a (simple) cycle y or (b) starts at a point on the boundary of the steep domain.In case (a) the trajectory qi (z) cannot separate the points attainable and unattainablefrom the set S in a sufficiently narrow neighborhood of the cycle y because the motionalong the limiting lines of the family III (F) in such a neighborhood mixes attainableand unattainable points. Consequently, case (a) is impossible.

Generally, in case (b) the trajectory ilr (z) can start either at a regular point 0 orat a folded singular point 0 on the boundary of the steep domain. A regular point onthe boundary of the steep domain of a system in general position belongs to the localtransitivity zone of the system. Consequently, in a neighborhood of such a point, thetrajectory #l, (z) cannot separate the points attainable and unattainable from the setS. Hence, the subcase of a regular point is impossible. Let 'j (z) start at a foldedsingular point O. If 0 is a folded singular point of one of the following types: a saddlepoint, a monkey saddle, or a stable saddle-node, then the trajectory 17; (z) can only bean outgoing separatrix of this singular point. The point 0 cannot be a folded stablenode because no limiting line can start at such a point.

To complete the proof of the theorem it remains to show that the boundary ofO+(S) contains no folded foci, folded unstable nodes, and folded unstable saddle-nodes.

A folded focus belongs to the interiors of both its positive and negative orbits and,consequently, cannot belong to the boundary of 0+ (S) (we note that for # U = 2 thisis not true because for a system in general position a folded saddle and a folded focuscan exist simultaneously, see the explanations after Theorems 1.7 and 1.8).

A folded singular point z which is an unstable node can be approached only fromthe side of the LTZ. Hence, for this point to belong to the boundary of the orbit O+(S)it is necessary that points of the LTZ lying in O+(S) exist arbitrarily close to it. Butthen we have z E Int O+(S). Consequently, the boundary of the orbit O+(S) cannotcontain folded unstable nodes.

Similarly, if our folded singular point z is an unstable saddle-node and there existpoints of the LTZ that are arbitrarily close to it and can be attained from the setS, then z E Int O+(S). Consequently, there are no such points for z E OO+(S).Hence, the only remaining way of approaching the point z is to move along theseparatrix 77 (z) entering the point z or along a trajectory lying arbitrarily close toit. It follows that (OO+(S), z) = (7; (z) U 77 (z), z), where 77 (z) is also a separatrixof this singular point. Therefore, the trajectory r7; (z) must be a double separatrix.


(a)

FIGURE 3.8

(b)

However, condition (A) of Theorem 1.3 does not permit such separatices. Thus, afolded unstable saddle-node cannot lie on the boundary of the orbit 0+ (z).

Note that the first assertion of Theorem 1.7 is also proved.Theorem 1.7 is proved.

5.2. Proof of Theorem 1.8. According to the first assertion of Theorem 1.7, thepoint z can be a folded singular point of one of the following four types: a saddlepoint, a stable node, a monkey saddle, or a stable saddle-node. We consider these fourcases in succession.

A saddle point (2a). By assertion (3) of Theorem 1.7, the germ at a point zof a separatrix entering it cannot belong to the germ (00+ (S), z). Consequently,(80+(S), z) = (r7i (z) U rig (z), z). In a neighborhood of the point z the separatricesstarting at it can be either closer to or farther from the LTZ than the incomingseparatrices (Figure 3.8a and b, respectively; in addition to the notation in Figure 3.1,in Figures 3.8-10 the LTZ is tinted and the orbit O+(S) is shaded). We first show thatsubcase shown in Figure 3.8b is impossible. Indeed, if in this subcase the orbit 0+ (S)lies under the outgoing separatrices, then z E Int O+(S). This is a contradictionbecause z E 80+(S). However, the orbit O+(S) cannot lie above these separatriceseither because otherwise it would be impossible to approach point z closely. This alsocontradicts the condition z E 8O+ (S). Hence, the subcase in Figure 3.8b is impossible.

We now consider the subcase shown in Figure 3.8a. Here the orbit O+(S) is locatedbelow its boundary because otherwise we would have z V O+(S). In particular, (LTZ,z) C (O+(S), Z).

Thus, assertion (2a) of Theorem 1.8 is proved, and its first assertion is true in thecase under consideration.

A stable node (2b). In a neighborhood of the point z the orbit O+(S) has nopoints lying above the separatrices entering this point because otherwise z V 80+(S).Consequently, (80+(S), z) = (r7i (z) U q2 (z), z), and, locally, in a neighborhood ofthe point z, the orbit 0+ (S) lies above these separatrices (Figure 3.9). Assertion (3)of Theorem 1.7 implies that the trajectories q, (z) and r72 (z) are not separatrices ofthis node. Hence, assertion (2b) of Theorem 1.8 is proved, and its first assertion is truein the case under consideration.

A monkey saddle (2c). In the vicinity of the point z the boundary of the or-bit O+(S) is determined by two outgoing separatrices because, by assertion (3) ofTheorem 1.7, we have (r7; (z), z) V (80+(S), z). This pair of separatrices cannotbe the left and upper separatrices (see Figure 3.10; the double lines are the limitingdirections at the point z, and the vertical and horizontal lines approaching the pointz are the saddle separatrices of the singular point z of the velocity field determinedby the corresponding isolated values of the control parameter) because otherwise theupper separatrix would belong to Int 0+ (S). Also, the boundary 80+(S) cannot be

§5. THE STRUCTURE OF ORBIT BOUNDARIES 123

FIGURE 3.9

FIGURE 3.10

determined by the two right separatrices because otherwise either z E Int O+(S) orit would be impossible to approach the point z closely from the set S. Consequently,(a0+(S), z) = (q, (z) U q2 (z), z), where q, (z) and q2 (z) are the pair of lower out-going separatices, and assertion (2c) of the theorem is true. The orbit O+(S) is locatedabove these separatrices because otherwise z E Int O+(S). Thus, in the case underconsideration the first assertion of the theorem holds.

A stable saddle-node (2d). According to assertion (3) of Theorem 1.7, the germat the point z of a separatrix entering it does not belong to the germ (190+ (S), z).Furthermore, we have (LTZ, z) c (O+(S), z) because z E aO+(S), and the part ofthe LTZ adjoining the point z is attainable from any point lying sufficiently close tothe point z. Hence, in this case the first assertion of the theorem holds, which is nowproved completely. It follows that the germ at the point z of an outgoing separatrixbelongs to either (80+(S), z) or (0+ (S), z). Accordingly, we obtain (8O+ (S), z)(r7; (z) U , (z), z) and (80+(S), z) = (q, (z) U t72 (z), z), where only the trajectory17; (z) is a separatrix of this saddle-node.


5.3. Proof of Theorem 1.9. The first assertion of the theorem is obvious becauseeach of the orbits 01 (Z) is open. The third assertion follows directly from the firstassertions of Theorems 1.8 and 1.8'.

We prove the second assertion. Let z E aO+(Z) n a0-(Z). If z is a pointon the boundary of the steep domain, then, by the first assertion of Theorem 1.7(1.7') this point can be a folded singular point of one of the following types: a saddlepoint, a stable (unstable) node or a stable (unstable) saddle-node, or a monkey saddle.Consequently, the subcases of a node and a saddle-node are immediately excluded.In the subcase of a saddle point, assertion (2a) of Theorem 1.8 (1.8') implies that


o-(Z) \ \ \ / / / o+(Z)

FIGURE 3.11

the separatrices nearest to the LTZ must be those starting from the point z (enteringthe point z). Therefore the subcase of a saddle point is also impossible. Thus, zis a singular point of the type of a monkey saddle. According to assertions (2c) ofTheorems 1.8 and 1.8', the germ of the boundary of the LTZ at the point z coincideswith the germ (7l; (z) U q (z), z).

If the point z belongs to the steep domain, then neither of the trajectories 71(z)and 112 (z) is a cycle. Indeed, if i , (z) is a cycle, then, by condition (B) of Theorem 1.3,it is simple. The lines of the family 11j (F) intersect it in one direction. It followsthat z V a0+ (Z) n a0- (Z), which is a contradiction. Furthermore, if at least oneof the relations (9O (Z), z) = (ql (z) U r/2 (z), z) is satisfied, then, according to thesecond and fifth assertions of Theorems 1.7 and 1.7', at least one of the trajectoriesrll (z) and 72(z) is a double separatrix. However, condition (A) of Theorem 1.3forbids such separatrices. Consequently, the pair of germs ((90± (Z), z) coincideswith the pair of first two germs in the second assertion of Theorem 1.7 becauseotherwise (0+ (Z) n 0- (Z), z) = (0, z), i.e., z V 9Z, whence we conclude that(Z, z) = (71 (z) U qj (z), z) if (aO+(Z), z) = (71; (z), z) and (aO-(Z), z) = (rl.i (z), z)(Figure 3.11; the positive and negative orbits of the point z are shaded with inclinedlines). Theorem 1.9 is proved.

§6. Stability

In this section Theorems 1.1 and 1.2 are proved. It suffices to establish theirvalidity for a system in general position, for which the results of the previous chapterand Theorems 1.5 and 1.7-1.9 hold. In particular, conditions (A)-(C) of Theorem 1.3are satisfied for each of the two branches of the field of limiting directions. It is quiteclear that for such a system the stability of the orbit D = O+(z) takes place if z E Int0+ (z). The conditions for asymptotic stability will be proved in four stages. We firstestablish them in the vicinity of the cycles lying on the boundary of the orbit in question(Section 6.1), then in some regions in neighborhoods of points of confluence, foldedstable nodes, and folded stable saddle-nodes belonging to this boundary (Section 6.2)and in the vicinity of the limiting lines lying on the boundary (Section 6.3), and,finally, in neighborhoods of folded saddles, folded monkey saddles, and folded stablesaddle-nodes belonging to this boundary (Section 6.4). Theorem 1.1 is proved inSection 6.5.

6.1. The first stage. Let a cycle y of the family of limiting lines II; belong toM. On this cycle the lines belonging to the family III enter D by intersecting y.Consequently, y is a stable cycle because otherwise it could not have been approachedfrom the point Z. The cycle y is a C' -submanifold of phase space. In a sufficiently small

§6. STABILITY 125

neighborhood Q of the cycle y there exists a function V E C 1(Q) such that V I, = 0,grad V IQ ; 0, and the derivative of V along L, is positive in Q fl D and negative inQ\D (V2 is a Lyapunov function for the cycle y). Furthermore, this cycle lies in thesteep domain, and, consequently, the cone of any point sufficiently close to the cycle isa sector with the angle smaller than 180° (recall that the cone of a point is the positivelinear hull of the set of feasible velocities at that point). The field of limiting directionsis continuous in the steep domain. This, together with the conditions VI,, = 0 andgrad VIQ # 0, implies that there is a neighborhood Q1 C Q of the cycle y such thatin the region Q\D the derivative of V along the field Lj is negative. Consequently, ateach point in this region the derivative of the function V along any feasible velocityis negative. Therefore, for any trajectory starting from this region the condition forasymptotic stability holds.

REMARK. We note that for a sufficiently small e > 0 the derivatives of the functionV along the two fields L1 and L2 on the level line V = e are negative. Hence,the limiting lines of the two fields intersect this level line transversally in the samedirection. Thus, the points on the level line can be used for indexing the limiting linesof each of the two branches of the field of limiting directions lying in M\D and closelyapproaching the cycle y.

6.2. The second stage. The subcases of a point of confluence, a folded stable node,and a folded stable saddle-node will be considered.

The subcase of a point of confluence. By the second assertion of Theorem 1.5,a point of confluence 0 E OD is a nonsingular point of each of the two branchesof the field of limiting directions. In particular, the values of these branches at thispoint are noncollinear. This fact, together with Lemma 3.1, implies that locally ina neighborhood of the point there exists a coordinate system x, y of class C' withorigin at the point 0 such that in this coordinate system the family of limiting lines isthe family of phase trajectories of the vector fields (1, 0) and (0, 1). The germ at thepoint 0 of the set M\D is the germ at this point of the third quadrant. The derivativeof the function V (x, y) = xy in this quadrant along each of the fields (1, 0) and (0,1) is negative, so that the derivative of this function along any feasible velocity is alsonegative. Consequently, for a sufficiently small e > 0 any trajectory starting from theregion Al = {(x, y) I -e < x < 0, -e < y < 0} satisfies the condition for asymptoticstability. Set A = {(x, y) 1 -e/2 < x < 0, -e/2 < y < 0} and denote by y each of thearcs x = -e/2, -e/2 < y < 0 and y = -e/2, -e/2 < x < 0.

The subcase of a folded stable node. Locally, in a neighborhood of a folded singularpoint 0 E 8D which is a stable node, there exists a coordinate system x, y of class COwith origin at 0 such that in this coordinate system the family of limiting lines at thispoint is the family of the integral curves of the differential equation y = (y' + x/10)2.In this coordinate system the germ ((9D, 0) is the germ at zero of the graph of afunction y = h(x), h E C1(R). Take e > 0 so small that in the closed region Qbounded by the axis of abscissas and by the limiting lines passing through the point(0, e) the indicated normal form takes place and the boundary of D coincides with thegraph of y = h(x). Set Al = Q\D. In the region Al the function V(x, y) = y - h(x)is positive and decreases to zero along every limiting line. Consequently, in this regionthe restriction of the function V to any trajectory also decreases to zero along thetrajectory. However, on the set Al the function V assumes the zero value only on thesubset OD l01. Therefore, for the trajectories starting from the region Al the conditionfor asymptotic stability holds. We define a region A in the same way as the region Al


with the only distinction that the limiting lines passing through the point (0, e/2) aretaken. Denote by y the intersection of each of the two limiting lines with the region A.

The subcase of a folded stable saddle-node. According to Theorem 1.8, the germ(aD, O) coincides with one of the germs (q (O) Ur/2 (O), 0) and (i (O) Urn; (O), 0),where the singular point 0 E OD is a folded stable saddle-node and q (O) is theseparatrix starting at this point. In the case of the first germ, the germ at the pointO of the family of limiting lines in the region M\D is homeomorphic to the germ atzero of the family of limiting lines in the region y > h (x) considered in the previoussubcase. Consequently, in the case under consideration the regions A and Al and thearcs y can either be defined by analogy with the previous subcase or can be obtainedfrom this subcase by means of an appropriate homeomorphism.

Consider the germ (ii (O) U rj; (O)). Locally, in a neighborhood of the point 0,we take a coordinate system x, y of class C 1 with origin at 0 that normalizes the familyof phase trajectories of the vector field v determined by an isolated value of the controlparameter and having a stable node at the point O. The axis of abscissas correspondsto the eigenvalue with the minimum modulus. The positive directions of the coordinateaxes are chosen so that the nonzero limiting velocities at the point 0 "face" the firstquadrant (i.e., have positive coordinates). Locally, in a neighborhood of zero, the lineq (0) is a part (for x > 0) of the graph of a function y = h(x), h E C' (R). Thefunction V(x, y) = h(x) - y is positive below this graph. For a sufficiently small6 > 0 the curve q (0, -e), the axis of ordinates, and the line q -( 0 ) bound a closedregion Q where the family of phase trajectories of the field v is normalized and where,except at the point 0 itself, the derivative of the function V along the field Lj is alsonegative. But in the region Al = Q\, (O) the derivative of the function V along thefield L; is also negative. Consequently, at any point in this region the derivative alongany feasible velocity is negative. Furthermore, when moving with the velocities of thesystem it is possible to leave the region Al only by intersecting the line 11 - (0). Onthis line the function V assumes its minimum value in the region. Consequently, thecondition for asymptotic stability holds for any trajectory starting from the region A1.We define a region A in the same way as AI with the only distinction that instead of thetrajectory q (O, -e) the trajectory ii (O, -e/2) is taken. Set y ='t (O, -e/2) n A.

6.3. The third stage. Let z E OD be a point in the steep domain that does notbelong to any of the regions A defined at the previous stages and satisfies the condition(aD, z) _ (qi(z), z). There is a single point z1 at which the trajectory q (z) entersone of these regions A. For a point z E 'i (z) sufficiently close to the point z thefollowing two assertions hold: (1) the trajectory q +(z) enters the region A at a pointit E y c i (z) and (2) the closed region Q bounded by the lines r7; (z, z 1 ) , i

q, (z, it ), and ii (zl , z1) belongs to the steep domain. For the field L, we take a rectifyingchart (W, g), Q C W of class C' such that in this chart the line qi (z, z I ) is the segment[-1/2,1/2] of the axis of abscissas and the region Q is located in the half plane y > 0.In the region Q n l y > 0} the derivative of the function V (x, y) = (1 - x) y alongthe field L; is negative. Furthermore, the cone of a point in this region is a sectorwith the angle smaller than 180°, which depends continuously on this point. This,together with the conditions V (x, 0) = 0 and grad V IQ 0, implies that the derivativeof the function V along the field Lj is negative at any point of the region l y > 0}sufficiently close to the arc ,i1(z, z). Consequently, for a sufficiently small e > 0 thederivative of the function V along any feasible velocity at an arbitrary point of the

§6. STABILITY 127

FIGURE 3.12

region Q1 = Qn {0 < y < e } is negative. It follows that, when extended, any trajectoryof the system starting from a point in this region reaches either the line ij1(z, z1) orthe line ,, (z1 , z1). However, when reaching the line qJ (z1, z1), the trajectory fallsinto the corresponding region 01 where the condition for asymptotic stability holds.Consequently, in both cases this trajectory tends to the orbit D, so that the conditionfor asymptotic stability is fulfilled in the region Q1. Thus, this condition holds in asufficiently small neighborhood of any point z c OD such that (OD, z) = (qi (z), z).

6.4. The fourth stage. We consider in succession the subcases of a folded saddle,folded monkey saddle, and a folded saddle-node.

The subcase of a folded saddle. Locally, in a neighborhood of a folded singularpoint 0 E aD which is a saddle point, there exists a coordinate system x, y of classC° with origin at this point such that in these coordinates the family of limiting lines isthe family of integral curves of the equation y = (y' - x)'. In this coordinate systemthe germ (aD, 0) is the germ at zero of the parabola 4y = x2. Take e1 > 0 and 62 > 0so small that in the closed region Q bounded by the axis of abscissas and the limitinglines passing through the points (0, e1) and (fee, 0) the indicated normal form takesplace and the boundary aD coincides with the parabola 4y = x2 (Figure 3.12; theregion Q is shaded). Set Al = Q\D. In the region Al the restriction of the functionV (x, y) = 4y - x2 to each of the limiting lines decreases either to zero or to its value atthe exit point of the limiting line from A. Consequently, the restriction of the functionV to any other trajectory of the system also possesses this property. It follows that theextension of any trajectory of the system starting from the region Al either tends tothe part of the boundary of the region determined by the parabola 4y = x2 or leavesthis region by intersecting the F-part of the boundary determined by the limiting linespassing through the points (fee, 0).

In a sufficiently small neighborhood of the intersection of F with the parabola 4y =x2 the condition for asymptotic stability holds (this was proved at the previous stage).This neighborhood contains F if e1 is sufficiently small. Therefore, the condition forasymptotic stability is fulfilled throughout the region Al as well.

The subcase of a folded monkey saddle. Let a singular point 0 E aD be a foldedmonkey saddle and let v be a smooth velocity field determined by an isolated valueof the control parameter and having a saddle point at 0. Locally, in a neighborhoodof the point 0, we select a smooth coordinate system x, y with origin at this point insuch a way that the outgoing (incoming) separatrices of the saddle point of the field vdetermine the axis of abscissas (the axis of ordinates) and the nonzero limiting velocitiesat this point "face" the first quadrant (Figure 3.10). We have (D, 0) = ({y > 0}, 0).For small e1 > 0 and 62 > 0 we set Al = {(x, y) I JxJ < e1, -e2 < y < 0}.


The derivative of the function V(x, y) = -y along each of the limiting directions isnegative at any point in the region Al provided that e1 and 62 are sufficiently small.Consequently, at such a point the derivative of this function along any feasible velocityis also negative, whence, by analogy with the previous subcase, we conclude that thecondition for asymptotic stability holds in the region Al for a sufficiently small E2.

The sarbcase of a folded saddle-node. Let 0 G OD be a singular point of the foldedsaddle-node type. The fulfilment of the condition for asymptotic stability in the case(aD, 0) = (r71 (O) U 1 7 2( 0 ) ) was established for a sufficiently small neighborhood ofthe point 0 at the second stage. In the case (aD, 0) = (rj; (O) U q (0)) we takethe same system of local coordinates in a neighborhood of the point 0 with originthere as in the corresponding subcase at the second stage. For a sufficiently small6 > 0 the derivative of the function V (x, y) = 0y2 - x at each point of the regionQ = {(x, y) 16y2 - 63 < x < 0} along any direction belonging to the cone of thispoint is negative. It follows that any trajectory of the system starting from this regioneither tends to the point 0 as t - +oo or reaches the segment [-6,6] of the axis ofordinates in a finite time. For the points of the negative semiaxis of ordinates lyingsufficiently close to zero the condition for asymptotic stability was proved at the secondstage because such points belong to the neighborhood A constructed there. We shallstudy the points of the positive semiaxis of ordinates. Consider the closed region Wbounded by the lines x = 0, x = 6 1 , q +( 0 ) , and # (0, 6). Let y = kx be the tangentline to r7; (O) at the point O. For a fixed e > 0 and sufficiently small 0 > 0 and 61 > 0the derivative of the function V (x, y) = y - (k +e)x at each point in the region W\Dalong any direction belonging to the cone of this point is negative. It follows that inthis region the extension of any trajectory of the system either tends to D or leavesthis region by intersecting the part IF of its boundary determined by the straight linex = 61. By the results proved at the third stage, in a sufficiently small neighborhoodI, n OD of the point the condition for asymptotic stability holds. For a sufficientlysmall d > 0 this neighborhood contains F. Consequently, the condition for asymptoticstability holds in both W\D and W, and, hence, in QI = Q U W as well.

The union D U A U Q1 contains a sufficiently small neighborhood of the point O.Therefore, the condition for asymptotic stability holds in this neighborhood.

Thus, we have proved in four stages that the condition for asymptotic stabilityis fulfilled in a sufficiently small neighborhood Q(z) of any point z belonging to theboundary OD. However, this boundary is a compact set covered by the union of allthese neighborhoods. Select a finite subcovering. The union of the neighborhoodsof this subcovering and the orbit D contains a neighborhood of the closure of thisorbit. Consequently, the closure of the orbit D is asymptotically stable. Theorem 1.2is proved.

6.5. Proof of Theorem 1.1. We shall use the following Lemma 6.1. For a sys-tem F in general position and a system P sufficiently close to it the set of limitinglines, the zones of nonlocal transitivity, the positive (negative) orbit of each of thesezones, and, outside the union of these zones, the family of the limiting lines of thesystem P are transformed into the same objects of the system F by a nearly identicalhomeomorphism of the phase space.

It is easy to show that Theorem 1.1 follows from Lemma 6.1. Therefore we discussthe proof of this lemma. For a system in general position the positive orbit O+(z) ofa point z belonging to the NTZ contains this point in its interior, and, consequently,

§6. STABILITY 129

by Theorem 1.2, the orbit is asymptotically stable. Therefore, for any 6 > 0 the 6-neighborhood of this orbit contains the positive orbit O+(z) of the point z for anysystem F sufficiently close to F [Fl]. On the other hand, the orbit O+(z) containsthe difference between the orbit 0+ (z) and the 6-neighborhood of its boundary if thesystem ,P is sufficiently close to the system F. Hence, the boundary8O+(z) lies in the6-neighborhood of the boundary 80+(z). According to Theorems 1.7 and 1.8, each ofthese boundaries has a specific structure. It follows, that if the system P is sufficientlyclose to the system F and the homeomorphism H1 o H in the second remark at theend of Section 4.2 is sufficiently close to the identical one, then it transforms the orbitO+(z) into the orbit O+(z) and, consequently, the zone O+(z) n 6- (z) into the zone0+ (Z) n 0-(z).

Furthermore, in view of this remark, the homeomorphism HI o H preservesthe family III (F) = III (F) everywhere except, possibly, in the neighborhoods Aconsidered in the third condition of the first remark (in the middle) of Section 4.2.However, each of these neighborhoods is a neighborhood of a regular point of theboundary of the steep domain. Thus, if this neighborhood is sufficiently small, then itbelongs to the interior of an NTZ of the system F (and of any other system sufficientlyclose to F).

Consequently, without loss of generality it can be assumed that a system F ingeneral position and any system F sufficiently close to it have the same families ofsingular limiting lines, zones of nonlocal transitivity, and, in the complement W ofthe union of these zones, the families of limiting lines of the first branch of the field oflimiting directions as well: III (F) = III (F).

Furthermore, the singular limiting lines of the families II2(F) and RAP) partitionthis complement into closed subregions. In each of the subregions W' of the partitionthe limiting lines of these families go from the same source to the same sink onintersecting the lines of the families III (F) = II2(F). Here a source (sink) of limitinglines can be either an unstable (stable) limit cycle of one of the families III (F) andII2(F) or an unstable (stable) folded singular point of one of the following two types:a node, or a saddle-node, or an arc of a limiting line of the family III (F) lying on theboundary of W (W') on which the limiting lines of the family 112(F) flow into W(flow out of W'). Let us index the lines of the families II2(F) and 112(F) in W' onthe sinks. In the first and fifth cases the points of the sink itself are taken as indices,in the second case the points of the level line V = s in the matrix in Section 6.1 playthis role, and in the third and fourth cases the points of the limiting line of the familyIII (F) intersecting W' sufficiently close to the sink are used.

Consider a homeomorphism that preserves the family 111(F) = III (F) in W,is identical outside W and on the set of singular limiting lines, and inside each ofthe subregions W' is defined in the following way. For each line y of the familyH2 (F) it carries its points of intersection with each of the limiting lines of the familyIII (F) = H1 (F) to the corresponding points of intersection of this limiting line with theline of the family H2(F) having the same index as y. It is clear that this homeomorphismtransforms the family of lines H2(F) in W into the family of lines H2(F). The degreeof closeness of this homeomorphism to the identical homeomorphism depends on thedegree of closeness of the system .P to the system F (this follows from the theorem aboutthe continuous dependence of a solution to a differential equation on the initial dataand on the right-hand side and the theorem about the continuous dependence of thefield of limiting directions on the system in general position) and the degree of closeness


of the earlier introduced homeomorphism HI o H to the identical homeomorphism.Lemma 6.1 is proved.

§7. Singularities of the boundary of the zone of nonlocal transitivity

The aim of this section is to prove Theorem 1.6. We study the NTZ's of a systemin general position. Let z be a point of such a zone. In Section 7.1 the singularities atthe angular points of the NTZ boundary are studied. According to the first assertionof Theorem 1.9 and the remark after Theorem 1.8, outside the angular points theclassification of singularities of the boundary of the positive orbit of a point coincideswith that of the singularities of the boundary of the positive orbit of a point belongingto the interior of this orbit. In Section 7.2 the singularities of the orbit are studiedin the steep domain, and in Section 7.3 the singularities on the boundary of the steepdomain are considered. In Section 7.4 the second and fourth assertions of Theorem 6are proved.

7.1. Angular points. By the second assertion of Theorem 1.9, an angular pointeither lies in the steep domain or is a folded singular point of the monkey saddle type.In the first case, according to the second assertion of Theorem 1.5, in a neighborhoodof this point the two branches of the field of limiting directions are smooth and, inparticular, have noncollinear values. This fact and assertion (2a) of Theorem 1.9 implythat the point under consideration is a type 1 singular point of the NTZ boundary.

In the second case, by assertion (2b) of Theorem 1.9 and assertions (2c) of Theo-rems 1.8 and 1.8', in a neighborhood of this point, the NTZ boundary coincides withthe closure of the union of the incoming and outgoing separatrices of a nondegeneratesaddle point of a smooth vector field. Consequently, in this case as well an angularpoint is a type 1 singular point of the NTZ boundary.

7.2. Singularities of the boundary of the positive orbit in the steep domain. Bythe second assertion of Theorem 1.7, the germ of boundary W+ (z) at each pointzo E aO+(z) in the steep domain coincides with one of the germs (?7j(zo),zo) or(ii (zo) U ,2 (zo), zo). According to the second assertion of Theorem 1.5, in thecase of the germ (ri; (zo), zo) the line r7, (zo) does not contain turning points, doublepassing points, and cutoff points of the field of limiting directions. Consequently, thegerm (00+ (z), zo) is C°°-diffeomorphic to the germ at zero of the set {y = 0} orl y = x I x I } (i.e., z is a nonsingular point or a type 2 singular point of the boundaryaO+(z), respectively) if the point z is not or is an i-passing point, respectively.

Locally, in a neighborhood of a point of confluence zo (i.e., for the germ (rji (zo) Urig (z), zo)) each of the branches of the field of limiting directions is smooth, and, inparticular, by the second assertion of Theorem 1.5, the values of these branches at thispoint are different. Consequently, zo is a type 1 singular point of the boundary of theorbit O+(z).

7.3. Singularities of the boundary of the positive orbit on the boundary of the steepdomain. By the first assertion of Theorem 1.7, the intersection of the boundary aO+ (z )with the boundary of the steep domain belongs to the set of folded saddles, foldedmonkey saddles, folded nodes, and folded stable saddle-nodes. We shall consider thesefour cases one by one.

A folded saddle. Let z E aO+(z) be a folded saddle. Then, according to assertion(2a) of Theorem 1.8, we have (aO+(z), zo) _ (;y (zo) U 1 ( z 0 ) , zo). By Theorems 4.4and 4.8-4.10 in Chapter 2, the point zo can be either a black zero-point, or a a-turning

§7. SINGULARITIES OF THE BOUNDARY OF THE ZONE 131

point, or a zero-passing point. By Theorems 4.8 and 4.10 of Chapter 2, in the case ofa black zero-point or a a-turning point the germ at the point z of the closure of theunion of the outgoing separatrices is C°°-diffeomorphic to the germ at zero of the setsl y = 0} or l y = I x 1 3 }, respectively, if z0 is a nonsingular point or a type 3 singularpoint of the boundary aO+(z), respectively.

Let z0 be a zero-passing point and let v be a smooth velocity field determined by anisolated value of the control parameter for which this point is a nondegenerate singularpoint. By Theorem 4.8 of Chapter 2, in the case under consideration this singular pointof the field v can be either a saddle point or a node because the point z0 is a foldedsaddle of the field of limiting directions. Here in both subcases only one eigendirectionof the linearization of the field v at the point z0 can be determined using the vectors ofthe cone of the point z0, and in the subcase of a node this direction corresponds to theeigenvalue with the smallest modulus. Furthermore, assertion (2a) of Theorem 1.8implies that locally, in a neighborhood of the point z0, the outgoing separatrices mustbe the closest to the LTZ. In the case of a node of the field v this is possible only whenthis node is stable, and in the case of a saddle point of the field v, this is possible onlywhen the eigendirection of the linearization of the field v corresponding to a negativeeigenvalue is determined by a vector of the cone of the point z0. In this situation thefollowing is true. In the case of a saddle point of the field v its outgoing separatricesare simultaneously the outgoing separatrices of a folded saddle, so that locally, in aneighborhood of such a point, the boundary a0+(z) is a smoothly embedded curve.In the case of a node of the field v the separatrices of a folded saddle are the phasetrajectories of two smooth vector fields issuing from the point z0, the values of the fieldbeing noncollinear at the point z0. Consequently, z0 is a type 1 singular point of theboundary a0+(z).

A monkey saddle. According to assertion (2c) of Theorem 1.8, locally, in a neigh-borhood of a folded singular point z0 E a0+(z) which is a folded monkey saddle, theboundary 00+(z) coincides with the closure of the union of the outgoing separatricesof the nondegenerate singular point z0 (a saddle point) of a smooth field. Therefore,in a neighborhood of this point the boundary aO+(z) is a smoothly embedded curve.

A folded stable node. By assertion (2b) of Theorem 1.8, the germ of the boundarya0+(z) at a folded singular point z0 which is a stable node, coincides with the germ(ilI (z0) U n2 (z0), z0) and, by the third assertion of Theorem 1.7, the lines q; (z0) arenot outgoing separatrices of this singular point. According to Theorems 4.4 and 4.8-4.10 of the second chapter, the point z0 is either a black zero-point or a zero-passingpoint. By Theorem 4.10 of Chapter 2, for a system in general position the germ ofthe family of limiting lines at a black zero-point is C°°-diffeomorphic to the germ atzero of the family of integral curves of the differential equation y = (y' + kx)2, wherek = a(a + 1)-2/2 and a > 1 is not an integer, so that it has a folded node at zero.In the case of general position the lines rji (z0) and i2 (z0) are not singular solutionsof the node because such a singularity can be easily removed by an arbitrarily smallperturbation of the system in an arbitrarily small neighborhood of the point, whereasthe singular limiting lines and the singular solutions of a node depend continuously onthe system in general position and the number of black zero-points of such a systemis finite. Consequently, in the case of a black zero-point, z0 is a type 6 point of theboundary aO+(z).

Let z0 be azero-passing point and let v be a smooth velocity field determined byan isolated value of the control parameter and having at this point a nondegenerate


stable node. In this case the vectors of the cone of the point zo can determine onlythe eigendirection of the linearization of the field v at the point zo with the greatestmodulus (see Theorem 4.8 in Chapter 2). The germ (q, (zo), zo) is the germ at thepoint zo of a phase trajectory of a node of the field v. For a system in general positionthe exponent of this node is not an integer and this trajectory is a nonsingular phasetrajectory of the node. Consequently, in the subcase of a zero-passing point, zo is atype 5 singular point of the boundary ,90+(z).

A folded stable saddle-node. Let zo E 8O+ (z) be a singular point of the type of afolded saddle-node. By Theorems 4.4 and 4.8-4.10 in Chapter 2, z is a zero-passingpoint at which the vector field v determined by the corresponding isolated valueof the control parameter has a nondegenerate stable node. In this case none of theeigendirections of the linearization of v at the point zo can be determined by the vectorsof the cone of this point. By assertion (2d) of Theorem 1.8, the germ (8O+(z), zo)coincides with one of the two germs (?1j (zo) U qa (zo), zo) or (q; (zo) U q (zo), zo),where the line, (z) is an outgoing separatrix. As in the previous section, in the caseof general position the exponent of a node of the field v at the point zo is a nonintegralnumber and, locally, in a neighborhood of the point zo, each of the lines

17(z)

belonging to 8O+(z) is a nonsingular phase trajectory of this node. Consequently,in the case of the first germ, zo is a type 5 singular point of the boundary 8O+(z).The line 17 (z) is a phase trajectory of a smooth feasible velocity field issuing fromthe point zo, the value of the field at the point zo determining the limiting direction atthis point. Hence, in the case of the second germ, zo is a type 4 singular point of theboundary 8O+ (z).

Thus, the first assertion of Theorem 1.6 is proved.

7.4. The completion of the proof of Theorem 1.6. It is clear that two NTZ's coincideif they have at least one common point. Consequently, the common points of theclosures of two different NTZ must lie on the boundary of each of them. By thefirst assertions of Theorems 1.7 and 1.7', the NTZ boundary belongs to the unionof the steep domain and the set of folded singular points (lying on the boundaryof this domain). A common point of two different NTZ's cannot be a point of theboundary of the steep domain because otherwise, by the third assertion of Theorem 1.9,these zones would have common points and, consequently, would coincide. Such apoint cannot be a point of a cycle of the limiting directions field either. Indeed,according to Theorem 1.2, in a sufficiently small neighborhood of this cycle, the NTZto whose boundary the cycle belongs is asymptotically stable, and, consequently, thecomplement of this zone in the neighborhood cannot contain points belonging tosome other NTZ. Finally, such a point cannot be a point of the steep domain notlying on a cycle of the limiting directions field. Indeed, otherwise, by the second andfifth assertions of Theorems 1.7 and 1.7' and the first two assertions of Theorem 1.9,one of the limiting lines passing through this point would be a double separatrix. ByTheorem 1.5 and condition (A) of Theorem 1.3, there are no such separatrices.

Hence, for a system in general position the closures of two different zones ofnonlocal transitivity are disjoint. The second assertion of Theorem 1.6 is proved.

Let us prove the third assertion. A nearly identical transforming homeomorphismexists by Lemma 6.1. From the stability of a system in general position with respectto small perturbations (Theorem 4.16 in Chapter 2) and the stability of the field oflimiting directions (Theorem 1.5) it follows that the singularities of the boundariesof the NTZ's of the system are also stable. Furthermore, the singularities of the first

§7. SINGULARITIES OF THE BOUNDARY OF THE ZONE 133

three types of the NTZ boundary do not contain moduli in their normal forms relativeto smooth transformations of coordinates, whereas the singularities of the last threetypes contain them (the number a is a modulus). Consequently, the transforminghomeomorphism can be chosen so that it is a C°°-diffeomorphism everywhere except,possibly, at the singular points of types 4, 5, and 6 of the NTZ boundaries.


CHAPTER 4

Attainability Boundary of a Multidimensional System

In the chapter we showed that for a typical system on a surface the closure of theattainability (controllability) set is a manifold with (possibly empty) boundary, and thefinite list of singularities that this boundary can have was found. For multidimensionalsystems the classification of the typical singularities of the attainability (controllability)set on its boundary is not yet known. However, in the case of general position theprospect of such a set looks rather good. In this chapter we show that for a typicalmultidimensional system the closure of the attainability (controllability) set is also amanifold with (possibly empty) boundary. It turns out that this boundary is a locallyHolder hypersurface in the phase space even for a typical bidynamical system, i.e.,when the number of different values of the control parameter is the smallest possible.The main results are stated in §1, and later they are proved.

§1. Definitions and theorems

We first define the class of systems under study and then state the basic results.

1.1. The class of systems. As in the two previous chapters, the phase space of asystem is a smooth manifold M without boundary. However, in this case its dimensioncan exceed unity. The set U of values of the control parameter is a topological subspaceof the space R" with the conventional topology and, as before, it consists of at leasttwo different points. The control system itself is determined in the neighborhood ofeach point in the phase space (as in the two previous chapters) by a mapping

(z, u) - f (z, u),

where z is a point in this space, u is the control parameter, and f (z, u) is a vectorbelonging to the tangent space T: M. Let k be a nonnegative integer. A controlsystem is of class Ck, k > 0, if the mapping f is k times continuously differentiablewith respect to the variable z (which exactly means that there exist derivatives of thismapping up to order k inclusive, and each of them is continuous with respect to theset of variables z, u). We endow the space of systems with a fine Ck-topology. Thecloseness of two systems in this topology means the following: their derivatives withrespect to the phase variable up to order k inclusive are close at all points of thespace of the variables z, u, and the closeness of these derivatives can be arbitrarilywell controlled at infinity. A typical system or a system in general position is one thatbelongs to an open everywhere dense set in this space with the indicated topology. Wenote that for polydynamical systems (i.e., for those whose sets of control parametervalues consist of a finite number of points) the introduced topology coincides with thefine Ck-topology.

Feasible controls and motions, the attainability of a point from another point,and the positive and negative orbits of a point are defined as before (see Section 1.1

135

136 4. ATTAINABILITY BOUNDARY OF A MULTIDIMENSIONAL SYSTEM

in Chapter 2). The positive orbit of the start set (i.e., of the set of points where themotion may start) is called the attainability set, and the negative orbit of the objectiveset (i.e., the set of points the system should reach) is called the controllability set.

ExAMPLE. Consider a swimmer drifting in a planar sea 1182,y carried by a watercurrent with velocity field (-x, -fly), fl > 2 (see the example in Section 1.3 of Chap-ter 1). The swimmer can swim in still water in any direction with the unit velocity.The possibilities of the swimmer are described by the control system x = -x + cos u,y = -y + sin u, where u is a circular angle. If the start and objective sets are in theneighborhood of zero (e. g. in a fl-neighborhood of zero), then the attainability setcoincides with the nonlocal transitivity zone, and the controllability set is the entirephase space of the system.

REMARK. All the results in this chapter are stated for the attainability set. Theycan automatically be extended to the controllability set. The point is that when thedirection of motion changes (time reversion), the attainability and controllability setsinterchange if the start and objective sets coincide.

1.2. The Lipschitz chartacter of the attainability boundary. Recall that a function gin the space R" satisfies the Lipschitz condition (or is Lipschitzian) if there is a constantC > 0 such that for any two points z(1) and z(2) in the space the inequality

lg(Z(2)) - g(Z(I))I < CIz(2) - Z(1)l

holds, where 1Z(2) - z(1)I = ((z 2) - zj1))2 + 1222) - z2I))2 +... + (Z(11 2) -By the epigraph of a function y = g(z) is meant the set y > g(z) in the space of

the variables y, z.

THEOREM 1.2. For a control system, the germ of the closure (interior) of the attain-ability set at a point z° of the boundary of this set is C°°-diffeomorphic to the germat zero of the epigraph (interior of the epigraph) of a function satisfying the Lipschitzcondition if the linear hull of the velocity indicatrix of the point coincides with the tangentspace, i.e., has maximum dimension.

This theorem is proved in Section 3.1. We note that the class of smoothness ofthe system is not indicated. For the assertion of the theorem to hold it suffices thatthe fields of the system be continuous. No constraints are imposed on the start seteither. It can be any nonempty subset in the phase space. A hypersurface in the phasespace is said to be locally Lipschitzian if in the neighborhood of each of its points itcoincides with the graph of a function z,,, = g (zI, Z2, ... , z,,,_ 1) satisfying the Lipschitzcondition in a suitable smooth system of local coordinates z1, Z2, ... , z,,, with origin atthat point. Theorem 1.1 immediately implies

COROLLARY 1.2. For a control system the attainability boundary is a locally Lip-schitzian hypersurface in the phase space if at each point of this boundary the linear hullof the velocity indicatrix coincides with the tangent space.

1.3. The Holder character of the attainability boundary. We shall say that a functiong in the space R_' satisfies a quasi-Holder condition (or is quasi-Holder) if there exist

(1) a set a = (a1, a2, ... , a,,) of positive exponents,(2) a set c = (c1, c2, ... , of nonnegative constants, and(3) a continuous family of smooth coordinate systems z1(z), z2(z), ... , z (z) in

this space with origin at the point z and parametrized by z such that for any

§1. DEFINITIONS AND THEOREMS 137

two points z(') and z0) in the space the inequalityit

Ig(z(2)) - ci 1(z, )(z(1))lar

i=1

holds.It is clear that a quasi-Holder function is Holder (i.e., Ig(z(2))-g(z(1))I < cIz(2)-z(1) I0for a constant c > 0 and an exponent a, 0 < a < 1) and vice versa. However, for n > 1the sets of exponents and constants reflect the properties of the function more preciselythan the Holder exponent and constant. For example, the functions g(x, y) = x2/3

and h(x, y) = (x2 +y2)1/3 have the same Holder properties but different quasi-Holderproperties.

l

Denote by ad) X = [Y, X] the Poisson bracket of two differentiable fields X andY. For a control system of class CZ a point z in the phase space is said to be regular ifthere exist pairwise distinct fields of feasible velocities X°, X2,. .. , Xi (correspondingto some pairwise distinct values of the control parameter) such that the dimension ofthe linear hull of the set of vectors

(1) {X°(z);(ad''X,.)(z), 0<i <k, 1 <r <1}

is equal to the dimension of phase space.It turns out that the boundary of the attainability set in the neighborhood of each

of the regular points of a system belonging to the boundary is the graph of a quasi-Holder function in a suitable system of local coordinates. More precisely, we shall showthat in the neighborhood of such a point the set of quasi-Holder exponents can bedefined by the following construction. Substitute the index s for k into (1) and denoteby,u. the dimension of the resulting set of vectors less by unity, 0 < s < k. Let m be thedimension of phase space. Forgo = m - 1 the linear hull of the velocity indicatrix ofthe point z has maximum dimension. Consequently, by Theorem 1.1, the attainabilityboundary in the neighborhood of this point is a locally Lipschitzian hypersurface in thephase space (and, in particular, it satisfies the quasi-Holder condition with exponentset a(z) = ( 1 , 1, ... , 1) and some set of constants). For po < m -1 we define µ_ 1 = 0,(ap, _ 1 + 1 = ap, _, + 2 = = a,, = (i + j) -1, 0 < i < j, where j is the smallestindex such that p3 = m - 1 ; a(z) = (al, a2, ... , a,,,_I).

THEOREM 1.3. For a control system of class Ck the germ of the closure (interior)of the attainability set at a point z° of the boundary of this set is Ck-diffeomorphic tothe germ at zero of the epigraph (interior of the epigraph) of a function satisfying thequasi-Holder condition with the set of exponents a(z°) and the set of constants if thispoint is a regular point of the system.

Theorem 1.3 is proved in Section 4.1. When the conditions of this theorem aresatisfied, we shall say the attainability boundary in the neighborhood of the point z°is a quasi-Holder hypersurface in phase space. Theorem 1.3 immediately implies

COROLLARY 1.4. The closure of the interior of the attainability set coincides withthe closure of the set itself and is a manifold with (possibly empty) boundary, which is aquasi-Holder hypersurface in the phase space in the neighborhood of each of its points ifall the points of the boundary of this set are regular points of the system.

1.4. Regular systems. A control system is said to be regular if each point in itsphase space is a regular point of the system.


THEOREM 1.4. Regular systems form an open set in the space of systems of class Ck.This set is dense in this space if the number of different values of the control parameter isno less than 1 + 1 and k > (2m - 1)/1 - 1.

The assertion of this theorem is analogous to the results of Lobry [L2] and Reb-huhn [Re] but differ in the class of systems and the topology in the space of systems.Thus, for #U > 1 + 1 and k > (2m - 1) /l - 1 regular systems are typical in thespace of systems of class CZ. In particular, they are typical in the space of systems ofclass C° if the number of different values of the control parameter is at least twice thedimension of the phase space. Theorem 1.4 is proved in §2.

§2. Typicality of regular systems

Here we prove Theorem 1.4. We first prove (Section 2. 1) the set of regular systemsis open and then (Section 2.2) that it is the dense.

2.1. The openness of the set of regular systems. It suffices to show that any systemsufficiently close to a regular one is also regular. Vectors in (1) depend continuouslyon the point z because each of the velocity fields X°, X1, ... , X1 is k times continuouslydifferentiable. Consequently, if for a system f the linear hull of these vectors at apoint z = z° has a maximum dimension, then it also has the greatest dimension at anypoint z belonging to the closure of a neighborhood V(z°) of the point z°. However,the vectors (1) depend continuously on the fields X°, X1, ... , Xl as well. Therefore,there is a 8(z°) > 0 such that for the system f the linear hull of the correspondingvectors in (1) has maximum dimension at each point of the neighborhood V (z°)if in this neighborhood the derivatives up to order k inclusive of the velocity fieldsX°, X1, ... , Xi, of this system differ from the corresponding derivatives of the velocityfields X°, A'1,. .. , Xj by at most 6(z°). Let us select at most countable locally finitesubcovering V (z 1), V (z2), ... , V (z"), ... , from the covering { V (z°) } of the phasespace. Define a function A: A(z) = min{b(z`)Iz E V(z')}. The function 0 is positive.We consider the systems whose derivatives with respect to the phase variable up toorder k inclusive differ from the corresponding derivatives of the system f at any pointof the space of the variable z, u by less than 0(z). They form a neighborhood of thesystem f. By the construction of the function A, all the systems belonging to thisneighborhood are regular.

Hence, regular systems form an open set in the space of systems of class Ck (in afine Whitney Ck-topology), and the first assertion of the Theorem 1.4 is true.

2.2. The density of the set of regular systems. It suffices to prove the assertion ondensity for polydynamical systems with 1+ 1 values of the control parameter. Indeed, onfixing 1 + 1 pairwise distinct values u°, u 1, ... , ul of the control parameter we can define,proceeding from a control system f °, a polydynamical system X = {X°, X1, ... , X, }.

Here X; is the feasible velocity field corresponding to the value u' of the controlparameter. If the perturbed polydynamical system X + v = {X + v°, Xl + vl , ... , Xl +V11, where v°, vl, ... , v1 are vector fields of class Ck, is regular, then the control system

f = f° + Ev,w(IU - u;12/e)-o

is also regular, where ,u is a smooth function on the real line and which vanishes outsidethe closed interval [-2,2], equals one on the interval [-1, 1], and is strictly monotone

§3. THE LIPSCHITZ CHARACTER OF THE ATTAINABILITY BOUNDARY 139

on each of the intervals [-2,-1] and [1,2]; e > 0 is a number small enough for the e-neighborhoods of the points uo, U!,. .. , ui to be pairwise disjoint. The absolute valuesof the derivatives of the difference f - f of the systems with respect to the phasevariable do not exceed the sum of moduli of the corresponding derivatives of the fieldsvo, v1, . . . , vi. Consequently, the set regular systems is dense (in fine Ck-topology)in the space of control systems of class CZ with at least 1 + 1 values of the controlparameter if it is dense (in fine Ck-topology) in the space of polydynamical systemswith 1 + 1 feasible velocity fields.

Furthermore, it suffices to show that in the space of smooth polydynamical systems(with fine Ck-topology) the set of regular systems is dense because a vector field ofclass Ck can be approximated arbitrarily accurately in Ck-topology with a smoothvector field [Hi].

Now we consider the space of smooth polydynamical systems X = {Xo, X1,Xl }, where Xo, X1, . . . , X1 are smooth feasible velocity fields. We define two types

of submanifolds in the space of k-jets of these systems. A submanifold Q, , of thefirst type is defined by the equations X, = 0 = Xj and corresponds to the case whenthe fields X; and Xj, 0 m, where m is the dimension of the phase space. Fori c {0, 1, ... ,1} we consider the matrix A, (of dimension m x (1k + 1 + 1)) of thevectors X,, ad', Xj, 0 < s < k, 0 < j < 1, j i. Denote by Q, the set of all k-jetssuch the rank of this matrix is less than m and X, 0. In the set of jets with X, 0,Q, is a closed Whitney stratified submanifold of codimension (k + 1)1 - m + 2. Thiscodimension exceeds m because, by the condition, (k + 1)1 > 2m - 1. This Q, is asubmanifold of the second type.

By the jet transversality theorem, the image of the k-jet extension of a systembelonging to an everywhere dense set in the space of polydynamical systems in fineCk+ I -topology and, consequently, in fine Ck-topology, does not intersect any of thesesubmanifolds. But such a system is regular because the rank of at least one of thematrices A0 and AI is equal to m. Consequently, the set of regular systems is dense inthe space of polydynamical systems of class Ck with at least 1 + 1 values of the controlparameter if k > (2n 1) - 1.

§3. The Lipschitz character of the attainability boundary

In this section we prove Theorem 1.1. We first define two fields of cones, onepositive and one negative, in a neighborhood of a point of the attainability boundaryand then prove the theorem using the properties of these fields.

3.1. Proof of Theorem 1.1. Let the linear hull of the set of feasible velocities at apoint z of the attainability boundary have the maximal dimension. The assertion ofthe theorem being local, we assume, without loss of generality, that M = R"' and thatthe point z coincides with the origin 0 of the coordinate system. It is clear that thereare values u 1 , u2, ... , u,,, of the control parameter such that the values at the point 0of the corresponding feasible velocity fields (VI, V2, ... , v,,,, respectively) are linearlyindependent.

For a point z we denote by R(z) the intersection of the positive linear hull of theset of vectors vI (z), v2(z), ... , v(z) with unit sphere in the tangent plane TR... = R"'.In a neighborhood of zero the set R(z) depends continuously on the point z becausethe fields VI, v2, ... , v,,, are continuous and linearly independent. Moreover, the setR(0) as a subset of the unit sphere has a nonempty interior. Consequently, there is


an e > 0 such that the intersection R of the sets R (z) over all points z belonging tothe e-neighborhood B£ of zero contains in its interior the closure of an open ball Kon the unit sphere with center at a point v of this sphere and radius r > 0. The pointsof the ball K are unit vectors in the space IR"'.

For a point z of the ball BE we denote by K+(z)(K-(z)) the intersection of thisball with the set {z + tw I w E K; t > 0 (accordingly, t < 0)}. By the choice of the setK, the cone K + (z) (K - (z)) belongs to the interior of the positive (negative) orbit ofthe point z. We shall use the following

LEMMA 3.1. For a point z belonging to the intersection of the attainability boundarywith the ball BE, the cone K+(z) belongs to the attainability set and the cone K- (z)belongs to the complement of this set.

This lemma is used to prove Theorem 1.1. Draw a hyperplane A passing throughthe origin and orthogonal to the vector v. For a point 2 E A consider the line segmentI (A) = {2 + tv I It I < e/2}. It can easily be seen that there is a 6 > 0 such that forIAI < 6 the point 2 + ev/2 (A - ev/2) belongs to the cone K+ (0) (K- (O)). However,by Lemma 3.1, the cone K+(O) belongs to the attainability set and the cone K-(O)belongs to the complement of this set. Since the attainability boundary is closed, weconclude that for I A I < 6 the intersection of the line segment I (A) with the boundary isnonempty. Take a point z (A) belonging to this intersection. According to Lemma 3.1,the cone K+(z(2)) belongs to the attainability set and the cone K-(z(2)) does notintersect this set. Hence, z (A) is unique, and the mapping 2 - z (A) is Lipschitzian forIAI < a.

Theorem 1.1 is proved modulo Lemma 3.1.

3.2. Proof of Lemma 3.1. The positive orbit of a point 2 of the attainability setbelongs to the attainability set. Consequently, for z E BE the cone K+(i) belongs tothe attainability set. However, the field K+ (z) is continuous on BE , the cone K+ (z)itself is open, and the point z belongs to the attainability boundary. Consequently,for any point E K+ (z) there is a point 2 E BE of the attainability set such that( E K+(z). Thus, the point ( and, consequently, the entire cone K+(z), belongs tothe attainability set.

Furthermore, if the intersection of the cone K - (z) with the attainability set con-tains the point C, then the open set K+(() f1 K- (z) belongs to the attainability set.Therefore the point z is contained in the interior of the attainability set. This contra-dicts the condition of the lemma. Consequently, there is no such point , and the coneK-(z) lies in the complement of the attainability set.

Lemma 3.1 is proved.

§4. The quasi-Holder character of the attainability set

In this section Theorem 1.3 is proved. We first define in the neighborhood of apoint z° two fields of cones, one positive and one negative, and then prove the theoremusing the properties of the cones belonging to these fields.

4.1. Proof of Theorem 1.3. For j = 1 this theorem follows directly from Theo-rem 1.1 proved above. Assume that j > 1 (and consequently k > 1).

Since the assertion of Theorem 1.3 is local, we can assume, without loss of gener-ality, that M = IR"' and that the point z° coincides with the origin 0 of the coordinatesystem. We shall need the following

§4. THE QUASI-HOLDER CHARACTER OF THE ATTAINABILITY SET 141

LEMMA 4.1. Under the conditions of Theorem 1.3 in a neighborhood of the point z°there exist (1) a continuous family of coordinate systems of class C':

,-1x(z1) =x(z°,') -x(z°,z1), yi(z1,.) = Eaij(zI)(yj(z°) -yi(zo,z1)),

j=1

where i, j E 11, 2, ... , m - 1 } and ai j are continuous functions parametrized by the pointz 1, and (2) constants C > 0 and E > 0 such that the positive and the negative orbit of anypoint z I E BE contain the cones K+ (z 1) and K- (z 1), respectively, where

BE = {Z I x2(z°, z) + yl (z°, z) +... + y2,_1 (z°, z) < E2

KI±1(z1) = {z I lyi(zl, z) I"' < cIx(z1, z) 1, 1 0} n BE.(<)

We note that the origin of the coordinate system x (z 1, ), yl (z 1, ), ... , y,,,_ I (z 1, )

is at the point z 1. Hence, the vertex of the cone KI±1 (z 1) lies at this point. We use thisfamily of coordinates, the constants C and e, and the ball BE introduced in Lemma 4.1.

LEMMA 4.2. For a point z belonging to the intersection of the attainability boundarywith the ball BE the cones K+(z) and K-(z) belong to the attainability set and itscomplement, respectively.

We do not give the proof of Lemma 4.2 as it is almost literally coincides with theproof of Lemma 3.1. Lemma 4.1 is proved in Section 4.2. We now use Lemmas 4.1and 4.2 to prove Theorem 1.3.

Take a point y in the plane x(z°, z) = 0 and a number 6 > 0 so small that fora pointy c B,5 the point (+ e/2, yl (z°, y), y2 (z°, y), ... , y,,,_ I (z°, y)) lies in the cone

KI-1(z°) and, by Lemma 4.2, it belongs to the attainability set (the complement of theattainability set). Since the attainability boundary is closed, we conclude that the linesegment {z I yi (z°, z 1) = yi (z°, y), 1 < i < m - 1, Ix (z°, z') I < e/2} intersects it.Take a point z(y) in this intersection. According to Lemma 4.2, the cones K+(z(y))and K- (z (y)) belong to the attainability set and its complement, respectively. In viewof the choice of the coordinate system and the construction of the cones, this impliesthat z (y) is unique and for a sufficiently small y the mapping y H z (y) satisfies theHolder condition with the sets of exponents a(z°) and constants c = ( 1 , 1, ... , 1)/c.

Theorem 1.3 is proved modulo Lemma 4.1.

4.2. Proof of Lemma 4.1. Set 8i = ,u1 - ui _ 1, 0 < i < j. By the definition of thenumbers ,u,, there exist sets of indices Ii = {ri, r2, ... , rd. } (where rs E {0, 1,... ,1})such that the vectors X,, (z°), (ad',1 X,.) (z°), r E Ii, 0 < i < j, are linearly independent.

Set V (z) = E;=0 /3i Xi (z). We shall need the following

LEMMA 4.3. There are positive numbers /3i, 0 < i < 1, such that their sum is equalto one and the vectors

(2) V(z°), (ad' X,.)(z°), r c Ii, 0 < i < j,

are linearly independent.

We do not prove this lemma, since it is an immediate consequence of the continuousdependence of the field V on the set 8 = {/3o, /31, ... , /3r } and the vector fields ad 'V X,.on the field V.


Take the numbers whose existence was established in Lemma 4.3. The field Vis of class Ck. At the point z° the value of this field is nonzero because all the vec-tors in (2) are linearly independent. Consequently, there exists a coordinate systemx 1, Y1, y2, ... , y»,-I of class Ck in R"' with origin at the point z° such that in a neigh-borhood of this point the field V is constant and is equal to a/ax [A3]. This coordinatesystem is not unique. We can select it in a more special way. As the unit coordinatevectors of the axes y, we consecutively take the remaining vectors in the list in (2). Letus show that this coordinate system can be included as one of the required family ofcoordinate systems and that the required constants c and a exist.

It follows from the choice of /3 that for sufficiently small numbers 2o, . . . , A, thevector

I

v(z,A) = V(z) +E2;(Xi(z) - V(z)),i=o

A = (20, Al' ... , Al), belongs to the convex hull of the set of feasible velocities at thepoint z. The fields X,, 0 1. Consequently,when admitting motion with velocity fields v(., A) for small A, we do not change theinteriors of the positive (negative) orbits of the points because under the transition tothe convex hulls of the sets of feasible velocities of a control system of class C' theseinteriors do not change [Pi]. We set Y, X, V Y = (Yo, YI, ... , Y,).

Take T, a > 0 and consider a motion starting at the point z 1. It begins at a velocityV(z') and continues for a time interval s(T + a) < t < s(T + a) + T(s(T + a) + T <t < (s + 1) (T +a)) at a velocity field V (V + YAs ), 0 < s < j. Denote by Z (Z", r, a, A),where A = (A°, A' , ... , 2j), the point at which we arrive at the end of the motion.

LEMMA 4.4. If the point z' is sufficiently close to the origin and the numbers T and aare sufficiently small, then

(3) z(zT) a,A) = z' + (j + 1)(T +a)V(0)

+ 11

(T(s i 1))1'a''Y(zi)+aY(z')s.a+O(Iarj)+o(rj)a

v a ax

i-I+

((r(s+1))" a' +o(Ti-1) a2 + O(IAIQ2) +0(0.2) 2s

I-0 2v! TX

s=0 L \V=0

We can use this expansion to prove Lemma 4.1. Set T = rj and rewrite (3) in theform

(4) z (z 1, T, Ti, A) = z' + (j + 1)(-c + T1) V (O)

I'' t 1 1+ j J T,,+JJa'

Y(z1)(s + l)L ax V.

S+ 2CS=0 J'=0

+0 (T2j) + O(IAIT2j)

where 8;' is the Kronecker delta (8;' = 1 for v = p and 8P = 0 for v 34 p). We put

((S_+61 s + - TJ - l S.

2=o

§4. THE QUASI-HOLDER CHARACTER OF THE ATTAINABILITY SET 143

For T > -1 this linear transformation of the A-space is a change of variablesbecause it has a matrix with nonzero determinant which differs from the Vandermondedeterminant of the numbers 1, 2, ... , j + 1 by a nonzero multiplicative constant. Thetilde over the coordinates is omitted.

Set .1;' = 0 f o r t- 54 I,,, 0 < v < j. The functions v (.) and r (.) on { 1 , 2, ... , m - 1 }

are defined in such a way that the unit vector of the axis y; is ad<(') Y.(1)(0), where

r(i) E I,,(;) (see (2) and the definition of Y). Set y; = y = (yI, y2, ... , y,,,_1).

Expansion (4) is now rewritten as

z(zI,T,Ti,Y) =z' + (j + 1) (T +T')V(0)nr-I ,.(i)+ Tt,(i)+j a Y,.(;) (zI) + 0(T2J) + 0(IYIT2i) Y;.

=1

The vector fields V, ad;/(') Y,.(;), 1 < i < m - 1, are continuous, and their valuesat the point z° form a basis in the space R"'. Consequently, the values of these fields atany point z' sufficiently close to the point z° also form a basis of this space. For suchpoints z' the last expansion can be rewritten in the form

z(zT,Ti,y) =z'((j + 1)(T+ri)+o(Ti))V(0)

+ T,(,)+ a y(' +oT (1)IYI + 0(1Y12)).t(1)

i=1

We now define in R'} a family of affine coordinate systems parametrizedby a point z' sufficiently close to z°. The unit coordinate vectors of such a coordinatesystem with origin at the point z' are the vectors V (O), 1; (z' ), where 1(z1) is theprojection along the axis x of the vector ad i $') Y (') (z 1) to the hyperplane x = 0 in

 0 and a+ > 0

such that the positive orbit of any point z I sufficiently close to the origin contains allthe points z representable as

(5)

,»-1

z = zI +TV(0)+ T''(')+jy;1;(Z1),;=1

where 0 < T < T+, yi + y2 + + y2i_1 < a+.Similarly, there are positive constants T- and a- such that the negative orbit of

any point z 1 sufficiently close to the origin contains the points z representable in theform of (5), where -T_ < T < 0, y? + y2 + + y2,_1 < a2 . However, we havea; = (v (i) + j) -'. Consequently, the constructed family of coordinate systems is thedesired one. It is easy to see that the required constants c and 6 exist.

REMARK. While proving Lemma 4.1 we used the methods applied in [He] to theanalysis of the attainability sets of analytic polydynamical systems.

4.3. Proof of Lemma 4.4. We take T and a so small that, when starting the motionfrom any point z I sufficiently close to the origin, we always remain in a neighborhoodof the origin where the field V is rectified. Consequently, the expansion term in (3)outside the summation has the required form. To obtain expansion (3) it suffices toobtain exactly the same expansions for one "step aside", when all the 2"s except one


are zero, and then to combine these expansions. All the terms resulting from theinteraction of two or more "steps aside" give the contribution representable in theform =o (O, (1AIU2) + 0, (a2))22. As to a single "step aside", the desired expansionis obtained for it by direct differentiation.

Let A ; 0, A' = 0 for i s. For a "step aside" z(t), s(T + a) + r < t <(s + 1) (r + a), we have the equation i = V (0) + Y(z)A'. Consequently,

4(S(T + a) -{- T) = V(0) + Y(Z1 + (S (C + a) + T) V(0))A5,

Z(s(T+U)+T) ax (Z1 +(S(T+U) +T)V(0))25+O(I2IS)As

Hence, by the Taylor formula, we obtain

z((s + 1)(T +U)) =z1 + (s + 1)(T +a)V(O) +aY(z1 + (S (-r +U) +T)V(0))As2

+ 2 ax(z1 +(s(T+U)+T)V(0))2s

+ O(I2IsU2)2s + o(U2)AS

Substituting in the above relation the

expansions

Y(z1 + (S (T +a) + T) V (O)((s + I )T)" a" Y (Z1) + aY (z1)sU

v ax axJ'=0

+ O(U2 + UT) + o(Ti),

aY(z1 + (s(T +U) +T)V(0)) JE at

tl+1( 1)(T(S + l))` + o(zf -1) + O(a),ax ax

zv

y=0

we get the required expansion for one "step aside".Lemma 4.4 is proved.

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