14
8. J.-L. Lions, Some Methods of Solving Nonlinear Boundary Value Problems [Russian trans- lation], Mir, Moscow (]972). 9. G. W. Mackey, Stochastic Integrals [Russian translation], Mir, Moscow (1972). 10. M. Viot, "Solutions faibles d'dquations aux derivdes partielles stochastiques nonlin- ~aires," Th~se, Paris (19?6). 11. I. I. Gikhman and A. V. Skorokhod, Introduction to Theory of Random Processes [in Rus- sian], Nauka, Moscow (1965). 12. Yu. L. Daletskii,"Infinite-dimensional elliptic operators and related parabolic equa- tions," Usp. Mat. Nauk, 22, No. 4, 3-54 (1967). 13. N. Dunford and J. T. Schwartz, Linear Operators, Wiley (1958). 14. W. Rudin, Principles of Mathematical Analysis [Russian translation], Hir, Moscow (1966). HATERIALIZATION OF RESONANCES AND DIVERGENCE OF NORMALIZING SERIES FOR POLYNOMIAL DIFFERENTIAL EQUATIONS Yu. S. II'yashenko and A. S. Pyartli UDC 517.925 I. Introduction In the present article, we consider differential equations with polynomial right-hand side in a neighborhood of a fixed point in the space C n. We will prove (Theorem I) that in the case of resonance the normalizing series is divergent for most of these equations. We will also prove (Theorem III) that for most of polynomial discrepancies of fixed degree s ~ 2 and a thick set in the space of linear parts (this set consists of the points that are patho- logically close to an infinite number of resonances) each neighborhood of the fixed point contains an infinite number of holomorphic invariant manifolds that obstruct the convergence of the formal substitutions, reducing the equation to linear normal form. This article serves to emphasize a general principle, according to which the phenomena, observed near the singu- lar points for analytic differential equations, must, as a rule, be observed also for alge- braic differential equations. This article uses the results and methods of [6, 7]. In order that it can be read in- dependently, let us recall some results, definitions, and notation. Let • denote the space of germs at 0 of holomorphic vector fields with singular point 0 and zero linear parts: f~$=~f(0)=0, f,(0)=0, and set d={(k, ])l]~il, n), k~Z%, Ikl~2}. For integral vectors k=(k I ..... kn), k~Z n , let us set Ik[ = Ikzl + ... + ]knl and let [zl = ~i12+...+I~I ff for a vector z ~G n . Let us consider the analytic differential equation = Az + f(z), (I. 1) [~,.r z~(C%0),* A=diag~, ~=(~ ..... ~)e~C ~. Definition I. A vector ~C n is called a resonance if (Z, k) -- Xj = 0 for some (k, j) ~ J. In the contrary case the vector is said to be nonresonant; correspondingly the linear part of Eq. (1.1) is said to be resonant and nonresonant. Poincare Theorem [9]. Equation (1.1) with nonresonant linear part is reduced to the normal form w = Aw in the class of formal power series by the substitution z = M (m) = w + h tin), h (0) = 0, h. (0) = 0. This substitution is unique and is called the normalizing substitution or the normalizing series in the sequel. *(C n, o) denotes a neighborhood of the origin in the space C n. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 8, pp. 111-127, 1982. inal article submitted June 25, 1980. Orig-" 300 UO90-41U4/86/3203-O300512.5U 1986 Plenum Publishing Corporation

Materialization of resonances and divergence of normalizing series for polynomial differential equations

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8. J.-L. Lions, Some Methods of Solving Nonlinear Boundary Value Problems [Russian trans- lation], Mir, Moscow (]972).

9. G. W. Mackey, Stochastic Integrals [Russian translation], Mir, Moscow (1972). 10. M. Viot, "Solutions faibles d'dquations aux derivdes partielles stochastiques nonlin-

~aires," Th~se, Paris (19?6). 11. I. I. Gikhman and A. V. Skorokhod, Introduction to Theory of Random Processes [in Rus-

sian], Nauka, Moscow (1965). 12. Yu. L. Daletskii,"Infinite-dimensional elliptic operators and related parabolic equa-

tions," Usp. Mat. Nauk, 22, No. 4, 3-54 (1967). 13. N. Dunford and J. T. Schwartz, Linear Operators, Wiley (1958). 14. W. Rudin, Principles of Mathematical Analysis [Russian translation], Hir, Moscow (1966).

HATERIALIZATION OF RESONANCES AND DIVERGENCE OF NORMALIZING

SERIES FOR POLYNOMIAL DIFFERENTIAL EQUATIONS

Yu. S. II'yashenko and A. S. Pyartli UDC 517.925

I. Introduction

In the present article, we consider differential equations with polynomial right-hand side in a neighborhood of a fixed point in the space C n. We will prove (Theorem I) that in the case of resonance the normalizing series is divergent for most of these equations. We will also prove (Theorem III) that for most of polynomial discrepancies of fixed degree s ~ 2 and a thick set in the space of linear parts (this set consists of the points that are patho- logically close to an infinite number of resonances) each neighborhood of the fixed point contains an infinite number of holomorphic invariant manifolds that obstruct the convergence of the formal substitutions, reducing the equation to linear normal form. This article serves to emphasize a general principle, according to which the phenomena, observed near the singu- lar points for analytic differential equations, must, as a rule, be observed also for alge- braic differential equations.

This article uses the results and methods of [6, 7]. In order that it can be read in- dependently, let us recall some results, definitions, and notation.

Let • denote the space of germs at 0 of holomorphic vector fields with singular point 0 and zero linear parts: f~$=~f(0)=0, f,(0)=0, and set

d = { ( k , ])l]~il, n), k~Z%, I k l ~ 2 } .

For integral vectors k=(k I ..... kn), k ~ Z n , let us set Ik[ = Ikzl + ... + ]knl and let [zl =

~i12+...+I~I ff for a vector z ~G n .

Let us consider the analytic differential equation

= Az + f(z), ( I . 1) [ ~ , . r z~(C%0),* A = d i a g ~ , ~ = ( ~ . . . . . ~ ) e ~ C ~.

Definition I. A vector ~C n is called a resonance if (Z, k) -- Xj = 0 for some (k, j) ~ J. In the contrary case the vector is said to be nonresonant; correspondingly the linear part of Eq. (1.1) is said to be resonant and nonresonant.

Poincare Theorem [9]. Equation (1.1) with nonresonant linear part is reduced to the normal form w = Aw in the class of formal power series by the substitution

z = M (m) = w + h tin), h (0) = 0, h. (0) = 0.

This substitution is unique and is called the normalizing substitution or the normalizing series in the sequel.

*(C n, o) denotes a neighborhood of the origin in the space C n.

Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 8, pp. 111-127, 1982. inal article submitted June 25, 1980.

Orig-"

300 UO90-41U4/86/3203-O300512.5U �9 1986 Plenum Publishing Corporation

The normalizing substitution is convergent or divergent depending on the degree of closeness of the vector ~ to resonant vectors. Various convergence theorems have been proved by H. Poincare, C. I. Siegel, V. I. Arnol'd, and A. D. Bryuno. We are interested in the cases of divergence of the normalizing substitution.

Bryuno Theorem [4]. Let a nonresonant vector ~ be pathologically close to a countable number of resonances; namely, for arbitrary positive numbers C and ~ let there exist a count- able set M ~ J such that [(~, k)--~jl<Ce-~!~! for (k, ])~M. Then there exists an f~ such that the normalizing substitution for Eq. (1.1) is divergent.

A strengthening of this theorem, valid for "most" of nonlinearities f~ , has been proved by Ii'yashenko [7]. A program to obtain geometric divergence theorems has been sug- gested by Arnol'd [I, 2] and has been realized in [6] for analytic differential equations (see also the earlier works of Pyartli, cited in [6])~ The geometric divergence theorems are valid for nonlinearities that belong to a thick subset of a certain functional space (a thick set is a countable intersection of dense open sets), but essential restrictions are imposed on the linear parts: They must be close to a countable number of the so,called opportune resonances. The problem of whether vector-valued polynomials belong to the above-mentioned thick set of nonlinearities and if so, then how much "massive" set do they form in the space of all vector-valued polynomials of a fixed degree has remained unsolved in all these works. The present article is devoted to the solution of this problem.

Definition 2. A resonance s n is called a Poincare resonance if 0 is an isolated point of the set {(k, s163 j)~J} and is called a Siegel resonance in the contrary case.

It can be easily proved that the Poincare resonances split into two (intersecting) sets:

Type I. All the numbers %j belong to a discrete subgroup with two generators of the group C o-T--complex numbers with respect to addition.

Type 2. The numbers ~j split into two sets; the elements of the first set lie on a straight line and are commensurable and the elements of the second set lie on one side of this straight line.

Let us set

(~) = { r ~ Z ~l(r, ~) : 0 , ~/:(r+ej, ] )~J}.

Definition 3. Monomials of the form z r, r~R(X) ; are called resonant monomials and those of the form zjzrej are called resonant terms.

Definition 4. The dimension of the linear hull of the set R(%) over the set of real numbers is called the multiplicity of the resonance %.

It is easy to verify that the Poincare resonances of type I have multiplicity n -- 2.

Definition 5. The number minlrl, r~R(s is called the Siegel order of the reso- nance %.

LEMMA 1.1. Let M(%) be the Z-module generated by the set R(%). Then the module M(%) has a basis that is contained in R(%).

Definition 6. The basis r1,...,r K, mentioned in Lemma 1.1, is called a basis of the resonance %. The matrix R with the columns rl,..~ K is called a basic matrix of the reso- nance.

Definition 7. An integral matrix is said to be prime if all its invariant factors are equal to I. A basis of a resonance is said to be prime if the corresponding basic matrix is prime.

LE~I~A 1.2. Each basis of a resonance is prime.

Lemmas 1.1 and 1.Z follow easily from the theory of integral modules [5]~

Let ~s denote the set of vector-valued polynomials f~ of degree at most s. We can easily introduce a coordinate system in the space ~8 by taking the coefficients of the poly- nomials as coordinates. We introduce the Lebesgue measure in this space in the natural man- her.

Let % be a resonance in Eq. (1.1). Then the formal change of coordinates

z = w + h @ ) ,

301

h(O)=O, h,(O)=O, (~o~)

reduces the equation to the normal for o

m = A m + Wg @), ( ~t. 3) where the series g(w) contains only resonant monomials and W = diagw.

The substitution (I.2) will be said to he normalizing.

THEOREM I. Let X be a Siegel resonance of order ~. Then there exists a set ~ {%) of full measure in the space ~s, s>~, such that if f~(%), then the normalizing substitution (1.2) is divergent.

Let V denote the set of the vectors %~C n such that %~/%j~R, for i ~ j, where the vec- tors %3,..-,%n lie between the vectors --%1 and --%= on the plane C.

Definition 8. A resonance %~C n is said to be

a) hyperbolic if %i/~i~R for i x j,

b) positive if R{~)~Zn+,

c) primitive if the set R(%) is generated over Z+ by a basis of the resonance~ This basis and the corresponding basic matrix are said to be primitive.

t Definition 9. A positive primitive Poincare resonance is called an opportune resonance.

Convention. In the sequel, by a basis and a basic matrix of an opportune resonance we will mean a primitive basis and the basic matrix corresponding to it.

LEMMA I .3. If a hyperbolic resonance ~C n is opportune, then under a suitable renum- bering of the components it turns into a resonance %~V. A basic matrix of such a resonance

has the form I ! 1 p!

R=~r~ r~ Or~ . . .0 I '

~ t l ~2 ~ . . . . . . t n /

(1 .4) ~r~+~@0, ]~(1 n--2). FI /'2

This is Lemma 3.3 of [6].

In the sequel, all the opportune resonances are assumed to belong to the domain V.

LEMMA 1.4. The set of opportune resonances is dense in V.

This lemma is a quite weakened form of Lemma 8.1 of [6].

By (X, Y) we will denote a neighborhood of a subset Y in the space X,

Definition I0. A resonance X is materialized in a K-dimensional family (~(e)} of equa- tions of the form (1.1), e~(C K, 0) , if the equation ~(s) has invariant manifold M(e) that is given in a certain (not depending on a) map w on (C n, 0) by the equation z = ~(w), where

(~) = ~ ~,~', ~, ~ c,~ r ~ R ( ~ , )

is a series in resonant monomials and ~:(C n, u) ~ (cK, 0) is a local epimorphism.

Let us consider th~ family of equations

z = A z + ZFs +F(z , ~), (i .5)

where Z~diagz, ~(C K, 0), F]~=oon~t~, X is a hyperbolic opportune resonance of multiplicity K = n- 2 with basic matrix R, and F be an n • (n -- 2)-matrix such that det RF z 0. Let us

set f = F!g=0.

THEOREM II. For each hyperbolic opportune resonance X in the space 9~s, s>~2, there exists a set Q(~) of full measure such that if [~Q(%), then the resonance % is materialized in the family (1.5).

THEOREM III. In the space ~, s>~2, there exists a set ~ of full measure and for each discrepancy f~ ~ there exists a thick (in V) set ~f such that for f~ ~ and ,%~:-'.~Pf, Eq. (I..I) has, in each neighborhood of the origin, infinitely many invariant manifolds, each of

302

which has the form wR=c, c~C n-2, in a suitable map w [the map w and the matrix R of the form

(1.4) are the same for each manifold; w R = (wr1~~ These manifolds obstruct the convergence of the normalizing series.

The proofs of Theorems II and III are based on the theorem on the materialization of resonances [6]. In this article, we give three statements, each refining the preceding one. The following theorem is the simplest of them.

Theorem on Materialization of Resonances (the first statement)_. Each opportune reso- nance ~ of multiplicity K is materialized in a K-dimensional family of Eqs. (1.5) of general position.

This is a particular case of Theorem la of [6].

2. Divergence of Normalizing Substitutions (Resonant Case)

This section will be devoted to the proof of Theorem I. For simplicity of expression, we will assume % to be fixed and will not indicate dependence on k. Moreover, % is a reso- nance throughout this section. Theorem I follows from the following two lemmas.

LEMMA 2.1. If the space ~s contains a set Q of nonzero measure such that the normal-. izing substitution for Eq. (1.1) is convergent for ~Q , then the normalizing substitution is convergent for arbitrary f from ~.

LEMMA 2.2. Let ~ be a Siegel resonance of order T. Then for each s > T there exists a discrepancy f~s such that each normalizing substitution of Eq. (1.1) is divergent.

To prove Le~ma 2.1 we need additional information about the dependence of the coeffi- cients of the normalizing substitution and the normalizing form on the right-hand side of the equation.

Let us recall that a function f of variables zl,.~ N is said to be quasihomogeneous of degree m with weight mj of the variable zj if

f . . . . . = tmf i . . ,

In Eqs. (1.1), (I.2), and (1.3) we, respectively, set

f (z) = ~ L, /%; (p,i)eJ (2.1)

n

/ = 1 rER(X) (q,l)ed

LEPLMA 2.3. Suppose that Eq. (1.1) is reduced to the normal form (1.3) by the substitu- tion (1.2). Then the coefficients gr,i (or h a 7) are quasihomogeneous polynomials in the coefficients fp,j for Ip] < Jr[ + I (or 3pl ~=Tqi) of degree Ir] (or lq] - I) with the weight IpJ -- I of the variable fp,j

Proof. The fact that the coefficients g~.i and hA ~ are polynomials in fn + follows from a~-n ~Tementary analysis of =" ~ . . . . the well-known process of computation of the coefficients of the normal form and the normal substitution [2]. Let us establish the quasihomogeneity of these polynomials.

Let us set f(t, z)= E tlPJ-~fP'/zPe~" If we find the normal form and the normalizing sub- (P,I) EJ

stitution for the equation

= Az + f~t, z), ( 2 . 2 )

then the coefficients of this no,rmal form and substitution will be the values of the poly- nomials gr,i and hq~l, at which each of arguments fp,j is multiplied by the corresponding

weight factor t ipl'1. The formal mapping H:w § w + h(w) corresponds to the normalizing sub- stitution (I.2); it is called the normalizing mapping for Eq. (1.1).

Let us set Lt(w) = wt = t-Zw. The mapping L t transforms Eq. (1.1) into (2.2) and Eq. (I .3) into the equation

�9 . , .

w~ = Aw t + W ~ ~ (m~),

303

where

# (w) = ~ F'g,w', g, ~ ~c~. (2 .3 ) r~R(s

AS b e f o r e , t h e s e r i e s g t c o n t a i n s o n l y r e s o n a n t m o n o m i a l s . T h e r e f o r e , t h e f o r m a l m a p p i n g L t o L~ 1 i s n o r m a l i z i n g f o r Eq. ( 2 . 2 ) , and ( 2 . 3 ) i s t h e n o r m a l f o r m f o r ( 2 . 2 ) . I t i s o b v i o u s t h a t

L t o H , L71 (w) = w + h t (w),

where

= t (q,l)EJ

This means that if each of the variables fp,j, on which the functions gr,i and hq,~ depend, is multiplied by the corresponding weight factor, then gr,i and hq, l would be multiplied by t [rl and tlq I-l, respectively, which was desired to be proved.

Proof of Lemma 2.I. This proof is analogous to the arguments of [7]. Without loss of generality, we can assume that the set Q lies in a closed ball with center 0.

Let f and h have the form (2.1). By Lemma 2.3, the coefficients hq,l of the substitu- tion are polynomials in the coordinates of the element f of ~, of degree at most lql -- 1. We denote them by hq,l(f). Let QN denote the set of ~Q such that ]hq,~(f)l < Nlql, where N is a positive integer. The sets QN are measurable since hq, I are polynomials, and UQ~=Q" Therefore, mes QN ~ 0 for some N; let us denote this QN by E. The following lemma is the com- plex variant of Lemma 2 of [8]. To prove this complex variant, we require only small changes.

LEMMA 2.4. Let K be a closed ball in C n, E be a measurable subset of K, and g be a poly- nomial of degree n in C m such that Igl E ~ s, where ~ > 0. Then

max,gl< e(c mesK ) ~, K mesE

w h e r e c d e p e n d s o n l y on m.

Using this lemma, we see that for f~K

N 'q' (c rues < (c lN) '< ( 2 . 4 ) I hq,t ff) l < \ , mes E /

The i n e q u a l i t y ( 2 . 4 ) i m p l i e s t h a t t h e n o r m a l i z i n g s u b s t i t u t i o n i s c o n v e r g e n t f o r a l l f ~ K and i t s r a d i u s o f c o n v e r g e n c e i s n o t l e s s t h a n ( e l N ) - Z . S i n c e we c a n t a k e any b a l l t h a t c o n - t a i n s Q a s K, t h e n o r m a l i z i n g s u b s t i t u t i o n i s c o n v e r g e n t f o r a l l f ~ . Lemma 2.1 i s p r o v e d .

The p r o o f o f Lemma 2 . 2 u s e s a r e s u l t o f A. D. B r y u n o ; we b e g i n w i t h i t s s t a t e m e n t .

A. Let L A be the operator of commutation with the vector field Az that acts on the space of formal vector fields. Let o(f) denote the spectrum of the operator LA, bounded on the space of all formal vector fields, in whose expansion only those monomials occur which also occur in the expansion of f. In other words,

if) = {61 ~ (k, 1) ~ J : fk, 1:7 e O, 6 = (k, ~) - - s

Definition. A set g~C is said to be totally nonlinear if {51 ..... 5m}~a implies that

5 i + . . . + 5 m ~ , (m > 1 ) .

Definition. The system

=Az + zg (~ + f(z) (2 .5)

is said to be special if the series f does not contain linear and free terms, the series g consists of only resonant monomials, and o(f) is a totally nonlinear set.

Let ('}~ be the projection of the space of formal vector fields on the eigensubspace of the operator L A corresponding to the eigenvalue 6 along the complementary invariant subspace

of this operator:

'r =6

304

Definition. The normalizing substitution (].2) for Eq. (~.I) is said to be distinguished

if {h}o = 0.

The following theorems are valid.

THEOREM I [4, p. 204]. If a distinguished normalizing substitution for Eq. (I.I) is divergent, then each normalizing substitution is divergent.

THEOREM 2 [4, Chap. III, Sec. 6]. Let H = id + h be a distinguished normalizing sub- stitution of the special system (2.5). Then this substitution reduces the system (2.5) to a normal form of the type

& = A w + ~ g (w),

the formal series {h}~ is the unique formal solution of the Moreover, for each 6 ~ a ( f ) system

~ (Aw + Wg) -- 0 (Aw+ ~g){h}~ + {F}+. ,, ( 2 . 6 ) Ow Ow

B. We pass to the proof of the lemma.

Let us recall that i is a Siegel resonance of order T. By Definition 5 of Sec. ], there exists an r~R(%) ~Z~ such that (r, %) = o and Ir[ = T. Without loss of generality, we can assume that rl ~ O. We take an arbitrary a~Zn+ such that ~I ~ 0, (~, ~) -- Ii = @ z 0, and [al < T. Let us set

g = r e 1 , f = z=el

and consider the system (2.5) with these f and g:

= Az + Zl fe 1 + z~el . ( 2 . 7 )

I t i s o b v i o u s t h a t t h e s e t o ( f ) c o n s i s t s of t h e s i n g l e p o i n t ~ a 0 a nd i s , c o n s e q u e n t l y , totally nonlinear. Therefore, the system (2.7) is special, and Theorem 2 can be applied to

it. Let H = id + h be a distinguished normalizing substitution for the system (2.7), h(w) =

Ehqjmael, The series (h} 6 satisfies Eq. (2.6); for ~ > I the l-th component* of this equation J

has the form

~.+ hqdwq [(q' ~) § qlw~] = ~' Z hq,,w+, ~,~--~=5 (q,~--zt=~ . . . .

or

side of Eq. (2.8).

hq,,w q [6 + q~w'] = O. ( 2 . 8 ) (q,~)--s

We fix an arbitrary vector ~~ such that ~0--r~Z~ . Let us set

6 ! = 6 ~ h~f,l=b i. We t a k e a l l t h o s e t e r m s whose e x p o n e n t s a r e c o n g r u e n t t o B ~ m o d u l o r f r o m t h e l e f t - h a n d

We get

bow~ ~ + ~ , w~ ] [6bj + ~{bi_~] = O. 1=1

Hence b i = O. Consequently, (h}6~ = u for ~ > O.

Now, analogous computation for the first component of Eq. (2.6) gives

~.~ hq, a w q [8 + (ql - - 1 - - q ) ~ ] = ~'r "

(~,q)--M=6

Let us set ~ = 0 ~J = ~ + jr, and h~ix=aj+. Selecting all those terms whose exponents

are congruent to ~ modulo r from the last equation, we get

ao6W ~ + _~ w ~] [6aj - - (1 + ra - - a{) a i _ j = me. 1=t

*Let us recall that the ~-th component of a vector or a vector-valued function~is denoted by~.

305

Therefore, since ~J = a l + jrl, we get

i ao = 6 - l , a~ = 6 - ( i + l ) [ - I (1 - - a 1 - - r , ( i - - 1)) .

Since rz ~ to, t he s e r i e s EaaW a l i s d i v e r g e n t . By the same t o k e n , t he d i s t i n g u i s h e d n o r - J m a l i z i n g s u b s t i t u t i o n f o r Eq. (2 .7 ) i s d i v e r g e n t , and, by Theorem 1, each n o r m a l i z i n g s u b s t i ' t u t i o n f o r Eq. (2 .7 ) i s a l s o d i v e r g e n t . Lemma 2.2 and , w i t h i t , Theorem 1 a r e p r o v e d .

3. Materialization of Resonance for Polynomial Equations

In this section, we prove Theorem II.

I. l-Nondegenerate Discrepancies. We formulate explicitly the requirement of general position for the family (1.5) in the theorem on the materialization of resonances. To this end, we require the following preliminary considerations.

Let us consider the family of equations

z = A z + ZFs + F (z, s), (1 .5 )

in which I is a hyperbolic opportune resonance with basic matrix R of the form (1.4) such that det RF z 0. Let f = F1a=0,

~=Az+/(~ (3.~) be the equation of the family (1.5) corresponding to the zero value of the parameter ~, and

= A w + ~ g g ( w )

be the normal form of Eq. ( 3 . 1 ) . Let us s e t u = w R. S ince 1 i s an o p p o r t u n e r e s o n a n c e , each r e s o n a n t monomial i s a p o s i t i v e power of u; in p a r t i c u l a r , t h e r e e x i s t s a f o rma l s e r i e s

such t h a t g(w) = ~ ( u ) . The f o l l o w i n g m a t r i c e s p l a y an i m p o r t a n t r o l e i n t h i s s e c t i o n :

A~ ([) = -- (RF)-*R ~ (0), 0 / 2

A (~., f) = R ~ (0).

(3.2)

Definition. A discrepancy f ~ ( S ~ 2 ) is said to be l-nondegenerate if the matrix Al(f) [or, what is the same thing, the matrix A(I, f), since det RF ~ 0] is nonsingular.

Theorem on Materialization of Resonance (second statement). Let I be a hyperbolic op- portune resonance and the discrepancy f = Fla= 0 be l-nondegenerate in the family (1.5). Then the resonance I is materialized in the family (1.5).

Remark. It has been proved in [6] that each analytic discrepancy f is turned into a l- nondegenrate one by a small perturbation; under this perturbation the resonant terms of the Taylor expansion of f vary. It is obviously impossible to construct a l-nondegenerate (with respect to a countable number of resonances %) polynomial vector field of fixed degree s in this manner. The following lemma constitutes the main content of this section.

2. LEMMA 3.1. If s ~ 2, then for each fixed hyperbolic opportune resonance s in the space ~s there exists an algebraic hypersurface LI, s such that det A(I, f) ~ 0 for f~L~,~.

Remark. It follows from Lemma 3.1 that if ~ s \ L ~ , s for the family (1.5), then the resonance I is materialized in this family.

For fixed I, det A(%, f) is a polynomial on ~s (this follows from Lemma 2.3). There- fore, to prove Lemma 3.1 it is sufficient to find one discrepancy f~s such that detA(%, f) x 0. The following lemma ensures the existence of such a discrepancy.

LEMMA 3.2. Let % be a hyperbolic opportune resonance with a basis {ri}, i~(l, n--2). "Let {qJ} be a collection of vectors from C n such that (r i, qJ) = Gij , where 6ij is the Kronecker symbol. Set

r [~/=f(%I=X, /(& = X 4 8q l, ! ~ ( l , n - - 2 ) , ( 3 .3 )

i ' l = l

306

Then the matrix A(X, f(6)) is a diagonal matrix for arbitrary ~ and is nonsingular for

sufficiently small ~.

Proof. A. Let H(~) denote a distinguished normalizing substitution w = H(~)(z) that transforms the equation

to the normal form

r~ = W (L + g(~)(w))

= Az § f(a) (z). ( 3o 4 )

For each formal series r and each natural number m let ~(m) denote the segment of ~ which consists of terms of degree at most m. We denote the unit vectors of the space C n-2 by ej, j ~ (I, n -- 2).

We introduce a partial order in the set Z~ as follows: p~q and q~p if q--p~Zn• p-~ q if p=2~q and p ~ q.

Let us denote the differentiation by d/dt 6 by virtue of the system (3.4). Let R be a basic matrix of the resonance % and set • max Irfl. Let us set

]E(l,n--2)

~<6) = t,,(~), , A~ = A (k, f6). The following proposition is the first step in obtaining explicit formulas for the ele-

ments aij(6) of the matrix A 6. For brevity, we will delete below an explicit indication of dependence on 5.

Proposition 3 1 I Let ~(z) ~ C n-~ �9 . ~ =N~ ,~p~ . If p-~ri for a certain ]~(i, n--2),

then #p = 0 and ~rj = ej.

2 ~ . Let ~ = (d/dt)~ and #(z) = 2~pzP, where #p ~ C n-2. If p-~ri+ri for certain i, j ~ (I, n- 2), then ~p = O o

~/+/= a~je~ + a~e) br g ~ 7; (3.5) p

~ : aiie i.

have

Proof. I ~ We have

HR, H-I= w R.

Since H is a substitution with identity linear part, the statement I ~ follows~

2 ~ We use the invariance of the differentiation along a (formal) vector field.

d Wm= (diag w R) Rg(wR); df

d H R = (diag H e) R'g (HR). dt

Hence t h e s t a t e m e n t 2 ~ f o l l o w s f r o m Eqs . ( 3 . 2 ) ~

B. Le t us c o n t i n u e t h e p r o o f of Lemma 3 . 2 . We h a v e

E (z) = -77- ~ (z) = % (p, ~, + Y, Pz3 z ~. P

We

Hence

~go = ~p (p, k) + i ~p- ' l (P - - et, fg , (3.6)

assuming that ~p-el = 0 for p--el~Z$.

For arbitrary i, ] ~ (I, n-- 2) we have

n

~ / + / = ]~ r (ri + r i - - e~, p), l = 1

(3.6')

since R% = 0.

307

Here we fix i, ]~(l,n--2), and consider vector p~Zn+, subjected to the following restrictions :

p . ~ ri + r i, p ~>.- r ~ for p >- rL ( 3 . 7 )

or

p..~rZ + r i, p~,-r i ~r p~>iri ( 3 . 7 ' )

We have (p, X) ~ O and Up = u under the condition (3.7') by virtue of Proposition 3.1. Then, from (3.6), under the condition (3.7') we get

(I)p = - - (p, ~)--1 ~ (~p_el (p--el, fO. ( 3 , 8 ) t=t

The last equation is a recurrent formula for the computation of the coefficients ~p. In order to write the final solution, we introduce the following notation. Let us consider the set F of all oriented polygonal lines with knots in the integral lattice of the space R n whose limbs are vectors that take one of the values el,...,e n. We will call a polygonal line y~1 ~ admissible if its origin is situated at one of the points rJ. Let Pk,p denote the class of all admissible polygonal lines with the origin r k and endpoint p, k ~ (I, n). Since

r k = r~e I + r~e2 + r~+2e~+2 ,

t h e c l a s s rk, p i s n o n e m p t y u n d e r t h e c o n d i t i o n ( 3 . 7 ) , p r o v i d e d k = i o r k = j . We f i x an a r b i t r a r y p o l y g o n a l l i n e y~Fk,p . We a s s u m e i t s o r i g i n t o b e t h e z e r o v e r t e x , f r o m w h i c h t h e f i r s t l imb i s s u e s ; we number t h e v e r t i c e s and t h e l i m b s in t h e o r d e r o f p a s s a g e r o u n d t h e polygonal line. Let us denote the s-th vertex of the polygonal line u by pS and the length of y by [y'[ (p0 = rk and p[Y[= p) and its s-th limb by el(s) or e~y(s ). Under the condi- tion (3.7'), for each polygonal line y~l~,p we set

Iv[ I-I v = ( - - 1)lvt H (p~ --et(~), fq~))

It is easy to prove by induction from Eq. (3.8) that under the condition (3.7')

E E n, (3.9) v~ri,p vEri, p

Hence @p = 0 for i ~ j. Indeed, the first factor in the product ~y is equal to zero for an arbitrary polygonal line y~F~.pU~j.~ under the condition (3.7'). This follows at once from the definition (3.3) of the discrepancy f and the form of rJ. It now follows from Eq. (3.6') that Pri+r j = 0 for i z j. Consequently, aij = aji = 0, and the matrix A is diagonal.

C. Let us now compute the right-hand side of Eq. (3.6') for i = j. If ~Fi,2/, then we denote the polygonal line ~el(ivl) by y' (assuming that a limb does not contain its first vertex, but contains the last vertex). We set

H$ = (2r t - et(IVl), fl(lvl)) ~?,. Simple computation gives

Ivl-I r

Fir ---- ( - - 1) 1~1-1 (r l , / t(o) 1-1 (pS, % +0(6)) (p% ~,)

t t IJv~(--l)l~l-16(l+O(6))for I v ( 1 ) = i + 2 ; 1-Iv----O for l v ( 1 ) = / = i + 2 . On the other hand, it follows from Eqs. (3.9) and (3.6') that

i

z Z I-Iv; vEF i,2rl

together with (3.5), this gives ....... ]

a . = ]~ ( - - 1 ) v~t-x6(1 + 0 ( 6 ) ) .

3?EFi,2r l l~(i) =t'-l- 2

It is obvious that aii ~ 0 for small 6. Lemma 3.2 and, with it, Lemma 3.1 and Theorem II are proved.

308

4. Infinite Number of Invariant Manifolds

This section is devoted to the proof of Theorem III.

I. Heuristic Arguments. We use a nontraditional approach to the geometric divergence theorems whose idea is due to Taraeev [1o]. In the traditional approach (see, e.g., [6]) we describe the set ~ of "almost resonant" (i.e., pathologically close to a countable number of resonances) linear parts, which turns out to be thick in a certain domain of the space C n. Further, for each linear part %~, we construct a dense set ~ that is thick in the space of discrepancies and is such that for %~ and [ ~ the equation

Z=AZ +f(z) (4. I)

has a countable number of invariant manifolds, obstructing the convergence of normalizing series, in each neighborhood of the origin. Each of these invariant manifolds arises as the materialization of one of the resonances, close to %. This approach requires the overcoming of substantial technical difficulties (see [6, Secs. 8, 9]), but enables us to describe the set 2 explicitly.

Tareev's approach, carried over to our situation, is as follows. At first, a discrepancy f is fixed. Then we take the set of all the opportune resonances ~, for which this discrep- ancy is %-nondegenerate. Lemma 3.1 ensures the existence of a discrepancy f, for which the indicated set is dense in a domain V~C n. This set is contained in a countable union of two- dimensional planes, each of which consists of resonances of multiplicity at least n -- 2. Each such plane R has as domain of influence a neighborhood (in C n) of a dense subset N*~H..

For each % in this domain of influence, Eq. (4.1) (the discrepancy f in this equation has been fixed above) has an invariant manifold in a neighborhood of the origin; it is the materialization of a, close to %, resonance ~ in the family of equations that contains (4.1). The existence of such a domain of influence is easily deduced from the theorem on the materialization of resonances. Of course, here we cannot find a lower bound for the "thickness" of this region. The set of the points that belong simultaneously to a countable number of domains of influence from different planes form ~f . For ~ / , Eq. (4.1) has a countable number of invariant manifolds, obstructing the convergence of normalizing series, in each neighborhood of the origin.

Now, it follows from Lemma 2.1 that for each %~f the normalizing series for the equa- tions z = Az + f(z) are divergent for almost all f~s.

Tareev's approach has the advantage that in the analytic case it enables us to overcome many technical difficulties and in the polynomial case it is the only possible method at pre- sent. The deficiency of this approach is that it does not enable us to indicate any element of the set ~Ff explicitly.

Theorem III is proved according to the following plan. At first, in Lemma 4.1 we impose sufficient conditions on the countable set of invariant manifolds under which this set ob- structs the convergence of normalizing series.

Then we construct the set ~ s , indicated in Theorem III, and fix an arbitrary dis- crepancy f~ ~. Further, we formulate the theorem on the materialization of resonances with estimates and construct domains of influence (Lemma 4.2). Finally, we construct the set ~f and prove that the invariant manifolds of Eq. (4.1) for %~I satisfy the conditions of Lemma 4.1 (Lemma 4.2 and the discussion following it).

~. Nonlinearizability. Let C n be the standard number space, C n = {~}, ~ = (~l,...,~n), and c and 6 be positive constants. Let us set

A = { ~ l ~ . . . . . ~ = o},

~ : @ ~ A , ~ ( ~ 1 , ~ ) ,

O ~ = { ; [ i ; ,1~(6% 6), i = l , 2; I ~ j l < 6 , 1 = 3 . . . . . n}. A~ = D ~ n A.

It is obvious that wD 5 = A6"

Each analytic map ~ on (C n, 0) gives a biholomorphie mapping of (C n, 0) onto a domain of the standard space C n, ~(0) = 0. Let A~ T~,c, D~,6, and s denote the inverse images

309

of A, Tc, D~, and A~, respectively, under this mappingo~ Let ~ be the germ of the mapping (C n, 0) + (s U), defined by the equality ~ o v~ = T o ~. The restrictions of an arbitrary representation of the germ ~ to subsets of (C n, O) are also denoted by ~.

Definition. A two-dimensional analytic manifold, embedded in a neighborhood (C n, 0), is said to be resonant in a map ~ defined on (C n, U) if it (more precisely, its real two- dimensional submanifold) is a covering without edge over the torus T~, c for a certain c under the projection ~.

THEOREM [6, Sec. 4]. Let the equation w = (diag%)w have an invariant manifold that is resonant in the map w. Then the components of the vector X are linearly dependent over Z.

LEMMA 4.1. Let z be a map on (C n, u), {~s} be a sequence that converges to zero, 6 s > 0, and D s = Dz,~s. Let M s be a sequence of two-dimensional manifolds that are e~Dedded in (C n, 0) and are such that MsnDs is a finite-layered covering without edge over the domain A~=D,n& under the projection ~z" In addition, let each tangent plane to each of the mani- folds Ms lie in the cone

Idzj[<ldz~l§ 7:8 . . . . . n. (4 .2 )

Then in each map w on (C n , 0)

z(~)=m+h(m), h(O)=O, h,(O)=O, and all the manifolds Ms, beginning with a certain manifold, are resonant.

The proof of this lemma is contained in the proof of Theorem 11 of [6, Sec. 8].

COROLLARY~ If Eq. (4.1), the spectrum of whose linear part is linearly independent over Z, has a countable number of invariant manifolds Ms that satisfy the conditions of Lemma 4.1, then each normalizing substitution for Eq. (4.1) is divergent.

Proof. By the condition of the corollary, the vector % is nonresonant and the normal form is linear. If at least one normalizing substitution is convergent, then each normal- izing substitution with the same linear part is also convergent. It now follows from Lemma 4.1 and the nonlinearizability theorem that the components of the vector % are linearly de- pendent over Z, which is a contradiction�9

2. Construction of the Set ~. Let G~ denote the set of all basic matrices of the hyperbolic opportune resonances from the domain V. For each matrix R~,Y~ we define a plane

E R as follows:

n~={x~_c~lRx=o}. For each discrepancy ~ , let us consider detA(%, f) as a function on ~R [the matrix

A(%, f) is defined by Eq. (3.2)].

Proposition 4.1. The set

~ = { f ~ s V R ~ N : d e i A ( s f) ~ 0 as a function of ) ~ R } (4 .3 )

c o n t a i n s a s u b s e t of f u l l measure t h a t i s t h i c k in ~s.

Pr__oof. We f i x an a r b i t r a r y o p p o r t u n e r e s o n a n c e X(R ) in each p l a n e 2 R. By v i r t u e of Lemma 2 . ] 7 the f u n c t i o n d e t A ( X ( R ) , f ) i s a p o l y n o m i a l in f ~ s . By v i r t u e of Lemma 3 . 1 , t h i s p o l y n o m i a l i s no t i d e n t i c a l l y equa l to z e r o . The s e t

has f u l l measu re , i s t h i c k , and i s c o n t a i n e d in ~. , which was d e s i r e d to be p r o v e d .

Remark. A n a l y z i n g the method of s u c c e s s i v e a p p r o x i m a t i o n , we can e a s i l y p rove t h a t the f u n c t i o n de tA(%, f ) i s r a t i o n a l w i t h r e s p e c t to s f o r each R ~ .

3. Theorem on the M a t e r i a l i z a t i o n of Resonances and Domains of I n f l u e n c e . A. We f i x a discrepancy f~ and construct a set ~f that satisfies the requirements of Theorem III. By virtue of the preceding remark, for each R ~ the domain H'R= {%e~HRli is a hyperbolic opportune resonance and detA(X, f) ~ 0} is Zariski-open$ in ~R"

Let us renumber the matrices R~:~={RII/~N} �9

~They are defined for sufficiently small 6. SA set is Zariski-open if it is the complement of an algebraic manifold.

310

o * LE~iMA 4.2. 1 For each matrix R ~ there exists a neighborhood U R of the set ~R in the space C n such that for each )~UR\IIR Eq. (4.1) has an invariant manifold M~(%) that is the materialization of a resonance V~II*~ in an (n- 2)-dimensional family of equations, containing (4.1).

2~176 Let {67} be an arbitrary sequence that converges to zero. Then the neighborhoods U R can be selected such that for each ~ there exists a 6 < ~ such that for %~U~ the in-

tersection M~(%)~Dz,~ is a finite-layered covering without edge over the domain Az,~ under the projection ~z; ~II*~,

The domain UR is the domain of influence of the plane H R .

Theorem III follows immediately from Lemma 4.2.

Indeed, let us set ~ = U U~; .~= ~ '~s" l>~s s

The set of opportune resonances is dense in V (this follows from Lemma 8.1 of [6]). Consequently, the set ~s is open and dense, and the set ~f is thick in V.

Each point %~f belongs to a countable number of domains of influence of different resonant planes. Equation (4.1) with this % and the discrepancy f, fixed above, has a count- able number of invariant manifolds, which satisfy, by virtue of Ler~na 4.2, the conditions of Lemma 4.1. By virtue of the corollary of Lemma 4.1, these manifolds obstruct the convergence of the normalizing series. Theorem III is proved.

Proof of Lemma 4.2. At first, let us recall the definition of materialization of a resonance, slightly varying the notation.

Definition. A resonance ~ is materialized in the family of equations

= (diag ~) z § Z F~ § f (z), (~ (~) )

if Eq. (~(8)) has an invariant manifold that has the form M(~, r = {w]~ = ~(w)} in a special,

independent of s, map w on (C n, 0), where ~: s ~r~f, ~r~C a-2 is a series of resonant rER(~)

monomials; and ~:(C n, O) § (C n-2 , O) is a l oca l epimorphism.

Remark. If z = H(w) = w + h(w), then

M(~, e)=H(M(~, e))

is an invariant manifold of Eq.(~(~))o

Sometimes we write w~, h~, and ~ to emphasize dependence on ~. The discrepancy f is, as above, fixed.

We need the following theorem.

Theorem on the Materialization of Resonances with Estimates [6, Secs. 5-7]. Let ~V be a hyperbolic opportune resonance with basic matrix R and the discrepancy f b~ ~-nonde- generate. Let ~ V be an arbitrary domain with compact closure in V, and b~Q. Then the resonance ~ is materialized in the family {~(s)}. In addition, there exist constants A and B, depending only on f, R, and ~, such that

I ~ . the substitution H = id + h and the vector-valued function 6 are defined in the domain lwl < A;

2 ~ lh(w)[ < Blw[ 2 for [w[ < A; (4.4)

3 ~ . ]~(w) l < Biwl 2 for lwl < A. (4.5)

These inequalities follow from the inequalities (7.2') and (7.3) of [6], where the de- pendence of A and B on f and R is indicated, but this is immaterial for our approach.

B. We construct the domain U R as the union of neighborhoods of points of H~ (as before, the discrepancy f is fixed). We fix an arbitrary point ~0~*R and a domain ~, containing ~0, such that ~ V and

Until the end of the paragraph B we assume that

311

all the estimates, "uniform with respect to D," are carried out for these ~; and the expres- sion "for all ~" means "for all ~R*RnQ ." The preceding theorem is applicable to all these

and the constants A and B do not depend on ~. As above, we define the mapping

n~ : (C.-2, O) ~ (C~-2, 0)', n~ (w~) = ~ (w~). The mapping ~ is a local holomorphism, since

~, (~ = -- (R O -~ A (~, f);

the matrix A(p, F) is nonsingular by the definition of ~. Let us set ~p = q~l. The mani- fold M(~, s), arising during the materialization of the resonance p in the family {~(s)}, has, by definition, the form

M (p, ~) = H~ (/~/(p, ~)), ( 4 . 7 )

,~ (~, 0 = (~ = ~. ( ~ } = ( ~ = * . (~)}. ( 4 . 8 )

We prove that for all ~ the vector-valued function ~ is defined in the same domain and admits, for a certain s0 > o, the following uniform (with respect to p) estimate:

I r ~ I~1<%. (4.9)

Indeed, by virtue of (4.6) there exists a C' such that for all

II ~7~ (o)II < c' . (4. ~ 0)

The estimate (4.9) now follows from the inequalities (4.5) and (4.10). The corresponding reduction (the inverse function theorem with estimates) is a standard lemma of analysis; it is given in [6] with additional details (Lemma 8.4).

Now, we prove that the manifold M(p, s) is "compressed" to the plane Awp for a fixed and ~ § 0, More precisely, if a~-(0, A) and bE(0, A),a<b , are arbitrary constants and

~ M ( g , e),'is such that Iw~]~ (a, b), i = 1, 2; ~ T ~ M , then

I ~ [ < ~ ( ~ ) , I ~ j l ~ ( ~ ( l ~ ] + l ~ = l ) , y = 3 . . . . . n, (4.11)

where the function B depends on a and b, but not on ~, and B(e) + 0 as s + 0.

This follows immediately from Eqs. (4.8), the inequality (4.9), and the fact all the elements of the matrix R are positive.

Now, let {6z} be the same sequence as in Lemma 4.2 and R = R I. Let Is] < so and 6~(0,81) (the numbers so and ~ will be chosen below). Let us set

By v i r t u e of the e s t i m a t e ( 4 . 4 ) , the mapping H~ is un i fo rmly (with r e spec t to ~) c lo se to the i d e n t i t y mapping in the C~-topology in a s u f f i c i e n t l y small neighborhood of the o r i g i n . Now, wi th the help of (4 . !1) we can e a s i l y prove t h a t the numbers so and 6 can be chosen so small that for all ~ the intersection M'(~, ~NDz.~ is a finite-layered covering without edge over the domain Az,~ under the projection ~z" By the same token, if

~ = ~ + F d , g ~ H ~ n ~, t 8 [<%, (4.12)

then the manifold M(~) = M(p, s) satisfies all the requirements of Lemma 4.2. The set of all the ~, satisfying (4.1Z), is the desired neighborhood of the point p0; let us denote it by

UR,~ 0 �9

We define the domain of influence U R by the equality

u~= U u~,~o. , ~o6HR

Lemma 4.2 and, with it, Theorem III are proved.

I.

LITERATURE CITED

V. I. Arnol'd, "Remarks on singularities of finite codimension in complex dynamical systems," Funkts. Anal. Prilozhen., 3, No. I, I-6 (1969).

312

2. V. I. Arnol'd, "Lectures on bifurcations and versal families," Usp. Mat. Nauk, 27, No. 5, 119-184 (1972).

3. V. I. Arnol'd, "Bifurcations of invariant manifolds and normal forms of neighborhoods of elliptic curves," Funkts Anal Prilozhen. 10 No 4 1-12 (1976)

4. A~ D. Bryuno, "Analytic form of differential equations," Tr. Mosk. Mat. Obshch., 25, 119-262 (1971).

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divergence of normalizing series," Tr. Sere. im. I. G. Petrovsk., No. 7, 3-49 (1981). 7. Yu. S. II'yashenko, "Divergence of series, reducing an analytic differential equation

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�9 s

9. M. Polncare, Oeuvre, Paris (1916)�9 10. V. P. Tareev, "On the conditions for the countability of the number of cycles for a

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