On the geometry of transport in phase space I. Transport in k-degree-of-freedom Hamiltonian systems, 2≤k

Physica D 44 (1990) 471-501 North-Holland

ON THE GEOMETRY OF TRANSPORT IN PHASE SPACE I. TRANSPORT IN k-DEGREE-OF-FREEDOM HAMILTONIAN SYSTEMS, 2 < k < oo

Stephen WIGGINS 1 Center for Nonlinear Studies, M.S. B258, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Received 15 February 1989 Revised manuscript received 2 March 1990 Accepted 3 March 1990 Communicated by R.S. MacKay

Over the past several years the effect of geometrical structures such as KAM tori, resonance zones, and cantori on phase space transport in two-degree-of-freedom Hamiltonian systems has been studied. In three-degree-of-freedom systems and higher, these structures no longer form complete or partial barriers to transport in the sense that they or their associated stable and unstable manifolds are no longer codimension one in the level set of the Hamiltonian. In this paper we show that in the ( 2 k - D-dimensional level sets of the Hamiltonian of a class of k-degree-of-freedom Hamiltonian systems the ( 2 k - 2)-dimensional stable and unstable manifolds of normally hyperbolic invariant ( 2 k - 3)-dimensional spheres form partial barriers to transport. Under certain conditions, we show that lobes and turnstiles can be formed from such geometrical structures in a way that is exactly analogous to the situation for two-degree-of-freedom Hamiltonian systems. Considering a foliation of the level set of the Hamiltonian into disjoint regions separated by such partial barriers, we can then give exact formulas for the transport of volumes of phase space throughout the different regions based on the dynamics of these generalized lobes and turnstiles. On the other hand, despite the fact that these partial barriers are codimension one, for k-degree-of-freedom systems, k > 3, they may not intersect in such a way so as to divide the phase space.

1. I n t r o d u c t i o n

Global transport in phase space in classical and quantum Hamiltonian systems is a rapidly evolv- ing subject which impacts on many different disci- plines. For example, such questions are involved in the study of reaction rates in molecular dynamics [14] and in the study of chaotic advection in fluid mechanics [17, 20].

Over the past 30 years remarkable progress has been made in understanding the geometrical structure of the phase space of multi-degree-of- freedom Hamiltonian systems. For example, the celebrated KAM theorem [1] states that the phase space of systems that are perturbations of integrable Hamiltonian systems contains families of

1Permanent address: Applied Mechanics, 104-44, Caltech, Pasadena, CA 91125, USA.

0167-2789/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)

invariant tori that are densely covered with orbits. Resonance zones consisting of alternating hyperbolic and elliptic periodic orbits may exist near the KAM tori. Associated with the stable and unstable manifolds of these hyperbolic periodic orbits may be complicated "stochastic" dynamics similar to those of the Smale horseshoe; see refs. [10, 11, 15, 24] for more background. The fact that the phase space can contain such geometric structures implies that theories of transport of phase space relying on assumptions of ergodieity are incorrect and must be modified to include the effects of these geometrical barriers in the phase space.

Over the past several years work along these lines has been carried out by MacKay, Meiss and Percival [12, 13] and Rom-Kedar and Wiggins [19]. All of this work, however, has been in the

472 S. Wiggins / The geometry of transport in phase space I

context of two-degree-of-freedom Hamiltonian systems. Such systems may, by a standard proce- dure, be reduced to two-dimensional area-preserving Poincar6 maps. For these maps, KAM tori are manifested as invariant circles which are complete barriers to transport in the phase space (i.e. points starting "inside" the area bounded by a KAM torus will never get out). The stable and unstable manifolds of the hyperbolic periodic orbits associated with resonance zones are one- dimensional and may intersect forming heteroclinic tangles. These heteroclinic tangles form partial barriers to transport in phase space. Addi- tionally, a relatively new type of invariant set called a c a n t o r u s (since it has the geometrical structure of a cantor set and the dynamics of a KAM toms) creates a partial barrier to transport in phase space. The work of MacKay, Meiss and Percival [12, 13] and Rom-Kedar and Wiggins [19] is concerned with the transport in phase space between regions separated by partial barriers.

Attempts to extend these ideas to k-degree-of- freedom Hamiltonian systems, 3 < k < oo, have not proven successful since in these systems KAM tori do not divide the phase space (i.e. the level sets of the Hamiltonian) into an inside and an outside, the dimensions of the stable and unstable manifolds of hyperbolic periodic orbits are too small for the resulting heteroclinic tangles to form partial barriers, and a general theory of cantori in multi-degree-of-freedom Hamiltonian systems does not at present exist. Yet there is a desperate need to extend these techniques to higher dimensions since, for example, in the study of molecular dynamics the most simple physically realistic model of a molecule has at least three degrees of freedom [14].

In this paper we develop the mathematical framework necessary for extending the techniques of transport in two-degree-of-freedom Hamiltonian systems to a large class of k-degree- of-freedom Hamiltonian systems, 3 < k < oo. We show that normally hyperbolic invariant (2k - 3)- dimensional spheres having (2k - 2)-dimensional stable and unstable manifolds in the ( 2 k - 1)-

dimensional level set of the Hamiltonian play the same role in k-degree-of-freedom Hamiltonian systems, 3 _< k < o0, as hyperbolic periodic orbits (i.e. 1-spheres) and their associated stable and unstable manifolds in creating partial barriers to transport in two-degree-of-freedom Hamiltonian systems.

This paper is organized as follows. In section 2 we review the essential ingredients from the theory of transport in two-dimensional area-preserving maps. In section 3 we discuss the mathematical framework for transport in k-degree-of- freedom Hamiltonian systems. In sections 3.1-3.3 we discuss a class of systems that gives rise to partial barriers to transport that are analogous to those arising in the theory for two-dimensional area-preserving maps. We define these partial barriers explicitly in section 3.4 and in section 3.5 we state the transport problem and formulas for the transport in phase space due to the dynamics of these partial barriers in a way that is completely analogous to the transport theory for two-dimensional area-preserving maps discussed in section 2. In section 4 we discuss an analytical technique for detecting such partial barriers to transport. In section 5 we give a three-degree-of- freedom example that illustrates the theory. In section 6 we discuss our need to consider perturbations of completely integrable Hamiltonian systems; in section 7 we discuss how the systems we consider and the associated partial barriers to transport may arise near resonances and in section 8 we discuss the relationship with Arnol'd diffusion. Most of the mathematical background as well as the existence, persistence, and differ- entiability theorems for invariant manifolds can be found in ref. [24].

2. Transport in two-dimensional area-preserving maps

The study of transport in externally forced time periodic one-degree-of-freedom (1-d.o.f.) Hamil- tonian systems and 2-d.o.f. Hamiltonian systems

S. Wiggins / The geometry of transport in phase space I 473

can be reduced to the study of transport in an associated two-dimensional area-preserving Poincar6 map via standard techniques [10, 11]. Transport in two-dimensional area-preserving maps has been recently studied by MacKay, Meiss and Percival [12, 13] and Rom-Kedar and Wiggins [19]. Our purpose here is to recall the general points of the two-dimensional theory recently developed by Rom-Kedar and Wiggins [19] and from that extract the key elements that we would need in order to develop a higher-dimensional theory.

The setup is as follows: we consider a map

F: ~,~--* , f , (2.1)

q I .... - - ~ L I

,wp,

Fig. l . qo, ql , q2 are pips. q3 is not a pip.

Definition 2.3. Let qo and ql be two adjacent pips, i.e., there are no pips on U[qo, q 1] and S[qo,ql], the segments of Wp~ and Wp~ which connect q0 and ql. We refer to the region bounded by the segments U[qo, ql] and S[q o, ql] as a lobe. See fig. 1.

where F is a C r, r>_ 1, orientation-preserving diffeomorphism of the differentiable two-manifold J into itself (note: recall that a map is orientation preserving if its Jacobian is everywhere positive on J . Poincar6 maps always pre- serve orientation, see ref. [24] for a proof). Let Pi, i = 1 . . . . . N denote N hyperbolic fixed points of F and we denote their stable and unstable manifolds by Wp~ and WpT, respectively, i-- l , . . . , N. (Note: This theory also applies to hyperbolic periodic points by considering the appropriate iterate of F for which they are fixed.) Before getting to the issue of transport we need to get some preliminary definitions out of the way.

Definition 2.1. A point q0 ~ J is called a heteroclinic point if it belongs to both a stable and an unstable manifold, namely q0 ~ Wp] n W." for Pj some Pi and pj. The point q0 is called homoclinic if i = j .

Definition 2.Z Consider a heteroclinic (or homoclinic) point qo ~ W~ N Wp~ and let S[pi, q o] denote the segment of W~ from p; to q0 and U[pj, q o] denote the segment of Wp~ from pj to q0. Then q0 is called a primary intersection point (pip) if S[p i, q0] and U[pj, qo] intersect only in qo (and possibly at Pi if i = j ) . See fig. 1.

We now consider the dynamics of lobes and how they move across a boundary formed by a piece of a stable manifold and a piece of an unstable manifold of a single hyperbolic fixed point or a pair of hyperbolic fixed points (note: we will use the case of two different hyperbolic fixed points for illustrative purposes).

2.1. Lobe dynamics and the mechanism for transport

Consider two hyperbolic fixed points Pl and P2 for which the unstable manifold of pl and the stable manifold of P2 intersect. Choose a pip qo ~ Wp~ N We2 and consider the segment defined by U[p~, %] U S[p2 , q0] - ~ - We label the two sides of ~ R 1 and R E (see fig. 2). We now want to discuss the dynamics of lobes defined by pips on ~ , and show that it is the dynamics of these lobes that is responsible for points crossing ~ .

We consider q0, F-l(q0), and F(qo). In ref. [8] it is shown that iterates of pips are still pips; therefore F-1(%) and F(qo) are pips. In general, between q0 and F-l(qo) there will be k pips ( k > 1) along U[pl, qo], denoted ql, q2" 'qk , which define k + 1 lobes, denoted L 0 up to L k.

This is illustrated in fig. 3 for k = 3. Without loss of generality, we may consider k = 1 by con-


R1

U [Pl' q o ] / . / /

/ /

~ B:U [P l , qo] U S[P2, q O]

R2

f~2" /

Fig. 2. The boundary, ~, between regions R 1 and R 2.

Lo-, k / - F (L 3 ) L 2 - k \ ,., / ~ F (L1)

L3-k L \ I 3 LF (L o ) -

Fig. 3. Lobes between q0 and F-l(qo) and their images under F.

sidering the union of all lobes which lie to either side of ~ as one lobe (see ref. [19] for more details). Thus we consider the situation in fig. 4, where we label the two lobes formed by q0, F - l ( q0 ) and the pip between these two points by L1,2(1) and L2,1(1). Note that the first subscript in this notation refers to the side of ~ on which the

lobe lies. By using the fact that a pip maps to a pip, and the interior of a closed set maps to the interior of its image under a diffeomorphism, we can conclude that lobes map to lobes and, more specifically, that F(L1,2(1)) and F(L2,1(1)) appear as in fig. 4. Thus L1,2(1) moves into R 2, while L2,l(1) moves into R 1 under one iteration of F (note: this explains the second subscript in our notation for the lobes, as well as the number in the parentheses). (Note: in the language of MacKay, Meiss, and Percival [12, 13], the lobes LI,2(1) and L2,1(1) are said to form a turnstile. Thus the movement of points across ~ under one forward iteration of F only occurs for points in either L1,2(1) or L2,1(1). Moreover, by the orientation preservation of F, the only way in which points can cross ~.~ is if they are in F-n(L1,2(1)) or F-n(L2,1(1)) for some n >_ 0. This is the key observation that enables us to develop a theory of transport which is based only on the

R1 LI,2(I) \ / F (L2,1(I))

L2'I(I)\ ~ qo -"~'~") /F(LI'2(1))

7"N /' X R2

Fig. 4. Dynamics of the turnstile lobes.


~'0

~Fql,, s ~

j / e

Fig. 5. Intersection of the turnstile lobes and redefinition of the turnstile.

dynamics of a small number of lobes in phase space, and appears to have been first pointed out by Channon and Lebowitz [5] and Bartlett [3].

Before proceeding further let us address a technical point. Suppose that the lobes between q0 and F - t ( q 0) intersect as in fig. 5. In this case we merely redefine the lobes by excluding the intersecting pieces which results in the new "lobes" labelled L 0 and /'1 in fig. 5. (See Rom- Kedar and Wiggins [19] for more discussion of this point.) Before formulating the main transport problem, we need a final definition.

Definition 2.4. A region is a simply connected domain of phase space with boundaries consisting of boundaries of the phase space and segments of stable and unstable manifolds starting at hyperbolic fixed points and ending at either pips or at the boundary of the phase space (which can be infinity). See fig. 6.

2.2. Formulation of the transport problem

Suppose J is divided up into N R disjoint regions, denoted Ri, i - - -1 , . . . , N R. Suppose that initially particles of species S i are uniformly dis- tributed throughout region R;, i --- 1, . . . , N R.

Problem. Compute the area occupied by species Si in region R i (for any i and j) at any specified later time.

!

qs RI qM i I

I I" R 4 ~jJ / R 3 ~

R I q2

Fig. 6. Examples of regions and their boundaries.

Let us comment on the notion of the "species of a point." The species of a point indicates the region in which the point is located initially, i.e. at t = 0. In many transport applications it is important to know at any time t > 0 where a given point was located initially. An example of this would be in the study of chaotic advection of fluids (see ref. [17]) where the fluid mechanical Lagrangian point of view is essential. The general theory developed in this section incorporates this information.

We remark that in practice it is highly problem dependent how one divides the phase space into disjoint regions separated by pieces of stable and unstable manifolds of hyperbolic periodic points. For example, in the study of transport in 2-d.o.f. Hamiltonian systems one might choose a finite collection of resonance bands (see ref. [13]) and in the study of chaotic advection one might choose


regions of the flow separated by stable and unstable manifolds of certain "stagnation points" in the flow (see ref. [20]). In any case, it is important to realize that the only way that points can move between such regions is if they are in the lobes defined by the tangling of the stable and unstable manifolds which bound the regions.

2.3. The main results

We now summarize the main results of the transport theory for two-dimensional maps developed by Rom-Kedar and Wiggins [19] (note: here we only give results for area-preserving maps since in this paper we are concerned with Hamil- tonian systems; however, the theory works just as well for dissipative systems). First we need some more notation.

Lk, j(n) = the lobe that leaves R k and enters Rj on iteration n.

L~d(n) = the portion of the lobe that leaves R k and enters Rj on iteration n that contains species

Si, i.e. Lk,j(n) (q R i.

Note that by definition (see also fig. 4) F n- l(Lk,j(n)) = Lk,j(1) and Lk,j(n) n Li,m(n) = C~ unless k = i and j = m since F"- l (Lk , j (n) ) R,,, Fn - l (L i ,m(n ) )ERi , F"(Lk, j ( n ) ) ~ R j and F"(Li , , , , (n))~R m and the regions Rk, Ri, Rj, and R,, are, by construction, disjoint. The main quantity that we can compute by knowing the dynamics of the lobes is

T,d(n) = the area occupied by species S i in region Rj immediately after the nth iteration.

If we know T~,j(n) then we also know the flux of species S i into region Rj on the nth iterate which is given as follows.

Flux of S i into Rj on the nth iterate = change in the amount of species S i in Rj on the nth iterate = T/ , j (n) - T/,j(n - 1).

Before giving the formula for T/,i(n) derived by Rom-Kedar and Wiggins [19] using the lobe dynamics, we want to make a comment regarding

the idea of flux. Recall the discussion of the motion of the lobes L j,2(1) and L2,~(1) across the boundary ~ at the beginning of this section. If we are interested in the amount of R 1 that moves into R 2 on one iteration and vice versa, it should be clear that this is just the area of the lobes Ll,2(1) and L2,1(1), respectively (note: this was first explicitly pointed out by MacKay, Meiss, and Percival [13]). However, the quantities T~d(n) allow one to compute the long time flux of points of a specific species since at any later time a given turnstile may contain many different species. The quantity T/,j(n) accounts for this.

The main results found in Rom-Kedar and Wiggins [19] are the following two formulas:

T/d(n)

NR n = Ti,j(0) + Y'~ 2 [ / * ( L ~ d ( l ) ) - t z ( L S , k ( / ) ) ] ,

k = l l = l

(2.2)

where Tid(0) = 0, i 4=j and T~,i(0) = MRi) with the notation ~(A) denoting the area of the set A for any A ~ o ~.

. ( O ) NR I--1

= )-" y" /Z(Lk,j(1 ) C~Fm(L,,~(1))) s = l m = 0

NR t-1 - E E /z(Lkd(1) AFm(Ls, i (1)) )"

s = l m = l

(2.3)

Formulas (2.2) and (2.3) express the amount of species S i contained in region Rj solely in terms of the dynamics of the turnstiles controlling access to the regions.

Additionally, we have the following conservation laws for the Tcj(n).

Conservation of species

NR 2 - - 1 ) ] = 0 ,

j = l

i = 1 . . . . . N R. (2.4)


R 1

q+...

q-

R 1

F-l(q'~ q+

Wp,_

F -l(q-)

R 3 R3

IDENTIFY =- = IDENTIFY =

R 1

q+

/ R 2 I \ ' ) o

R 3

IDENTIFY - ~

Fig. 7. Construction of regions and turnstiles.

Conservation of area and

E [ T / , j ( n ) - Ti,j(n- 1)] i= l

=0 ,

j = 1 , . . . , N R. (2.5)

We now show how the general formulae (2.2), (2.3), (2.4), and (2.5) apply in a specific example.

As an example we consider an area-preserving map of the cylinder having a hyperbolic fixed point denoted by p. For geometrical clarity we will represent the cylinder by identifying the coordinates x=2nar , n = 0 , + l , + 2 . . . . on the Cartesian plane having coordinates (x,y). We denote the two branches of the stable manifold of /9 by W~ + and W~ _. We suppose that W: + intersects WpU+ and W:_ intersects Wp~,_ as shown in fig. 7. This is a very generic situation arising in, for example, the standard map and the periodically forced pendulqm. The geometry has the form of the generic 1:1 resonance for area- preserving maps.

Choose pips q + ~ W~ + ~ Wp ~, + and q - ~ Wp~ _ n Wp ~, _. Then the segments

S i p , q - ] u U [ p , q - ] - ~ -

divide the cylinder into three disjoint regions. We denote the region above ~ + by R t, the region bounded by ~ + and ~ - by R~, and the region below ~ - by Ra, see fig. 7. Associated with ~ + is a turnstile formed by the segments of W~ + and 1~ u bounded by q+ and F-l(q+). We p, + denote the two lobes comprising the turnstile by Lt,2(1) and L2. l(1). Similarly, associated with ~ - is a turnstile formed by the segments of Wp~_ and Wp~_ bounded by q - and F- l (q - ) . We denote the two lobes comprising this turnstile by L2.3(1) and L3,2(1), see fig. 7.

Our goal is to elucidate the general formulae by computing Ti, j(n), i , j = 1,2,3, for this example. First, note that if the map is invariant with respect to 180 ° rotations (i.e. F(x, y ) - - F ( - x , - y ) ) , as is the periodically forced pendulum and the standard map, then regions R l and R 3 are equivalent in the sense that we have

=

T:,,(.) =

,,) = n). S [ p , q +] U U [ p , q +] - ~+ (2.6)


These three relations along with the five independent conservation laws given in (2.4) and (2.5) provide us with eight independent equations for the nine unknowns Ti, j (n) -Ti , j ( n - 1 ) , i , j= 1, 2, 3. Hence we need only compute one of the T,,s(n), i , j = 1,2,3, since the remaining can be determined from these eight independent equations. So, for definiteness, we will discuss T3, t(n). From (2.2) and (2.3) we can immediately write down

T3,1(n)

= ~ (n -m){/z(L2,,(1 ) AF"'(L3,2(1)) ) m=l

-/~(L2,,(1) AFro(L2,3(1)))

-/.l,(Ll,2(1) n Fm(L3,2(1)))

+p.(Ll,2(1) n Fm(L2,3(I)))}. (2.7)

Thus transport across the resonance band is quantified completely in terms of intersections of the images of the turnstile associated with ~ - with the turnstile associated with ~ + as described in (2.7). In particular, the following four

quantities are required:

L2, I(a) n F~(L3,z(1)), L2,I(i) A F"( L2,3(1) ), L1,2(1) n vm( L3,z( a ) ),

Lt,2(a) N Fm( L2,3( I ) ).

(2.8a) (2.8b)

(2.8c)

(2.8d)

In fig. 8 we show part of the homoclinic tangle. Note in the figure that for some integer k > 0 we have

Fk(L2,3(1)) NL3,2(1 ) 4= O,- (2 .9a)

Fk(Lz, t(1)) ALl ,z(1 ) =g 0 , (2.9b)

rk (L3 ,2 (a ) ) AL2,1(1 ) 4= ~ . (2.9c)

This provides a route for each of (2.8a), (2.8b), (2.8c), and (2.8d) to be nonempty. In particular, (2.9e) implies that (2.8a) is nonempty for all m larger than some m = ~ , (2.9a) and (2.9c) imply that (2.8b) is nonempty for all m larger than some m = ~ , (2.9b) and (2.9e) imply that (2.8c) is nonempty for all rn larger than some m = ~ , and (2.9a), (2.9b), and (2.9c) imply that (2.8d) is nonempty for all m larger than some m = ~ . Eqs. (2.2) and (2.3) give an exact expression for

R 1 R1 R1 t ,' F2(L21(1)) I I I

L2,1(1) ~.~. 2(1)~i q +F(L21(1))'/ ~ll . " 1 ¢ ~ .zF3(L2,1(1))Ir,

" t "---7; / ' , I F3(L2'3(1)) I F2(L2,3(1)) I

I

R3 j R3 I I f R3 I I I - ' i ~ IDENTIFY --;= IDENTIFY --i= IDENTIFY -'--'

Fig. 8. Some iterates of the turnstile lobes.

s. Wiggins / The geometry of transport in phase space I 479

the route that points must follow in moving from region to region for any partition of the phase space by segments of stable and unstable manifolds of hyperbolic periodic orbits. We stress that (2.7) represents an exact solution to the problem of transport between the regions R~, R 2, and R 3 in terms of the intersection of the turnstiles controlling access to the regions. In particular, no assumption of chaos or rapid diffusion is needed in order to compute the transport rates as is required in the MacKay, Meiss, Percival [12, 13] models of transport in phase space. Indeed, in the study of chaotic advection in fluids (see, e.g., refs. [17, 19]) such assumptions would be unten- able since the goal is to study the motion of fluid particle trajectories under convective as opposed to diffusive processes. Moreover, it is not necessary to find a complete partition of R~, R 2, and R 3 into stochastic regions as in the MacKay, Meiss, Percival [12, 13] transport model since we take into account the long-term dynamics of the turnstiles.

Now the obvious question is: How does one compute the quantities in the sum of (2.7), i.e. the area of the intersection of the images of the turnstile associated with ~ - with the turnstile associated with ~ + ? At present, the most widely applicable method is numerical computation. Even so, since one need only follow small segments of the stable and unstable manifolds of p this method still affords a substantial savings in computational effort over brute force or even Monte Carlo methods (see ref. [4]). Recently, Rom-Kedar [18] has developed techniques which allow one to achieve upper and lower bounds on quantities such as (2.8).

2.4. Generalization to higher dimensions

We now want to look at what would be needed in order to make the above theory of Rom-Kedar and Wiggins [19] go through in higher dimensions. Recall that in formulating the problem of transport for two-dimensional maps the only place dimension enters the picture is in the fact that

hyperbolic fixed points or periodic points of C r (r >__ 1) orientation-preserving diffeomorphisms of two-manifolds have codimension-one stable and unstable manifolds. The fact that the stable and unstable manifolds are codimension one means that they separate the phase space and can therefore be used to form boundaries defining disjoint regions of the phase space which act as partial barriers to transport. From pieces of the codimension-one stable and unstable manifolds lobes which "trap" regions of phase space are formed. The dynamics of the lobes then completely determine the transport amongst the different regions of phase space. Once lobes and regions are defined in this way the derivation of formulas (2.2), (2.3), (2.4) and (2.5) is completely independent of dimensional considerations; it is just an arduous exercise in set theory and measure theory using invariance of the manifolds. Thus the main point is this; for k-d.o.f. Hamiltonian systems, k > 3, if we can find a normally hyperbolic invariant set (a higher-dimensional analog of hyperbolic periodic points for two-dimensional area-preserving maps) having codimension-one stable and unstable manifolds then it may be possible that regions, lobes, and turnstiles can be formed in (essentially) the same way as described above and formulas (2.2), (2.3), (2.4) and (2.5) will apply exactly for ( 2 k - 2)-dimensional volume-preserving maps derived from k-d.o.f. Hamiltonian systems (note: in this case/~(A) is interpreted as the volume of the set A in ( 2 k - 2)-dimensional space). It should be clear that the stable and unstable manifolds of hyperbolic periodic points are no longer the cor- rect structure on which to build the theory, since for a (2k - 2)-dimensional volume-preserving map a hyperbolic periodic point will have ( k - 1)- dimensional stable and unstable manifolds. These stable and unstable manifolds are codimension one (i.e. they divide the space) only for k = 2.

We now describe a class of perturbed completely integrable k-d.o.f. Hamiltonian systems, k > 3, which does have a normally hyperbolic invariant set (which is the analog of the hyperbolic periodic orbit in the 2-d.o.f. case) having


codimension-one stable and unstable manifolds which under certain conditions can be used to develop a transport theory which is identical in structure to that described above for two-dimensional area-preserving maps.

where (x, u, v) e R 2 x R" x R",/~ e R p is a vector of parameters, and 0 < E << 1. This Hamilto- nian gives rise to the Hamiltonian vector field

Y c = J D x H ( x , u , v ) + e J D x I 4 ( x , u , v , g ; ~ ) ,

3. The mathematical framework for transport in k-d.o.f. Hamiitonian systems, 3 < k <

it = D~,H( x , u , v ) + eDvt:I( x , u , v , l z ; E ),

= - Du l l ( x , u, v) - ED,/4(x, u, v,/z; e),

We now arrive at the main part of this paper. We begin by describing the mathematical structure of the perturbed k-d.o.f, completely integrable Hamiltonian systems, 3 < k < 0% that we are considering. In particular, we pay close atten- tion to the relationship between geometry and dimension. Our discussion will proceed as follows:

(a) Define the systems under consideration. (b) Describe the geometry of the unperturbed

phase space. (c) Describe how the k-d.o.f, systems under

consideration can be reduced to the study of an associated ( 2 k - 2)-dimensional volume-preserving Poincar6 map.

(d) Describe the geometry of the perturbed phase space and the mechanisms for transport (i.e. the analogs of hyperbolic periodic points, stable and unstable manifolds of hyperbolic periodic points, regions, lobes, turnstiles, etc. from the transport theory for two-dimensional area- preserving maps).

(e) Give analogs to definitions 2.1, 2.2, 2.3, and 2.4 and then state the main results of the transport theory for k-d.o.f. Hamiltonian systems.

We begin with (a).

3.1. The class of perturbed, completely integrable k-d.o.f. Hamiltonian systems under consideration

We consider a perturbed Hamiltonian of the form

=H(x,u,v)

where J is the 2 × 2 symplectic matrix defined by

j = ( 0 1 - 1 0)"

We make the important assumption that in a region of the phase space the (u, v) coordinates can be transformed to action-angle variables (I, O) ~ R" × T m so that the Hamiltonian has the form

/L(x, A =H(x,I)

with the transformed Hamiltonian vector field given by

k = J D x H ( X , I ) + eJDxtYI( x , I ,O, l~; E),

[ = --EDotYI(x, I, O, I~; ~),

O = D I H ( X , I ) + e D z I 4 ( x , l , O , I z ; e ) , (3.1),

with 0 < e < < l , ( x , I , O ) ~ R 2 × R ' ~ × T m, and tz ~ R p is a vector of parameters. Additionally, we will assume

Let V c R Z x R m and I'VCR2X~mX~PX[~ be open sets; then the functions

H: V ~ •1,

I~: W X T m ~ ~ l

are defined and C r+l, r_> 1. We will refer to (3.1), as the perturbed system.

s. Wiggins / The geometry of transport in phase space 1 481

3. 2. The geometric structure of the unperturbed phase space

The system obtained by setting e = 0 in (3.1), will be referred to as the unperturbed system.

mally hyperbolic invariant manifold of (3.1) 0. Moreover, . d has C" (2m + 1)-dimensional stable and unstable manifolds denoted WS(.d) and WU(.gd), respectively, which intersect in the (2m + 1)-dimensional homoclinic manifold

~ = J D , , H ( x , I ) ,

/ = 0 ,

0 = D i l l ( x , I ) . (3.1)0

F= { ( x t ( - t o ) , l , Oo) ~ R 2 X Rm X r " l

( to, l, Oo) ~ R 1 X U× Tin}.

We have the following two structural assumptions on (3.1) 0 .

(A1) There exists an open set U c W" such that for each I ~ U the x-component of (3.1)o, i.e.

~ = t D x H ( x , I ) (3.1)0.x

possesses a hyperbolic fixed point which varies smoothly with I, denoted y(I ) , which has a homoclinic orbit xt( t ) connecting the hyperbolic fixed point to itself (i.e. limt_~ +® x t ( t )= y(l)) . (Note: smoothness of the hyperbolic fixed point with respect to I follows from an application of the implicit function theorem; for details see ref. [241).

(A2) DIH(x, I) .# O. We remark that (3.1) 0 is a 2m + 2 - k-d.o.f, completely integrable Hamilto- nian system defined on R 2 x U x T m with m + 1 integrals given by H(x, I), I l . . . . . Ira. Now let us assemble these pieces into a geometric picture in the full (2m + 2)-dimensional phase space. We consider the set of points . d in R 2 x R m x T m defined by

. d = {(x, 1,0) E R 2 × ~m X Tmlx = y ( I )

where y ( I ) solves D x n ( y ( I ) , I ) = 0

subject to d e t [ D 2 n ( y ( I ) , I ) ] * 0,

VI ~ V, 0 ~ T m} (3.2)

Proof ref. [241. t3

It is easy to see that the unperturbed vector field restricted to . d is given by

i - - 0 ,

i¢ = D t H ( y ( I ) , I ) ,

( I , 0) E U× T m

with flow given by

I ( t ) = I -- constant,

O( t) = D I H ( y ( I ) , I ) t + 0o.

So . d has the structure of an m-parameter family of m-tori with the flow on the tori being either rational or irrational. Let us denote these tori as follows: for a fixed i ~ U, the corresponding m- torus on . d is

¢ ( i ) -- { (x ,1 ,0) ~ R 2 × U X Tml

x--v(i), 1=i}.

¢(i ) has (1 + m)-dimensional stable and unstable manifolds denoted WS('r(i)) and WU(¢(i)), respectively, which intersect along the (1 +m)- dimensional homoclinic manifold given by

G= Oo) R 2 x R x Trot

(to, 0o) ~ R l X T"}.

and we have the following proposition.

Proposition 3 . 1 . . d is a C" 2m-dimensional nor-

Additionally, , ( ] ) has a 2m-dimensional center manifold corresponding to the nonexponentiaily expanding or contracting directions tangent to

482 s. Wiggins/The geometry of transport in phase space I

~/. See fig. 9 for an illustration of the geometry of the unperturbed phase space.

Several remarks are now in order.

(1) We comment on the coordinates of (3.1) 0. We are considering an (m + 1)-d.o.f. completely integrable Hamiltonian system. As mentioned above, the m + 1 integrals are H(x, I), 11 . . . . . I m. Now these integrals are not everywhere independent since DxH(y(1), I ) = 0. Moreover if they were everywhere independent, then the phase space could not possess homoclinic orbits (since, in that case, the phase space would be completely foliated by (m + 1)-tori, see ref. [1]). Hence the coordinates of (3.1)0 are the most general for an (m + 1)-d.o.f. completely integrable Hamiltonian system possessing homoclinic orbits, i.e. only m of the (m + 1) integrals are independent. This is proved in ref. [16].

(2) It is possible for the phase space to contain many normally hyperbolic invariant manifolds, say ~/~, i = 1 , . . . , N, with the .~, having both homoclinic and heteroclinic connections. This is done by having many different m-parameter families of hyperbolic fixed points in (3.1)0, x having homoclinic and heteroclinic connections. If this is the case then we apply the following theory to each ~// individually.

(3) In the definition of ~ / given in (3.2) the condition DxH(x , I) = 0 is just the condition for (3.1)0. x to have a fixed point (since J is nonsingu- lar) and the condition det[Dx2H(x, I)] 4:0 is necessary for the fixed point to be hyperbolic.

(4) Roughly speaking, the term normal hyperbolicity means that the rate of expansion and contraction of tangent vectors normal to .d' un-

der the flow linearized about ~" dominates the expansion and contraction rates of vectors tangent to ~ (this should be clear in this case). For precise definitions as well as a proof that ~¢" is normally hyperbolic, see ref. [24]. The reason that normal hyperbolicity is important is that normally hyperbolic invariant manifolds persist under perturbation as invariant manifolds.

(5) Let us discuss the parametrization of W ~ ( ~ ) n W U ( ~ / ) - ~ e " given in proposition 3.1. First consider the notation x t ( - t o ). Let us consider I fixed and x t ( t ) as a homoclinic trajectory of (3.1)0, x. Then xi(0) is a unique point on the homoclinic orbit and t o is the unique time for the point x l ( - t o ) to flow to the point x1(0). (Note: uniqueness follows by uniqueness of solutions for ordinary differential equations.) Hence, for (3.1)0.x, x t ( - t o ) , t o ~ R, provides a parametrization of the one-dimensional homoclinic orbit. Hence in the full (2m + 2)-dimensional phase space, the expression

F = { ( x t ( - t o ) , l , Oo) ~ ~ × ~ " × Tin[

( t o , l , Oo ) ~ ~l × UX T m}

provides a parametrization of WS(.~)(3 WU(.d ') - ~ where varying the 2m + 1 parameters (t 0, I, 00) serves to label each point on WS(~/) ¢3 W°( .a ' ) - ~ .

(6) At this stage a consideration of the dimensions of ~ , WS(~/), and W U ( ~ ) may give the reader a hint of what is to come. The phase space is (2m + 2)-dimensional, ~" is m-dimensional, and W S ( ~ ) and WU(.~) are (2m + 1)- dimensional (i.e. codimension one). We will see

graph 7 ( I ) ~

T( i)-/~

w s (': ( i ) ) n w u ( ' : ( i ) )

T ~

Fig. 9. Geometry of .,¢', WS(.~'), WU(.~), r(1), WS(~-(l)), and WU(r(i)).

S. Wiggins / The geometry o f transport in phase space I 483

that .Z/ plays a role similar to the hyperbolic periodic points in the transport theory for two- dimensional area-preserving maps once we have reduced the study of our systems to the study of a 2m-dimensional Poincar6 map. However, we will first need a theorem showing that ,d' persists along with its stable and unstable manifolds in the perturbed system (3.1)~. This might be somewhat surprising due to the extremely degenerate flow on . ~ (i.e. rational and irrational flow on an m-parameter family of m-tori); however, we will see that it is the structure of the flow normal to .Z/(i.e. the "normal hyperbolicity") that is important for its persistence. Now since the system is Hamiltonian, the (2m + 2)-dimensional phase space is foliated by the (2m + 1)-dimensional level sets of the Hamiltonian which are invariant under the flow. This will be important when we construct lobes and reduce to a Poincar6 map. More specifically, the following lemma will be useful.

Lemma 3.2. WS(.lg) n WU(Jg) - ,¢g- F intersects H(x , I ) = h = constant transversely.

Proof. See appendix 1. []

We know dim(ToF) --- 2m + 1, dim(Tph) = 2m + 1, and, by transversality, d im(TpF+ T p h ) = 2m + 2. Hence, we have dim(TpF N Tph) = 2m.

Another key ingredient in our theory will be the nature of the intersection of . ~ with H(x , I ) --h. This is described in the following lemma.

Lemma 3.3. For h > H ( y ( I ) , I ) , where f = {I U I H ( y ( I ) , I ) is a minimum}, , d ' n h is diffeomorphic to S 2m ÷ x.

Proof The proof is accomplished in two steps. First we prove the lemma for a model Hamilto- nian system where the result is obvious. Then we show that the result obtained for the model Hamiltonian system is diffeomorphic to the general Hamiltonian system. We begin with step 1.

Step 1. Consider the following completely integrable Hamiltonian system

Y¢ = JDxHo( x ) ,

Ul = UI'

I)1 = --tO2Ul '

We remark that the proof of lemma 3.2 uses the assumption Di l l (x , I ) 4= 0.

Transversal intersections of F and H(x , I ) = h have two important implications.

(i) Transversal intersections persist under perturbation.

(ii) Recall (see ref. [2]) that two manifolds are said to intersect transversely at a point p if the vector space sum of the tangent spaces of each manifold at p is equal to the tangent space of the ambient space at p. This specifies the dimension of the intersection of the manifolds. The dimension of the intersection can be calculated from the dimension formula for intersecting vector spaces. In our case, denoting H(x , I ) = h by just h, for any point p ~ F N h we have

d im(Tpr + Tph) -- d im(Tpr) + dim(Tph)

- dim(TpF n Tph).

(lm ~- U m ,

13 m 2 = -- O) m Urn,

( X , U 1 . . . . , U m , U I . . . . . Um) ~_ R2 >( ~ m >( ~ m ,

(3.3)

which comes from the Hamiltonian

n ( x , u I . . . . , U m , U 1 . . . . . Urn)

+

i = 1

(3.4)

We assume that the x-component of (3.3) has a hyperbolic fixed point at x = x 0 with a homoclinic orbit, x( t ) , connecting x 0 to itself (i.e. lim t _. +® x ( t ) = x0). This is equivalent to assumption (A1) above. We also assume that to i > 0,


i - 1 , . . . , m. We will shortly see that this implies that assumption (A2) above is satisfied.

So for this completely integrable Hamiltonian system we have

Mg= {( x ,u I . . . . . u~ ,v 1 . . . . ,Vm) ~ •2 X R m X Rml

x =x0}. (3.5)

In this coordinate system we have

.d'= { ( x , I 1 . . . . . Im,O1,...,Om)

E ff~2 X ( ~ + ) m X Tmlx =Xo}.

Using (3.9) and (3.10) we obtain

(3.10)

Using (3.4) and (3.5) we obtain

.d'n h

== ( ( x , u I . . . . . u,,,,v 1 . . . . . v,,,) ~ R 2 X R" X Rml

½ ~ [(~oiui) 2+v/21 = h - H o ( x o ) }. (3.6) i=1

Clearly ~e 'nh is diffeomorphic to S 2'~-1 provided h - Ho(X o) > 0. We will see shortly that the requirement h - Ho(x o) > 0 is equivalent to h > H(y(I"), [) for the more general system.

Now we transform the (u ,v) components of (3.3) into action-angle variables with the transformation

~ t n h = { ( x , l 1 . . . . ,I,~,Ox,...,Om)

e R2x (R+)m x Tm I

m

lifo i =h -Ho(xo) . (3.11) i=1

Now since (3.7) is a diffeomorphism it follows that (3.6) and (3.11) are diffeomorphic for h -

Ho(xo) > O.

Step 2. Now we consider the general completely integrable Hamiltonian system (3.1). Using (3.2) we obtain

U i = ~ i / / l O i COS Oi,

v i = ~ s i n O i, i = l , . . . , m .

Under this transformation (3.3) becomes

/c = J D~Ho( x ),

/ ~ = 0 ,

ira=O,

Ol = 0")I '

(x, I i , . . . , Im,O, . . . . . Om ) ~ ~2 X ( ~ + ) m X T m

and the Hamiltonian (3.4) becomes

m

H ( x , I , . . . . . In) =Ho(X ) + E Iio0 i. i=1

(3.7)

( 3 . 8 )

(3.9)

~ ' n h = {(x, 1,0) ~ •2 × (R+) ,n × Tmi

H ( T ( I ) , I ) =h}

or,

. ~ n h = { ( x , ] , O ) E R2X (R+)m X Tml

H ( T ( I ) , I) - H ( y ( [ ) , D

= h - (3.12)

The proof of the lemma will be complete if we show that (3.12) and (3.11) are diffeomorphic. Now from (A2), D I H ( X , I ) ~ O, hence, by the implicit function theorem, H(y(I) , I) - h = 0 can be represented as a graph over any m - 1 components of I. Similarly, since to i 4: 0, i = 1 , . . . , m, Y~.im=lliCoi +no(x o) - h = 0 can be represented as a graph over the same m - 1 components of I. These two graphs are evidently diffeomorphic. This proves the lemma. []


The importance of this lemma lies in the fact that S 2m-1 is compact and boundaryless. This implies that ( F o h) o ( . dO h) separate s H(x, I) -- h into an inside and an outside. So we see that in terms of structures providing barriers to transport, a natural analog to hyperbolic periodic orbits (S 1) in 2-d.o.f. systems in k -- (m + 1)-d.o.f. systems is a normally hyperbolic invariant (2m - 1)-dimensional sphere ($2"-1) .

from a Hamiltonian vector field (see ref. [1]). The reduction of an additional dimension comes from the fact that the level sets of H(x, I )= h are invariant under (3.1) and that ~ and H(x, I )= h = constant are transverse (note: this follows from an argument exactly like that given in lemma 3.2). Thus if we denote

Zh n ( H ( x , I) = h)

3.3. Reduction to a Poincard map

We now want to describe how the study of (3.1) 0, which has k = m + 1 degrees-of-freedom, can be reduced to the study of a 2 k - 2 - - 2 m - dimensional volume-preserving Poincar6 map. The reason for doing this is to make the connec- tion with the theory of two-dimensional area-preserving maps described in section 2.

The construction of the Poincar6 map proceeds in the usual way. Choose any component of the coordinate of (3.1) 0 which is bounded away from zero, say 0,. for some 1 _< i < m. We note that by assumption (A2) 0i is nonzero for some 1 < i < m. Consider the following (2m + 1)-dimensional surface in ~2 X Rm X Tin:

~ = { ( x , I , O ) ~ R 2 X R m X T m I

0 i = 0, for some 1 < i < m}.

The requirement that //i is bounded away from zero for some 1 < i < m implies ~ is a cross-section to the vector field (3.1) o and that all trajectories starting on ~ return to ,S. For any point

( x , l , ' O ~ - - ( O 1 , . . . , O i _ l , 0 i + 2 , . . . , O m ) ) ~ we denote the first return time of this point to ~ by

= z(x, 1,0). Thus it is natural to consider a Poincar6 map of Z into ~, denoted P, which is defined as follows.

P: £ ---+~,

( x ( O ) , I ( O ) , O ( O ) ) ~ ( x ( T ) , l ( , ) , O ( z ) ) . (3.13)

This map preserves volume since it is constructed

then P restricted to ,S h, denoted Ph, is a 2m- dimensional volume-preserving Poincar6 map.

Now let us see how .d , WS(.d), and WU(.d) enter this picture. In lemma 3.2 we showed that .d , WS(.d), and WU(.d), are transverse to H(x, I ) = h -- constant. From the definition of ,~, it should be clear that .d , W~(.d), and WU(.d) are likewise transverse to ,S. Thus, following remark (ii) after lemma 3.2 we have

~¢'f'~ "~h is (2m - 2)-dimensional, W~( .d) O 2~ h is (2m - D-dimensional, W U ( l ) o "~h is (2m -- 1)-dimensional.

From lemma 3.3 and the remarks following its proof, it should be clear that . ~ o "~h is compact and boundaryless and that ( . d O "~h) U ( F n "~h) separate 2~ h into an inside and an outside. Thus ( . d n "~h) U ( F O "~h) is a barrier to transport.

Let us describe in more detail two specific examples.

2-d.o.f. systems This is the case that has been studied the most.

In this case we have m = 1 so that .,g has the structure of a one-parameter family of hyperbolic periodic orbits. From the above arguments, we can reduce the study of this system to the study of an associated two-dimensional area preserving Poincar6 map where . ~ n ~h is a hyperbolic fixed point and WS(.~ ') n "~h and WU(.~) n -~h are the respective stable and unstable manifolds of the fixed point.

3-d.o.f. systems In this case we have m = 2 and . ~ has the

structure of a two-parameter family of two-tori.

486 S. Wiggins/The geometry of transport in phase space I

The study of this system can be reduced to the study of an associated four-dimensional volume preserving Poincar6 map where ~/O,~h has the structure of a one-parameter family of one-tori with WS(.~/)n~ h and WU(.4C')n,~h each being three-dimensional.

3. 4. The geometric structure of the perturbed phase space

The main result that we need is the following.

Proposition 3.4. There exists % > 0 such that for 0 < 6 < % the perturbed system (3.1), possesses a C r 2m-dimensional normally hyperbolic locally invariant manifold

~ ' , = { ( x , I , O ) e R Z X R mxTml

x = ~ / ( I , O ; e ) =3'(1) + ~ ( 6 ) ,

I ~ lfl c U c [~ m, 0 ~ T m}

where / ) c U is a compact, connected m-dimensional set. .d' , has local C r stable and unstable manifolds, WiS(,¢'~) and Wt~(.~'~), which are of the same dimension and C r close to Wl~(.~" ,) and Wt~(.~',), respectively. Moreover, ,~'n h, is diffeomorphic to S 2m-1, where h, denotes the (2m + 1)-dimensional level set of H ( x , l ) + J-](x, I, o, ~; 6).

Proof. See ref. [24] for complete details. In the proof, the fact that .~', n h, is diffeomorphic to S 2"-1 follows from the fact that the perturbed manifolds are constructed as graphs over the normal bundle of the unperturbed manifolds. []

We remark that the reason we must make U slightly smaller (i.e. take any compact U c U) is to deal with the behavior of the boundary of U. This technical point is dealt with in great detail in ref. [24].

We note that the flow on ~ may be quite complicated. Indeed, Graft [9] has proven that

~ ' contains a Cantor set of KAM tori having positive measure; but the complexity of the dynamics on .Z/is not what is important to us. The main point is that .Z/ persists as a smooth manifold having smooth stable and unstable manifolds of codimension one.

Since the perturbed system is still Hamiltonian the (2m + 2)-dimensional phase space is foliated by the (2m + 1)-dimensional level sets of the Hamiltonian H(x, I) + d-](x, I, 0,/x; E) which we denote by h,. We also note that by persistence of transversal intersections under perturbations, h~ intersects WS(.~',) and WU(./g,) transversely, ,~ intersects WS(.~',) and WU(~',) transversely, and

intersects h, transversely. Let us now think in terms of the 2m-dimen-

sional perturbed Poincar6 map which we denote Ph, with 2h, ----" "~ N h,. We denote

~' , n,~h, = .~ , , (3.14a)

W~(,dt',) n,~h, = W~(.~,), (3.14b)

WU(~/,) n,~h, = WU(.~,). (3.14c)

In analogy with the usual set-up for transport in two-dimensional area preserving maps, .~', will play the role of the hyperbolic fixed point with the tangling of W~(.~/,) and WU(~',) possibly providing lobes and turnstiles.

Now suppose W~(.~,) and W U(~/,) intersect transversely in a ( 2 m - 2)-dimensional set, ~ , such that S[~/,, ~ ] u U[.~/, ~,~] separates "~h, into two disjoint components where S [ . ~ , , ~ ] denotes the segment of WS(.~'~) from . ~ to and U[.~,, ~ ] denotes the segment of W~(.~,) from .~/, to 9 ~.

This key sentence deserves further comments.

(1) WS(.~,) and W~(.~,) are both ( 2 m - 1)- dimensional manifolds in a 2m-dimensional ambient space (-~h,). Therefore by remark (i) following lemma 3.2, if they intersect transversely then the dimension of the set of intersection is 2m - 2.

(2) The requirement that the intersection set is such that S[ .~, , 9 ] U U[.~',, ~ ]


separates Zh, into two disjoint components is obviously very important. In k-d.o.f, systems, k > 3, the intersection of WS(.~/,) and WU(.~/,) may not have this property (we will see this explicitly in the example in section 5), thus it will be important to determine when 9 satisfies this condition. This will be discussed in section 4.

We will refer to 9 defined in this way as a transverse homoclinic manifold. In the context of 2-d.o.f. systems (i.e. two-dimensional area-preserving maps) we did not need to worry about these details since .~, was a point and the transverse intersection of the one-dimensional W s(.~,) and WU(.~,) in the two-dimensional ~h, was also a point with S[.~/, 9 ] u U[.~/, 9 ] obviously separating "~h, into two disjoint components. In forming lobes it will be important that 9 is compact, boundaryless, and has the same dimension as .~/,.

Now since WS(~/,) and W U(ffe',) are invariant, the existence of one transverse homoclinic manifold 9 implies the existence of a countable infinity of others under iteration by Ph • This leads to a tangling of WS(.~,) and W~(.~) of exactly the same character as that of the stable and unstable manifolds of hyperbolic fixed points of two-dimensional maps, see fig. 10 for a heuristic illustration of this phenomenon.

3.5. The general formulation of the transport problem

In sections 3.1-3.4 we demonstrated that a certain class of "near integrable" k - (m + 1)- d.o.f. Hamiltonian systems defined in (3.1), can possess geometrical structures in the phase space which are partial barriers to transport analogous to the lobes and turnstiles familiar from the transport theory for two-dimensional area-preserving maps. We now want to formulate the problem of transport in this system exactly along the lines of the transport theory for two-dimensional area-preserving maps due to homoclinic and heteroclinic tangles discussed in section 2.

We consider the map

Ph,: 2fh, --' £h,,

which is a C r (r >_ 1) 2m-dimensional orientation-preserving diffeomorphism. Let .~,.i, i - -1 . . . . , N, denote N normally hyperbolic invariant ( 2 m - 2)-dimensional manifolds which have the structure of .~/, defined in proposition 3.4. We denote the ( 2 m - 1)-dimensional stable and unstable manifolds of ~/.,i by WS(.(~.,i) and WU(~¢,i), respectively. We now give analogs to definitions 2.1, 2.2, and 2.3.

Fig. 10. An illustration of the possible breakup of W'(.~') WU(.~) under the perturbation (angle variables suppressed).

Definition 3.1. A ( 2 m - 2)-dimensional set, 9 is called a transverse heteroclinic manifold if

(1) ~ c WS(~ , . i ) N WU(~/, . j) .

(2) WS(.~,,i) and WU(.~,,j) intersect transversely in ~ .

(3) S[~,, i, 9 1 u U[~/,,/, 9 ] separates "~h,. 9 is called a transverse homoclinic manifold if (1), (2), and (3) hold and i --j.

We remark that it follows from this definition that ~ is compact and boundaryless. Also, it is not always true that S[.~., i, 9 ] u U[.~/,d, 9 ] separates -~h, (as we will see in the example in section 5).


Definition 3.2. Consider a transverse heteroclinic (or homoclinic) manifold ~ c W S ( , ~ , , i ) n Wu(.~/,/) and let S[.~, i, ~ ] denote the segment of W~( '~ , , i ) f rom ~',.'i to ~ and U [ . ~ , / , ~ ] denote the segment of WU(.~,,j) from .~,,j to ~ . Then ~ is called a primary intersection manifold (pim) if S [ .~ , , i ,~ ] and U[.~/,,/, ~ ] intersect only in ~ (and possibly at -~,,i if i =j).

Definition 3.3. Let ~0 and ~1 be two adjacent pims, i,e. there are no pims in U [ ~ 0, ~1] and S [~0 ,~1] , the segments of WS(~/,.i) and WU(.~,,j) that connect ~0 and ~1. We refer to the region in "~h, bounded by the segments U[~0, ~1] and S[~0, ~ l ] as a lobe.

We remark that the fact that ~0 and ~ t are compact and boundaryless along with the fact that U[~0, ~1] and S[~0, ~ l ] are codimension one in Zh, implies that a lobe separates "~h, into an inside and outside exactly like the lobes from the transport theory for two-dimensional area- preserving maps discussed in section 2. Unfortu- nately we cannot draw any diagrams illustrating definitions 3.1, 3.2, and 3.3 since, in the simplest nontrivial example (i.e. 3-d.o.f. systems) -~,,i is a normally hyperbolic invariant compact, boundaryless two-manifold having three-dimensional stable and unstable manifolds.

With definitions of lobes and pims at hand, the transport of phase space across a boundary consisting of pieces of the stable and unstable manifolds of the ~",,i intersecting at pims is due to the dynamics of lobes and occurs exact/y as in the discussion given in section 2.1.

Definition 3.4. A region is a simply connected domain of "~h,, with boundaries consisting of boundaries of the phase space and segments of

stable and unstable manifolds of the ~/e,i, i = 1 , . . . , N and ending at either pims or at the boundary of the phase space (which may be infinity).

Now the main transport problem can be stated exactly the same as for two-dimensional area-preserving maps.

Suppose "~h, is divided up into N n disjoint regions, denote Ri, i = 1, . . . , N n. Suppose that initially particles of species S i are uniformly dis- tributed throughout region Ri, i = 1 , . . . , N R.

Problem. Compute the volume occupied by species S i in region Rj (for any i and j) at any specified later time.

Using definitions 3.1, 3.2, 3.3, and 3.4 along with the Lk, y(n), Lik,j(n), and T,..j(n) given in section 2.2, eqs. (2.2) and (2.3) along with the conservation laws (2.4) and (2.5) immediately apply to give an answer to this problem (with/z(A) interpreted as the volume of a set A C,~h ).

Finally, we remark that it would have been possible to begin this paper with an arbitrary volume-preserving, even-dimensional C r ( r> 1) orientation-preserving diffeomorphism and ab- stractly proposing definitions 3.1-3.4. However, we feel that existence of normally hyperbolic invariant manifolds like .~,, i would have appeared somewhat dubious. Therefore we chose to take the approach of starting with a class of systems (i.e. eq. (3.1),) which possessed such structure and then develop the geometry of such systems in detail. In particular, our ideas of transport in k-d.o.f. Hamiltonian systems are not restricted to (3.1), provided one can find analogous structures to construct partial barriers and mechanisms for transport. We discuss some more aspects of this in section 5.

4. On the existence of transverse heteroclinic manifolds in (3.1),

For Hamiltonian systems, the existence of transverse intersections of WS(.~,,i) and WU(.~,,j) may not be difficult to determine. However, the fact that these intersections can be used to form geometrical barriers in phase space may be hard to verify. For (3.1), we can use the Melnikov-type theory developed in ref. [24].


In chapter 4 of ref. [24] it is shown that the function

M( I, Oo, to;IX ) = f_~[(D~H, JD, H ) - (DiH, Dot~)](q~(t),l~;O)dt

+(DlH(3'(I) , I ) , f~DoI~(qg(t) , lx;O)dt) , (4.1)

where

qg(t) =- (xt(t) ,+to , ,,,f DtH(x ( ' ) , / ) d ' + 0o),

is a measure of the distance between WS(~/,,i) and WU(.~,.) where x1(t) is the m-parameter family of heteroclinic orbits in (3.1)0. x and ( - , . ) denotes the usual Euclidean inner product. We note that M(l, Oo, to; ~) also depends on parameters /x; however, we will henceforth suppress this from the notation since it is not important for now.

Theorem 4.1. Suppose there exists a point (L 30, to) E 0 × T m × R 1 such that (1) M(I, 0o, to) = 0 and (2) DM(I, O o, t o) has rank 1 on the zero set of M containing (i, 3 o, to).

Then WS(~/,,i) intersects WU(~/,.) transversely in a (2m -2)-dimensional intersection manifold.

Proof. This is a simple application of the global implicit function theorem, see, e.g. ref. [6]. []

We make the following remarks concerning theorem 4.1.

(1) The reader should consult ref. [24] for a detailed discussion of the parametrization of the stable and unstable manifolds in terms of the independent variables of M(I, O0, t 0) and the relationship with Poincar6 maps. In refs. [24, 25] the fact that zeros of the Melnikov function form primary intersection manifolds is discussed in detail.

(2) If condition (2) of theorem 4.1 does not hold everywhere the conclusions of the theorem may still hold by a consideration of higher-order derivatives.

(3) Theorem 4.1 provides sufficient conditions for WS(.~, . i ) to intersect WU(.d',,). However, it does not tell us if segments of these manifolds can be used to form lobes, as we will see in section 5. Presumably, one can find conditions on the the Melnikov function for which this situation holds.

5. An example

In this section we give an example of a three-degree-of-freedom Hamiltonian system that illustrates the theory. Consider the Hamiltonian

H,(gb, V, Xl,Yl, X2, Y2) l ~ v 2 -- C O S ~ + ~yll 2 + ~¢a01Z 1 1 2 .2--1_i_ ~y22 + ~¢.O2X 212-2 + ~/_~(U, t~, Xl, yl, X2, y2)

(5.1)


with (~b, v, xl, Yl, x2, Y2) ~ $1 × R1 × RI × R1 × R1 × R1- This Hamiltonian describes a simple pendulum and two harmonic oscillators coupled through the e/4 term. We will consider three types of couplings:

1-'I( U, ~ , Xl, Yl, X2, Y z ) = ½e[ ~1( Xl -- ~ )2 .+. T2( X2 -- ~ )2 ] ,

I~(u,(b,x,, Yl,X2, Y2) = ½"[(Xl - x 2 ) 2 + Y(x , - q~)2],

I~(v,6, x~,y~,x2,Y2) = ½ , [ ( x ~ - x 2 ) 2 + ~'(x2 - 6 )2 ] ,

(5.2a)

(5.2b)

(5.2c)

where Yl, Y2 and y are parameters. The discussion will proceed as follows. We begin by discussing the geometrical structure of the unperturbed phase space. We then turn to the perturbed phase space and discuss the Melnikov functions for each of the three types of couplings. With the Melnikov functions in hand we then discuss the possibility of the construction of lobes and turnstiles for the three types of couplings. We end with a summary of our results.

5.1. Geometry of the unperturbed phase space

The Hamiltonian of the unperturbed system is given by

H(da v , x 1 Y l , X 2 , Y 2 ) = 1 2 1 2 1 2-2 ly2 1 2 . 2 , , 2V -- COS ~b + ~Yl + 2°)I'Ll + 2 2 + 2°12x2, (5.3)

which gives rise to the vector field

4~ = v , b = - s in ~b,

3~1 = Y l , ))1 = --¢'02XI' "~2 -----Y2, Y2 = - ~ x 2 . (5.4)

It is easy to verify that

,-~--- { ( ~ , V , X l , Y l , X 2 , Y 2 ) E S 1 X ~1 X ~1 X ~1 X ~1 X ~11~ = TI', V = 0} (5.5)

is a normally hyperbolic four-dimensional manifold invariant under the flow generated by (5.4). Moreover, . ~ has five-dimensional stable and unstable manifolds, denoted WS(Jt ') and WU(.~'), respectively, which coincide along two homoclinic manifolds that separate the six-dimensional phase space into three disjoint regions. These homoclinic manifolds have the following parametrization:

/ ' ± = { ( ~ , U, X1, Yl ' X2, Y 2 ) [ ~ = _ 2 s i n - ' [ t a n h ( - t o ) ] , v = _ 2 s e c h ( - t o ) , t o ~ ~'}. (5.6)

The level sets of the Hamiltonian are given by

1 2 1 2 1 2 2 1 2 1 2 2 h = ~v - cos ~b + ~Yl + ~¢'D1XI + ~Y2 + 20)2X2" (5.7)

Using (5.5) and (5.7), it follows that ~ ' n h is given by

h _ l = l 2 1 2-2 1 2 1 2 2 2Yl + 2¢OlXl + ~Yl + 2t02X2"

Thus for h > 1 ~/gN h is diffeomorphic to the three-sphere, S 3.

(5.8)


The reader should notice that the coordinates of (5.1) are not of the same form as (3.1) e. This can be remedied by transforming (xl, Yl) and (x2, Y2) into action-angle variables as follows:

x i = ~ s i n O i , yi=x/~itoicosOi, i = 1,2. (5.9)

Henceforth we will assume that this has been done. The reason that we did not immediately give the Hamiltonian (5.1) in these coordinates is that we feel that initially the geometry might be somewhat clearer in the cartesian coordinates (x 1, Yl, x2, Y2).

5.2. The Melnikov functions for the three types of couplings

Using (4.1), (5.2), and (5.6), the Melnikov functions for the three types of coupling are given by

M± (11,/2,010' 020' to)

= _+ 2"rr[y121~l/W 1 sech(lrrtOl)sin(tolt o + 01o)

+ 3 ' 2 ~ sech(½"rr to 2)sin( tOEt o + 02o)], (5.10a)

M± ( 11 ,/2,01o, 020, to) = -+ 2 ~ 3 ' ~ sech(½~r to1) sin( tOlt o + 01o ) , (5.10b)

M ± ( I l, 12, 01o, 020, to) = ___ 2 w 3 ' ~ sech(½rrto2) sin(toEt o + 02o ) . (5.10c)

Note that the zero set of M+ is the same as the zero set of M_ (where M+ denotes the Melnikov function computed on F+ and M_ denotes the Melnikov function computed on F_). Therefore in the following we will omit the subscripts ' + ' and ' - ' on the Melnikov functions.

We want to use these Melnikov functions to describe the geometry of WS(~'e) n h e and WU(.~'e) N h e. Now suppose the Poincar6 map

Ph.: "~h. --''~h,

is defined on the cross-section "~h, -- "~ N h, where

= { ( 6 , U, I1,01, 12, 02) ~ S 1 × R 1 X R + X S' X R + X sll02 = 0}, (5.11)

then we can set 020 -- 0 in (5.10). Also, recall that on .d , WU(.dd), and W S ( ~ ) we have

h - 1 = Ilto I +/20)2,

hence

12 = (h - 1 - IltOl)/to2.

Also, using (5.12), we note that

11 ~ [ 0 , ( h - 1) / to l ] .

(5.12)

(5.13)

(5.14)


So using (5.14), we rewrite (5.10) as

M( 11 , 010 , to;0, h)

= 2 r r [ y a ~ sech(½rrtOl) sin(toito + 01o)

+ y2gt2(h - 1 - 11001)/to£2 sech(1,rrw2)sin( to2to)],

M(Ii,Olo,to;O,h) = 2 - r r y ~ sech(½,rrtol) sin(tolt 0 + 010),

M( I 1 , 010,020 , to; 0, h) = 2rry~/E(h - 1 - IltO 1 ) / to 2 sech(½,rr to2) sin(tO2to).

(5.15a)

(5.15b)

(5.15c)

This new notation, M(I1, 01o , to; 0, h), explicitly denotes that we are on the 020 = 0 cross-section and the level set of the Hamiltonian denoted by h. The three variables 11, 01o , and t o provide a parametrization of W~(~/,) and WU(~/,). Thus the two-dimensional surface defined by M(II, 01o , t0;0, h ) = 0 (h fixed) describes the intersection set of the three-dimensional manifolds W~(~/,) and W " ( ~ / ) in the four- dimensional "~h,"

5.3. Construction of lobes and turnstiles

We now discuss the construction of lobes and turnstiles from WS(~/,) and WU(.~,) for the three different types of couplings. In order to do this we will need to understand the nature of the intersection of Ws(.~,) and W"(.~/,). This will be described by the zero set of the Melnikov function. The zero sets will be represented as two-dimensional surfaces in I1-01o-t o space. Since the two-dimensional zero sets may merge, it is often difficult to represent the situation in a three-dimensional figure. Therefore we will show the intersection sets in a series of fixed 11 slices of the I1-01o-t o space for 11 ranging from zero to (h - 1)/to 1.

The reader may wonder how this representation of the intersection of Ws(~/,) and W " ( ~ / ) is manifested in the two-degree-of-freedom case, i.e. where ~ ' , is a point and WU(~,) and W~(,~/,) are curves. In this case we are studying the zero set of M(to) = 0 (i.e. there are no I or 0 variables) so that the zero set of ~ is a set of discrete points on R 1. Hence this representation of the intersection of W~(.~',) and W"(.~',) for two-degree-of-freedom systems corresponds to collapsing the two-dimensional lobes to a curve in such a way that the two intersection points defining the lobe are preserved. Although this eliminates some information concerning the lobe (in particular, its volume) it preserves all information concerning the intersection of the manifolds that form the lobes. Hence, this representation of the intersection of W~(,~'~) and W~(~/~) is particularly useful for three-degree-of-freedom systems where the intersection set of WS(~ , ) and W"(~ ' , ) can be represented in three dimensions with no loss of information.

We now consider the first type of coupling.

sin 2+ sin 02-


From (5.15a) the Melnikov function is

M(11,010 , to; 0, h)

= 2,rr[ 2¢-~l/to 1 sech(½artol) sin(tolt 0 + 010)

+ y2~/E(h - 1 - lltol)/to 2 sech(½"rrto2) sin (to2t0)] . (5.16)

1 In fig. 11 we plot the zero set of (5.16) for h = 2, Yl = 3'2 and tol = to2 = 1. Note that at 11 = ~ the intersection sets merge. This implies that there exists no primary intersection manifold so that we can define partial barriers and turnstiles from WS(.~,) and WU(~/,). In fig. 12 we plot the zero set for (5.16)

1 for h = 2, to1 -- ~to2 = 1, and Yl = 3"2- Similar behavior occurs as in the previous case; for 11 = 0.011578 the intersection sets merge.

We now want to argue that this type of behavior, i.e. the impossibility of defining disjoint regions separated by the codimension-one WS(,2/,) and WU(.~) , always occurs for this type of coupling.

The condition for the merging of the intersection sets of (5.16) is given by

M(II , 010, t0;0, h) = 0, (5.17a)

0M 001o ( I 1, 01o, to; 0, h) = 0, (5.17b)

OM'I 0 ato l, 1, 10, to;0,h) = 0 - (5.17c)

(Note: from (5.17) it is clear that, for h fixed, this merging of the intersection sets generically only occurs at isolated points in I1-01o-t o space). Using (5.16), (5.17) is given by

a l ( I i , to1,3"J) sin(tolt0 + 010 ) + a2(11, to1, to2, h, Y2) sin(to2t0) = 0, (5.18a)

a l ( 11, tol, Yl) cos( tolto + 010 ) = 0, (5.18b)

al( I1, °91,3") °91 COS( tolt0 + 010) + a2( I1, t°l, °)2, h, 3'2) to2 cos(to2to) = 0, (5.18c)

where

a1(11, 601, 3'1) - 2,rr3'1 2/~/---~l/to l sech(1,rrtol),

a2( 11 , to,, to E , h, 3'2) ~ 2"rr 3'2~/2(h - 1 - Iltol)/to 2 sech(½"rr toE).

Now it should be clear that (5.18) can be satisfied if and only if

a1(I1, to1,3"1) = a2(11, to1, °)2, h, 3'2)- (5.19)

Now consider to1, to2,3't,3"2 and h > 1 as fixed. It is easy to verify that

Oa I Oa 2 Oi i > 0 and ~ < O.

I F J

r . . . .

i . . . . . , k ,

l

o

P~

~ 4

CT~

c~

' \ °l ~ \ \ \ \ ~

II

t= .

. . . . . . . . . . . . . . . . . . . . . . . . it ~

N

o

c ~

r =

e -

o ~

.o~

. . , b ,

J l , , , , i , I , , i , . . i i , , , k . . L o

o O

o o

i i ¢o

ff

I I

II c , " ~

8 i . .

[ . ~


y, h arbitrary, o 1= 0 ) 2 ~ 1

6 ' ' ' . i . , _

2 4 " 6

010

Vlle [o, h-l]

6 t01 o

- 2

- 4

- 6

y, h arbitrary, e)~= o ) 2 = 1

i i ,

i i i 2 4 6

010

"q'll~ [0, h-1 ]

Fig. 13. Zero set of the Melnikov function shown in a fixed 11 slice for coupling (5.2b). The zero set appears qualitatively the same for arbitrary Ii, y, tot, t0z, and h > 1.

Fig. 14. Zero set of the Melnikov fundtion shown in a fixed I] slice for coupling (5.2c). The zero set appears qualitatively the same for arbitrary 11, y, to1, to2, and h > 1.

Moreover,

a, ~ [0,2rryl] / /2(h - 1)/o91 z sech(½"rro91) ]

and

a 2 ~ [2avy2V/2(h - 1)/o9~-sech(l'rro92),0].

Hence there exists a unique value of I1, such that al(I 1, o91, Yl) = a2(I1, ool, o92, h, Y2) for any o91, o92, 71, Y2 and h > 1. So for this type of coupling, it is impossible to partition the phase space into disjoint regions using WS(a¢/,) and W U(.~/,).

We next consider the second type of coupling.

e f f I (v ,4~,I t, 12,01,02) = 1 6 [ ( ~ sin 0 1 - ~ s i n 0 2 ) 2 + ' y ( ~ sin 01 - - 1~)2] .

From (5.15b) the Melnikov function is given by

M( I, , 0,o, to; O, h) = 2 " r r y ~ sech(½"rr o9, ) sin( o91to + 0,o ) . (5.20)

In fig. 13 we plot the zero sets of (5.20) which are easily shown to be t o = (n~r - 010)/ /( ,O1, 11 ~ [0,(h - 1)/o91] , n = 0, + 1 . . . . . Hence they appear the same for each value of 11, y, o91, o92, and h > 1 and never merge. Nevertheless, lobes cannot be formed since there are only two connected intersection sets.


Finally, we consider the third type of coupling.

"l~(v,dp, I I , I2 ,Ol ,O2)= ½ " [ ( ~ sin 0 1 - ~ sin 02)2 + Y ( ~ sin 02-- ~b)2].

From (5.15c), the Melnikov function is given by

M( 11, 010, to; 0, h) -- 2,rr~2(h - 1 - Iltol)/to E sech(½-rr toz) sin(toEt0). (5.21)

In fig. 14 we plot the zero sets of (5.21) which are easily shown to be to=n~/ to2 , 010~[0,2~], I t ~ [0,(h - D/tot], n = 0, +_ 1 . . . . . Hence they appear the same for each value of I t, y, to~, toE, and h > 1, they never merge, and they are disconnected. So for this type of coupling primary intersection manifolds exist allowing for the construction of lobes and turnstiles for any y, to1, to2, and h > 1.

5.4. Summary

We end this example with several remarks and observations.

(1) Even though WS(,~,) and WU(.~/,) are codimension one in "~h,, segments of WS(.~,) and WU(.~/,) starting at .~, and ending at an intersection set of W~(.~,) and W~(.~,) may not partition 2~h, into disjoint components. This is a phenomenon that can only occur in k-degree-of-freedom systems, k > 3. In this example we saw that whether or not the phase space could be partitioned by segments of WS(~/,) and W~(.~',) depended on the nature of the coupling of the oscillators.

(2) We remark that if instead we had considered nonlinear oscillators, i.e. an unperturbed Hamiltonian of the form

H ( $ , v , 1 1 , 12) = ~v~ 2 +cos~b+ G l ( l l ) + G2(12).

Our results would be unchanged provided

i~G i O2G i ai i >0 , ~i----- T <0 , i = 1 , 2 .

6. On the need to consider near-integrable systems

One of the attractive features of the transport theory for two-dimensional maps is that it does not require the systems under consideration to be perturbations of integrable systems. This is be- cause the task of finding hyperbolic periodic points can be cast into a nondynamical framework, i.e. computing zeros of functions. Once the hyperbolic periodic points are found then general theorems (see ref. [24]) imply the existence of their stable and unstable manifolds which in turn can be used to form barriers to transport.

We have already seen that the stable and unstable manifolds of hyperbolic periodic points of higher-dimensional maps do not divide the phase space in such a way as to create barriers to transport. Therefore a more dynamically complex object is required for this purpose. In our case we have seen that an appropriate analog of the hyperbolic periodic point is a normally hyperbolic invariant manifold whose stable and unstable manifolds are codimension one. The dynamics on this manifold may be quite complex and given an arbitrary system it may be difficult to find such lower-dimensional manifolds. Our perturbation methods are one way of finding such manifolds.


However, one might expect that as • is increased normal hyperbolicity might be lost resulting in the manifold being destroyed in much the same way as KAM tori are destroyed as the strength of the perturbation to integrability is increased. For this reason one might expect that our methods are extremely limited. However, in the next section we discuss a setting where systems such as (3.1), may arise as a "normal form".

7. Transport near resonances

It has long been observed (see, e.g., refs. [13, 11, 14]) that resonances strongly affect transport in phase space. The obvious question then is: How does the structure that we have developed relate to resonances? This appears to be an area where more extensive research is required. How- ever, for now we make the following two comments.

(1) In our framework we require the existence of homoclinic or heteroclinic orbits which are associated with zero-frequency motions.

The next comment is less trivial.

(2) The study of periodic orbits near elliptic fixed points has been a subject of intensive study over the years (see, e.g., refs. [22, 23, 7]). In the case where the frequencies associated with the linearized motion are nonresonant, the Lyapunov subcenter theorem gives a characterization of the dynamics near the fixed point for arbitrary (but finite) degrees of freedom. The theorem states that n two-dimensional invariant manifolds pass through the elliptic fixed point and each two- manifold contains a continuous family of periodic orbits. The cases involving resonant frequencies are much more difficult and there is no general result analogous to the Lyapunov subcenter theorem.

The general technique used in the study of the resonant cases involves a change of coordinates "adapted" to the specific resonance with a further simplification of the Hamiltonian using the method of Poincar6-Birkhoff normal forms (see

ref. [21]). For 2-d.o.f. Hamiltonian systems the truncated normalized Hamiltonian is completely integrable and, for certain resonances, contains orbits homoclinic to families of periodic orbits. Thus the structures described in this paper should occur when the effects of higher-order terms in the normal form of the Hamiltonian are considered.

For 3-d.o.f. the situation is not as nice and is still an area of current research. Unlike the 2- d.o.f, case it is not always possible to find a truncated, normalized Hamiltonian that is completely integrable. In some cases, however, this can be done and our methods should then apply. We refer the reader to ref. [21] for more information.

In any case, the abovementioned work gives some hope that in certain cases we may be able to show that (3.1), is a "normal form" near some resonances. This would enable to show how the geometrical barriers to transport that we have constructed are related to certain resonances in multi-degree-of-freedom Hamiltonian systems.

8. The relationship to Arnord diffusion

We now want to describe how the transport via lobes is related to Arnol 'd diffusion. First consider the structure of the unperturbed phase space described in section 3.2. The normally hyperbolic invariant manifold .Z/ has the structure of an m-parameter family of m-dimensional tori denoted z(I) , I ~ U, where each torus has an (m + 1)-dimensional stable manifold, an (m + 1)- dimensional unstable manifold, and a 2m-dimensional center manifold, see fig. 9. Now for m = 1 (i.e. 2-d.o.f. systems) . ~ intersects the (2m + 1)- dimensional level sets of the Hamiltonian in isolated periodic orbits. In general, .~" intersects the level sets of the Hamiltonian in ( m - 1)- parameter families of m-tori. Thus for m > 2 (i.e. for systems with three or more degrees of freedom) the tori along with their stable and unstable manifolds are not isolated in the level sets of the Hamiltonian.

S. Wiggins / The geometry o[ transport in phase space I 499

1:8 ( I j + l ) - ~

"C 8 (I j) I ,

t e (Ij.

Fig. 15. Illustration of the geometry of Arnol'd diffusion (angle variables suppressed). T~(I/+ 0, ¢~(I/), and ~-,(I/_ 1) represent three of the tori in the Cantor set of tori on .~', that survive under the perturbation.

Now when the system is perturbed a generalization of the KAM theorem due to Graft [9] tells us that a Cantor set of tori on .~' having positive measure is preserved. At the same time, generically the stable and unstable manifolds will intersect transversely yielding a homoclinic tangle. Near the torus, the homoclinic tangle exhibits large amplitude oscillations due to the hyperbolicity. Since we have a Cantor set of tori, this set of tori is dense in itself, therefore the homoclinic tangle of each torus becomes intertwined with the homoclinic tangles of nearby tori leading to the possibility that orbits starting near any one of these tori may "diffuse" along this Cantor set of homoclinic tangles in a chaotic fushion. This picture was first described by Arnol'd [2] and has come to be called Arno l ' d diffusion; it can be verified for (3.1), using Melnikov-type arguments described in ref. [24]. We attempt to illustrate the geometry of this phenomenon in fig. 15.

Fig. 15 should give some indication of the nature of transport via Arnol'd diffusion versus transport via lobes. Roughly speaking, Arnord diffusion occurs in the I variables (i.e. along directions "tangent" to W'( ,¢ ' )and Wu(.Z/))and transport via lobes occurs in the x directions (i.e. along directions transverse to WS(,~ ') and WU(,¢')). An obvious question is: Which type of

transport is dominant? Results of Nekhoroshev (see ref. [1]) imply that the I variables vary by @(~1/c) on a time scale of length ~'(exp(1/E) 1/4c) where c - 4(m 2 + 2m + 2). Our results show that the x variables vary by ~'(E) on a time scale of length ~(1). One might expect from this that transport via lobes would dominate; however, more work needs to be done in order to verify this.

We remark that (3.1)~ can be viewed as a generalization of the special example studied by Arnol'd [2] in which he first demonstrated the existence of what is now called Arno l ' d diffusion.

However, in his example he did not point out the existence of lobes which could lead to much faster transport. Finally, we remark that using the non- hyperbolic techniques developed in ref. [24] for orbits homoclinic to normally hyperbolic invariant manifolds, it should be possible to show the existence o f chaotic dynamics similar to the Smale horseshoe.

9. Conclusions

Our goal in this work is to find codimension-one surfaces that are barriers to transport in k- degree-of-freedom Hamiltonian systems that are analogous to the stable and unstable manifolds of hyperbolic periodic orbits associated with resonance bands in 2-degree-of-freedom Hamiltonian systems. We have shown that the ( 2 k - 2 ) - dimensional stable and unstable manifolds of normally hyperbolic invariant ( 2 k - 3)- dimensional spheres are partial barriers to transport in the ( 2 k - 1)-dimensional level set of the Hamiltonian in a class of k-degree-of-freedom Hamiltonian systems. We have shown that under certain conditions these structures can be used to form lobes and turnstiles in precisely the same way as for two-dimensional area-preserving maps. This enables us to use the transport theory of Rom-Kedar and Wiggins [19] to obtain exact ex- pressions for the transport of phase space volume between regions separated by these partial barri-


ers. However, even though the stable and unstable manifolds of the spheres are codimension one in the level set of the Hamiltonian, segments of the manifolds and their intersection sets may not partition the phase space into disjoint regions. This is a phenomenon that occurs only in k- degree-of-freedom systems, k > 3. The generalized Melnikov function found in ref. [24] was introduced as a tool to study the intersection sets of the stable and unstable manifolds of the spheres.

Finally, we show how the transport due to lobes through turnstiles may have a much more dominant effect in mul t i -degree-of- f reedom Hamiltonian systems than Arnol 'd diffusion.

Acknowledgements

Conversations with Mark Muldoon helped clarify the proof of lemma 3.3. I would like to thank Darin Beigie for numerically computing the zero sets of the Melnikov functions for the example in section 5. Also, Greg Ezra and Richard Gillilan have been very generous in discussing their similar results related to coupled standard maps. Conversations with them have helped clarify the geometry. This research was partially supported by an NSF Presidential Young Investigator Award and an ONR Young Investigator Award.

Appendix

Lemma 3.2. W S ( ~ ") n W U ( ~ ) - ~¢'~ F intersects H(x, I ) -- constant transversely.

Proof Recall (see ref. [24]) that F and H(x, I ) = constant intersect transversely if the vector space sum of the tangent space of F and the tangent space of H(x, I)---constant at each point of intersection spans the tangent space of R 2 × R '~ × Tm. Since F and H ( x , I ) = constant are both codimension one, this will follow if the normal vector to F and the normal vector to H(x, I ) =

constant are both independent. Therefore we will compute the two normal vectors and show that they are independent. We begin with a special case.

Case 1. Suppose H(x, I) is of the form

H ( x , I ) = h ( x ) + G( I ) .

Then the unperturbed vector field (3.1) o becomes

Yc=JDxh(x ), /=0, O=DtG(I ), ( x , I , O ) ~ Z x R m x T m.

In this case it is easy to see that the vector normal to F is given by

Ur= (Dxh(x),0,0)

and the vector normal to H(x, I ) = constant is given by

Nt4 = (Dxh ( x ) , DtG( I ) , O).

Thus N r and N H are independent provided D I G ( I ) 4: 0.

"We now show that the general case can be reduced to case 1.

Case 2. The general case. Let

x=u+r(I) .

Under this transformation (3.1) o becomes

a -- J D x n ( u + r ( I ) , I ) ,

= DtH(u + y ( t ) , I ) .

1 = 0 ,

The argument then proceeds exactly as in case 1 since the fixed point of the u-component of this vector field is given by u = 0. Hence F intersects H(x, I ) = constant transversely provided D~H(x, I ) 4~ O. []


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Documents

On the geometry of transport in phase space I. Transport in k-degree-of-freedom Hamiltonian systems, 2≤k