Smooth Invariant Manifolds and Normal Forms (World Scientific Series on Nonlinear Science. Series a, Monographs and Treatises, V. 7)

NONLINEAR SCIENC E ~~'""10~ WORLD SCIENTIFIC SERIES ON

Series A Vol. 7

Series Editor: Leon O. Chua

SMOOTH INVRRIRNT MRNIFO~OS RNO NORMR~ FORMS

I. U. Bronstein and A. Va. Kopanskii Institute of Mathematics Academy of Sciences of Moldova

h

'

World Scientific

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SMoom INVARIANT MANIFOLDS AND NORMAL FORMS Copyright C 1994 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts tMreof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be inllented, without wriUen permission from tM PublisMr.

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ISBN: 981-02-1S72-X

v

CONTENTS

Introduction

vii

Chapter I. Topological properties of nows and cascades in the vicinity of a rest point and a periodic trajectory § 1. Basic definitions and facts

1

§ 2. Hyperbolic rest points § 3. Floquet-Lyapunov theory § 4. Hadamard-Bohl-Perron theory

8

Chapter

n.

13 21

Finitely smooth normal forms of vector fields and diffeomorphisms

c:c

§ 1. The problem on reducing a ~ vector field to normal form in the vicinity of a hyperbolic rest point § 2. Normalization of jets of vector fields and diffeomorphisms § 3. Polynomial normal forms

29 34 47

§ 4. Simplification of the resonant normal form via finitely

67

smooth transformations § S. A general condition for § 6.

c:c linearizability

c:c linearization theorems

§ 7. Some sufficient conditions for

76 89

c:c linearizability

c:c

§ 8. Theorems on normal forms § 9. Linearization of finitely smooth vector fields and diffeomorphisms § 10. Normal forms (a supplement to § 8) § 11. Summary of results on finitely smooth normal forms

108 136 143 178 182

vi

Chapter

m.

Linear extensions of dynamical systems

§ 1. Basic notions and facts § 2. Exponential separation and exponential splitting § 3. The structure of linear extensions § 4. Quadratic Lyapunov functions § S. Weak regularity and Green-Samoilenko functions § 6. Smooth linear extensions

193 200 206 224 230 251

Chapter IV. Invariant subbundles of weakly non-linear extensions § 1. Invariant subbundles and their intrinsic characterization § 2. The decomposition theorem § 3. The Grobman-Hartman theorem § 4. Smooth invariant subbundles

261 268 274 278

Chapter V. Invariant manifolds

§ 3. Necessary condition for persistence

290 293 300

§ 4. Asymptotic phase

304

§ 1. Persistence of invariant manifolds § 2. Normal hyperbolicity and persistence

Chapter VI. Normal forms in the vicinity of an invariant manifold § 1. Polynomial normal forms (the nodal case) § 2. Polynomial normal forms (the saddle case)

318 332

Appendix. Some facts from global analysis

343

Bibliography

364

Subject index

380

List of symbols

384

vii

INTRODUCTION

This book is related to the qualitative theory of dynamical systems and is devoted to the study of flows and cascades in the vicinity of a smooth invariant manifold. Much attention is given by specialists in differential equations to the investigation of invariant manifolds. There are several reasons for this. Firstly, the collection of all compact invariant manifolds (in particular, eqUilibria, periodic orbits, invariant tori etc.) constitutes, so to speak, the skeleton of the dynamical system. Therefore, one would like to know whether these manifolds persist under perturbations of the vector field, and what happens in their vicinity (for example, do the nearby solutions tend to the manifold, or stay nearby, or leave the neighbourhood?) Secondly, the existence, for example, of an exponentially stable invariant sub manifold permits one to reduce the investigation of nearby motions to that of points in the manifold itself and thereby to lower the dimension of the phase space. Thirdly, the possibility of reducing a dynamical system to normal form is intimately related to the existence of invariant manifolds. The following simple observation serves as an illustration. Two differential equations = ft.x) and y = g(y) are conjugate via a smooth change of variables y = h(x) if and only if the system = ft.x), = g(y) admits a smooth invariant sub manifold of the form {(x, y): y = h(x)}. Various interrelations between conjugacies of dynamical systems, on the one hand, and invariant sections of certain extensions, on the other hand, are repeatedly used in this work. The main purpose of the book is to present, as fully as possible, the basic results concerning the existence of stable and unstable local manifolds and the recent achievements in the theory of finitely smooth normal forms of vector fields and diffeomorphisms in the vicinity of a rest point and a periodic trajectory. Besides, an attempt is made to summarize the not numerous results obtained so far in the investigation of dynamical systems near an arbitrary invariant submanifold. The choice of material is stipulated by the wish to reflect, in the first place, the typical, generic properties of dynamical systems. That is why we consider normal forms relative only to the hyperbolic variables (i.e., in the direction transverse to the center

x

x

y

viii

manifold), whereas the subtle problem concerning further simplifications along the center manifold is beyond the scope of our considerations. The first two chapters deal with dynamical systems near an equilibrium and a periodic orbit. Several important results are stated here without proof because they easily follow from more general theorems concerning arbitrary compact invariant manifolds which are presented (with full proofs) in the last four chapters. This way of presentation has allowed us to essentially shorten the text, but, as can be expected, it will not be approved by readers interested only in classical topics. Let us note in excuse that, suprisingly enough, the proofs for a rest point are not much easier as compared with the general case (cf., for example, the papers by Takens [1] and Robinson [1]). There is a vast array of papers and books devoted to the questions touched upon in this book. When speaking about invariant manifolds, one should first of all mention the fundamental investigations of Lyapunov [1] and Poincare [2] mainly devoted to the analytic case. Further progress was achieved by Hadamard [1], Bohl [1], and Perron [13]. Hadamard [1] proposed a highly useful method for proving the existence of invariant manifolds now called tM graph transform method. Another approach close to the method of Green's functions was developed by Bohl [1] and Perron [1-3]. The Hadamard-Bohl-Perron tMory was further elaborated and extended by Anosov [1-3], Smale [1], Kelley [I], Kupka [1], Neimark [1-3], Pliss [1-2], Reizins [1], Samoilenko [1,2], Takens [1] and many others. A great number of theorems about integral manifolds was established by applying asymptotic methods due to Bogolyubov and Mitropolskii (see the book by Mitropolskii and Lykova [1]). Grobman [1] and Hartman [1] have shown that a vector field near a hyperbolic singular point is topologically linearizable. This result was extented by Pugh and Shub [1] to the case of an arbitrary normally hyperbolic compact invariant submanifold. On the basis of previous results obtained by McCarthy [1], Kyner [I], Hale [1], Moser [1] and others, Sacker [1,2] proposed a rather general condition sufficient for a compact invariant manifold to persist under perturbations. In the seventies, the Hadamard-Bohl-Perron theory was summed up and brought to its final form (see Hirsch, Pugh and Shub [1] and Fenichel [1-3]). Unfortunately, the style of presentation in these works can hardly be acknowledged as fully satisfactory because many proofs are only sketched and their accomplishment (left to the reader) needs in fact a deep insight into global analysis on manifolds. The method of normal forms founded by Poincare [1] was further developed by Dulac, Siegel, Sternberg, Kolmogorov, Arnold, Moser, Bruno and others (see the books by Arnold

ix [4], Hartman [3] and Bruno [2]). These investigations are chiefly devoted to formal, analytic, and infinitely differentiable normal forms. The problem on finitely smooth normal forms was studied by Belitskii [1], Samovol [1-10] and Sell [1-3]. Let us briefly review the contents of the book. In the first chapter, we present the well-known facts on the structure of flows and cascades near an equilibrium and a closed orbit. In § 1, we recall the relationships between differential equations, vector fields, and phase flows. The second section, § 2, is devoted to the Grobman-Hartman linearization theorem in the vicinity of a hyperbolic singular point. § 3 is concerned with the Floquet-Lyapunov normal form of a vector field near a periodic orbit. The next section contains the main results on the existence of local smooth manifolds in the vicinity of an equilibrium and a periodic trajectory (the so-called Hadamtlrd-Bohl-Pemm theory). These results are used to derive some theorems on preliminary normal forms which serve as the starting point of the next chapter. Chapter n, central to this book, deals with normal forms of vector fields and diffeomorphisms in the neighbourhood of a fixed point with respect to the group of finitely smooth changes of variables. In recent years, it was acknowledged that these nonnal forms are essential for the non-local bifUrcation theory (see Arnold, Afrajmovich, Il'yashenko and Shil'nikov [1], D'yashenko and Yakovenko [1]) because they are stable under perturbations, in contrast to the classical resonant normal forms. The first section serves as an introduction. We pose here the problem on reducing vector fields near a hyperbolic equilibrium to normal form and sketch the research objects pursued in this chapter. In § 2, we present the classical results due to Poincare and Dulac on normalization of jets of vector fields and diffeomorphisms at a rest point. The next section contains several important theorems on polynomial (weakly) resonant normal forms. We show, in particular, that if two vector fields have contact of a sufficiently high order at the equilibrium, then they are locally ~ conjugate with one another. In § 4, we discuss the possibility of further simplification of the resonant normal form and consider a number of examples which demonstrate that certain monomials entering the normal form can be killed by ~ changes of variables. In § S, we propose a new, very general condition, S(k) , imposed on a monomial xT that enables one

to delete xT out of the resonant normal form. This condition is used in § 6 to prove a deep theorem on d< linearization. Because the condition S(k) is rather involved, it is desirable to have some relatively simple conditions each implying S(k). Several spch

x conditions are established in § 7. The next section contains theorems on c! normal forms expressed in terms of the condition i!(k). These results are supplemented in § 9 and § 10 by some theorems based, besides i!(k) , on some other principles. The last section gives a survey of all the results obtained in Chapter II. The third chapter is concerned with linear extensions of dynamical systems. Such objects occur, for example, when linearizing a dynamical system near an invariant submanifold. In §§ 1-3, we give a brief review of the main results obtained in this area (for a detailed exposition, the reader is referred to the book by Bronstein [4]). Although these results may appear somewhat far from our subject, they are basic to many constructions and proofs in the sequel. In order to describe two important classes of linear extensions (namely, linear extensions satisfying the transversality condition and those with no non-trivial bounded motions), we use in § 4 quadratic Lyapunov junctions dermed on the underlying vector bundle. Various kinds of weak regularity of linear extensions are investigated in § S. Some relationships between weak regularity , are transversality, hyperbolicity, and the existence of a Green-Samoilenko function established. In particular, it is shown that a c! Green-Samoilenko function exists if and only if the k-jet transversality condition is fulfilled. In Chapter IV, we investigate invariant sub bundles of weakly non-linear extensions of dynamical systems. Some results on the existence of invariant subbundles of extensions close to exponentially splitted linear extensions are presented in § 1. In particular, a theorem which generalizes the classical result of Hadamard [1] is proved. In § 2, we show that any non-linear extension sufficiently close to an exponentially separated linear extension can be decomposed into a Whitney sum of two extensions. The GrobmanHartman linearization theorem is generalized in § 3 to weakly non-linear hyperbolic extensions. In § 4, we examine the question on smoothness of invariant subbundles. The proof of the main theorem is based on the now traditional graph transform method. It also makes use of the smooth invariant section theorem which is presented (with a detailed proof) in the Appendix. The application of global analysis methods and results enables us to avoid the use of local coordinates and to control all stages of the proof. Chapter V deals with smooth invariant sub manifolds satisfying the so-called normal k-hyperbolicity condition introduced in § 1. We prove in § 2 that such a submanifold is

c!

persistent under perturbations. We also establish the existence of its stable and unstable local manifolds. These results constitute the kernel of the general Hadamard-Bohl-Perron theory. Besides that, we present a theorem on topological

xi

linearization near the given submanifold which is a direct generalization of the Grobman-Hartman theorem. As it is shown in § 3, normal k-hyperbolicity is not only sufficient, but also necessary for a submanifold to be ~ persistent. The notion of

asymptotic phase for an exponentially stable invariant sub manifold is studied in § 4, and some theorems on smoothness of the asymptotic phase are proved. Besides, it is shown that the stable manifold W of a normally k-hyperbolic compact invariant submanifold A is invariantly fibered by ~ submanifolds~,

x e

A (of

course,

a

similar

result

is valid for the unstable manifold W'). These statements may be considered as a supplement to the Hadamard-Bohl-Perron theory. In the final part of this section, we present proofs of several theorems stated (but not proved) in Chapter I. Chapter VI is concerned with the question of whether two dynamical systems are smoothly conjugate to one another in the vicinity of their common smooth invariant submanifold. It is assumed that these systems have contact of high order at all points of the submanifold. In § I, we consider the case when this sub manifold is exponentially stable and prove a generalization of Sternberg's [1] theorem on linearization of contractions. The general case is handled in § 2, and a theorem due to Robinson [1] is presented which extends some results previously obtained by Sternberg [1,2], Chen [1] and Takens [1]. We deduce from these theorems some results (stated without proof in Chapter IT) concerning polynomial resonant normal forms of dynamical systems near an equilibrium and a periodic orbit with respect to finitely smooth changes of coordinates. The book is adressed to specialists in the qualitative theory of differential equations and, especially, in bifurcation theory. Although written for mathematicians, it may prove to be helpful to all those who use normal forms when investigating concrete differential equations. While research workers will find in the book an up-to-date account of recent developments in the theory of finitely smooth normal forms, the authors have tried to make the first part (Chapters I and II) accessible to non-specialists in this field. Such readers should use Chapter I as a summary (or, rather, a glossary) and take the classical results presented in this chapter on faith. The background material needed to understand Chapter II is differential calculus of several variables and ordinary differential equations. To be more precise, the main tools used here are the Taylor expansion formula and Banach's fixed point theorem for contractions (applied to operators in some special functional spaces). As to the second part, the reader is assumed to be familiar with the fundamentals of global analysis on manifolds (Bourbaki

xii [1], Leng [1], Hirsch [1]) and fiber bundle theory (Husemoller [1]). For the reader's convenience, at the end of the book an Appendix is given which contains some definitions and facts from differential calculus and the theory of smooth manifolds, as well as some more special results repeatedly used in the course of the book. We adopt standart notation: the group of real numbers is denoted by IR, the group of integers is denoted by Z, z+ is the set of non-negative integers. Given a mapping X -+ Y, graph(f) denotes the set {(x, j(x»: x E Xl. The symbol ~ marks the beginning and marks the end of a proof. The sections are divided into subsections, each numbered (within a chapter) by a pair of numbers, where the first one refers to the number of section and the second one

f.

refers to the number of the subsection in this section. If necessary, we add, in front of these two numbers, the number of the chapter. So, the triple m.2.4, for example, denotes subsection 4 of section 2 of chapter m. The Appendix is also divided in subsections numbered consecutively and marked with a capital A. The authors are thankful to G.R.Belitskii, A.D.Bruno, Yu.S.n'yashenko, and V.S.Samovol for many helpful conversations on subjects considered in the book. Special thanks are due to V.A.Glavan for a number of useful suggestions and comments on the text and to a. Yu.Demidova for the help rendered in preparing the camera-ready manuscript. We are extremely grateful to Leon a.Chua, the editor of the World Scientific Series in Nonlinear Sciences for his kind offer to include our work in this series. Finally, we should like to acknowledge our great debt to the late K.S.Sibirskii for the instruction, interest in our work, and encouragement he has offered over many years.

1

CHAPTER I TOPOLOGICAL PROPERTmS OF FLOWS AND CASCADES IN THE VICINITY OF A REST POINT AND A PERIODIC TRAJECTORY

§ 1. Basic Definitions and Facts

In this section, we recall the basic terminology and properties of differential equations, vector fields and flows. We establish relations between flows and cascades and discuss a general approach to the notion of normal form.

Differential equations, vector fields, flows and cascades 1.1. Differential equations. Let U be an open subset of IRn and f. U -+ IRn be a continuous map. A relation of the form

.

x. -

dx =j(x) dt

(x

E

(1.1)

U)

is called an ordinary autonomous differential equation. Let x E U and let I be an open interval of the real axis IR containing O. A differentiable function ,,: I -+ U is said to be a solution of the equation (1.1) with initial condition (x, 0) if the following equalities hold: d

dt ,,(t)

= j(,,(t))

(t

E

I),

,,(0)

= x.

According to the well-known Local Existence and Uniqueness Theorem, if the function

f. U -+

IRn

is continuously differentiable then for each point x

E

U there exists a

2 solution Ill: I -+ U with the initial condition (x, 0) and, moreover, if 1/1: J -+ U is also a solution satisfying the same initial condition then lIl(t) = I/I(t) for all t e l n J. Among the solutions of (1.1) with initial condition (x, 0) there is a solution IIlx: Ix -+ U defined on the maximal interval Ix c: lit The set {(x, t): x E U, t E Ix} is open and the map Ill: (x, t) 1-+ IIlx(t) is continuous. Moreover, the equality IIl(X, t + s) = lIl(ep(X, t), s) holds for all x, t, s such that both the right hand and the left hand sides are defined. If / e C-(U, IRn ), then the map ep is also of class C-. Strictly speaking, the right hand side of (1.1) is not a function. To see this, let us introduce a new variable y

= ~(x),

where ~: U -+ IR n is a diffeomorphism. Then (y

E ~(U)

whereas the function x t-+ j(x) after the coordinate change becomes y t-+ t(/(t- 1(y»). From the geometric viewpoint, the right hand side of (1.1) is an object different from a function. Namely, it is a vector field. 1.2. Vector fields. Let M be a smooth (boundaryless) manifold, (TM, tangent bundle and r be a positive integer. A vector field to be a

C smooth section

such that TM

0

I;

I; of the tangent bundle TM: TM -+ M, i.e., a

= idM • The set of all C

C topology is denoted by

0/ class C

TM'

M) be its

on M is defined

C map 1;: M -+ TM

vector fields 1;: M -+ TM provided with the

rl"(TM)'

1.3. Velocity vectors. Let I be an interval of the real axis and Ill: 1 -+ M be a differentiable map. The pair (Ill, 1) is called a (local) motion. Let Tep: TI I x IR -+ TM denote the tangent map. The tangent vector Tep(t, 1) e TM is called the velocity vector of the motion ep: I -+ M at the moment t E I. Denote

=

~(t)

= Tep(t,

1)

!!!

TIIl(t)·1

(t

E

I)

1.4. Motions of a vector field. Let ~ e rl"(TM)' A differentiable map ep: I -+ M is said to be a motion 0/ the vector field 1;, if the velocity vector of ep at t t=- I coincides with the value of the vector field ~: M -+ TM at the point ep(t), i.e., ~(t)

= ~(ep(t»

(I

E

1).

(1.2)

3

1.5. Global Existence and Uniqueness Theorem. Let M be a compact smooth manifold and ~: M ~ TM be a vector field of class

(1) Given a point x

E

C, r

1. The following assenions hold: M, there exists a motion IPx: IR ~ M of the vector field i!:

~

such

E C+ 1(1R,

M); that IPx(O) = x and IPx (2) If IP: I ~ M and 1/1: I ~ M are two motions of number to E I n I, then rp(t) = I/I(t) for all tel n 1.

1.6. Flows. Let M be a smooth manifold and {lP t :

C diffeomorphisms IPt: M ~ M, i.e., IP: M x IR ~ M by lP(x, t) = IPt(x) (x

1P 0 E

= id

M, t

E

t

E

and rpt

~

and lP(to) = I/I(tO> for some

IR} be an one-parameter group of 0

IPs

=

IPt+s (t, s

E

IR).

Define

IR). If the map IP: M x IR ~ M is continuous

then {lP t : t E IR} is said to be a continuous flow. The flow {lP t : t E IR} will also be denoted by (M, IR, IP). The function IPx: IR ~ M defmed by IPx(t) = cp(x, t) (t E IR) is called the motion of the point x E M. The set IPx(lR) is the trajectory (or the orbit) of x, and M is the phase space. The flow (M, IR, IP) is said to be smooth if for each x E M the motion CPx: IR ~ M is smooth. In this case one can define the velocity vector ~x(t) = Trp(t, 1) == Trp(t)·1 (t E IR). Thus, a smooth flow (M, IR, tp) gives rise to a vector field ~tp: M ~ TM, where ~cp(x) = ~x(O) (x

E

velocity vector field

M).

~tp

We say that a smooth flow (M, IR, IP) is of class C is

C smooth.

if its

It is easy to see that in this case the map

rp: M x IR ~ M belongs to the class C,r+l in the sense of Bourbaki [1] and, moreover, to

the class C. 1.7. Theorem. Let M be a compact smooth manifold and ~: M ~ TM be a vector field of class C, r i!: l.Define IP: M x IR ~ M by rp(x,t) = rpx(t) (x e M, t e IR), where rpx: IR ~ M is the motion of the vector field ~ with initial condition rpx(O) = x. Then (M, IR, IP) is a flow of class C.r+l.

Thus, if M is a compact (boundary less) smooth manifold, then there is a one-to-one correspondence between C vector fields

~:

M ~ TM and c!",r+1 flows (M, IR, IP).

1.8. Cascades. Along with flows, the theory of dynamical systems is concerned with

4 the study of cascades. Let M be a smooth manifold and {g": n e z} be a group of

C

diffeomorphisms g": M ... M homomorphic to the group of integers. In such a case we say that {g": n

E

z} is a C

determined by the

g-"

= g-I

0 ... 0

g-I

cascade. Clearly, the cascade {g": n

C diffeomorphism g _ gl:

E

z} is uniquely

g" = g

0 ... 0

g,

(in both cases the composition consists of n multipliers). Every

C

M ... M

j

namely,

smooth flow (M, IR, g) determines in a natural way a cascade on M, namely, {g": n E z}. The converse statement does not, in general, hold, i.e., not for each diffeomorphism g: M ... M there exists a flow

{l:

IE IR} on M such that

i = g.

For example, if the diffeomorphism g: M ... M is not homotopic to the identity mapping, then it cannot be embedded in a flow with the same phase space M. However, to any diffeomorphism g: M ... M we can put in correspondence a flow (M, IR, 11') defined on some enlarged manifold M. This manifold can be obtained from the Cartesian product M x R by identifying points according to the rule: (X, t + n) E. (g"(x) , t) (x E M, nEZ, t E IR). The shift flow (M x R, R, 11'0), where tpo(x, t, s) = (x, t + s) (x E M; I, s E IR), determines the required flow (M, IR, 11') on the quotient manifold M = (M x IR) I E. The submanifold

{O} of M is a global cross-section of the flow (M, IR, 11') so that 11'1: Mo ... Mo can be identified with g: M ... M. The flow (M, R, 11') is called the Smale suspension of the diffeomorphism g. Thus, flows and cascades are closely related with each other. It is reasonable to expect that results obtained for flows are usually valid for cascades and vice versa. In what follows, the presentation of material is carried out in parallel for flows and cascades, but, as a rule, proofs are given for only one kind of dynamical systems.

Mo

III

M

x

Conjugation of flows and cascades in the vicinity of an invariant manifold The main objects of investigation in this book are flows and cascades near an invariant manifold. 1.9. Dermition. Let M and N be smooth manifolds, A be a sub manifold of M. Let us define the following equivalence relation on the set of all maps F: M ... N: FI - Fl , if

6 there exists such an open neighbourhood U of A in M that F\(x) = F2 (x) for all x E U. The corresponding equivalence classes are called A-germs (or, simply, germs, when there is no chance of ambiguity). A A-germ is said to be a local C- diffeomorphism (local homeomorphism) if there are a representative F of this germ and a neighbourhood U of the submanifold A such that F: U ... F(U) c N is a C- diffeomorphism (homeomorphism, respectively). Sometimes, the mapping F: U... F(U) itself will be called a local diffeomorphism (or local homeomorphism). Let Diff,;(M) denote the set of all local C- diffeomorphisms F: M ... M satisfying FI A = id. Let HomeoA(M) be the class of all local homeomorphisms with the same property. 1.10. Def"mition. Let M be a smooth manifold and A be a sub manifold of M. Let

(M, R, fI) and (M, R, "') be flows such that fit (x)

= ,,l(x)

E

A (x

E

A, t

E

~), i.e. A is

their common invariant set. We say that the fI and '" are locally C- conjugate flows in the neighbourhood of the invariant sub manifold A if there exists an element

h

E

Diff,;(M) such

that the

A x {O}-germs

of

the maps (X, t)

(x,t) t-+ h(flt(x» coincide. If we replace the condition h we get the definition of local topological conjugacy.

E

t-+

Dif~(M) by h

E

",t(h(x» and HomeoA(M),

1.11. Notation. Let M be a smooth manifold, (M, R, ,,) be a flow and A be a submanifold of M invariant under fl. Let U be an open neighbourhood of A in M and x E M. Let 1x (" , U) denote the maximal connected interval of R containing the number 0 and such that

"E' E

1x ('" U) implies "T(x)

E

U.

1.12. Lemma. The flows (M, IR, ,,) and (M, IR, "') are locally topologically conjugate with one another near their common compact invariant submanjfold A iff there exist an open neighbourhood Uo of A and an element h E HomeoA(M) such that (1.3) An analogous statement holds in the case of local

• (X, t)

Suppose that the t-+

A x {O}-germs

of

C- conjugacy, as well. the

maps

(X, t)

t-+

",t(h(x»

and

h("t(x» coincide for some element h. HomeoA(M). Then there exist an open

6 neighbourhood U of A in M and a number E > 0 such that ,,/(h(x» = h(c/(x» (x E U. III < E) and (- E. E) c: Ix(cp. U) for all points x in a sufficiently small neighbourhood Uo c: U. Let x holds. Suppose III

ITI

2:: E.

E

Ua and

I

E

Ix(cp. Ua). If III


O. Then rplel2+ T (x) E Uo

:S

+ 't,

where n

E

l.

(I = 0..... n) and

I/IT(h(x» = h(cpT(x». Hence

Employing this argument n times. we get that (1.3) is valid. Conversely. suppose (1.3) holds. Choose a neighbourhood U of the compact sub manifold A in

M and a number E

= h(cpt(x»

I/It(h(x» (x, I)

t-+

I/It(h(x»

> 0 in

(x E U.

and (x. I)

such a way that cpt(U) c: Ua for all I E (- E. E). Then

III t-+




e.

C~(E, E) and IPe(z) = IP(z) whenever IIzll::S e/2. Set Fe = L + IPe' Then

C(E, E) and Fe(z)

= F(z)

sup {IIDFe(x) - LII:

for all z with IIzll::s e/2. It is easy to verify that

x

E

E} • sup {IiDIPe(x)II:

x

E

E} ::s Ce

where C is a constant. To end the proof, apply Theorem 2.7 to the function Fe, where e is sufficiently small. It 2.S. Remark. Theorem 2.7 gives a complete classification (up to the group

~(O,

id»

of mappings F E C(E, E) such that L II DF(O) is a hyperbolic operator and sup {IIDF(x) - LII: x E E} is sufficiently small. In fact, F is conjugate to L by a homeomorphism H E ~(O, id). On the other hand, if LI and ~ are hyperbolic linear operators and H

0

LI

=~

0

H for some homeomorphism H

E

~(O,

id) then LI =~. Indeed,

= x + h(x) where h E ~(E, E), hence Llx + h(Llx) = L,;c + ~h(x), (~ - LI)(x) = h(Llx) - L'J,h(x) (x E E). Clearly, h LI - ~ h E ~(E, (~ - L I ) E ~(E, E), but this is possible only when L'J, = L I .

H(x)

0

0

consequently, E). Therefore

Widening the class of admissible transformations allows further simplification of topological normal forms of local diffeomorphisms in the vicinity of a hyperbolic fixed point. In view of Theorem 2.5, the problem is really concerned with normal forms of linear operators. One can show that for every hyperbolic linear operator there exist uniquely determined L-invariant linear subspaces ~

~,~,

Et, Et

L: E -+ E

such that

• ~ = E', Et. Et = E'

and, moreover, the dimensions of these subspaces completely determine the topological structure of L. Namely, L is topologically conjugate with the

er,

operator (XI' x'J" YI' Y~ 1-+ (2x I , - 2x'J" y1/2, - Y2/2) (Xt E Yt E Eft, t = 1, 2). As a corollary, we obtain that the set of all topological normal forms of local diffeomorphisms near a hyperbolic fixed point is finite.

11 Vector fields 2.9. Dermition. Let ~: E -+ TE be a

C

smooth vector field and ~(O) = O. Denote

the linear operator D~(O): E -+ E by A (we assume that rl(E) is identified with C(E,E». The singular point 0 of the vector field ~ is said to be hyperbolic if the spectrum of A (i.e., the set of its eigenvalues) do not meet the imaginary axis. In this case one also says that the linear vector field y = Ay is hyperbolic. 2.10. Remark. As far as we consider the problem on topological classification of vector fields in the vicinity of a hyperbolic singular point, we may assume, without loss of generality, that the vector field ~ determines a flow (E, IR, ~~). In fact, it suffices to replace ~ by a vector field ~ such that ~(x) = ~(x) whenever IIxll:s 112 and ~(x) = Ax whenever IIxll ~ 1.

2.11. Grobman-Hartman Linearization Theorem for Singular Points. Let 0 be a hyperbolic singular point of the vector field ~ E rl(E). Then the flow generated lTy ~ is topologically linearizable in the neighbourhood of the point 0, i.e., there exists a local homeomorphism transfering motions of t; to motions of the linear vector field = Ay, A = Dt;(O).

y

• This assertion follows from the next theorem.

2.12. Theorem. Assume that 0 is a hyperbolic singular point of the vector field ~ E rl(E) ,. C(E, E)

and, besides, that sup {IIDt;(x) - All: x E E} is SUfficiently small. Then there exists an uniquely determined element H E H(O, id) that conjugates the flow (E, IR, ~~) with the flow of y = Ay (y E E) . • Theorem 2.12 is a particular case of Theorem N.3.S. 2.13. Remark. In addition to Theorem 2.11, let us state some results concerning the topological classification of linear vector fields. Assume that A is a linear operator with no pure imaginary eigenvalues. Then there exists an A-invariant splitting E = E+ e K

such that the spectrum of A I E+ (A I K) lies to the right (respectively, to

12 the left) of the imaginary axis. It turns out that the flow of the differential equation

x = Ax is topologically conjugate with the flow of the model vector field y = y, i = - z (y • E+, Z E). If the vector fields x = Ax and x = Bx are both hyperbolic then E

their flows arc topologically conjugate

with one another iff dim E+ (A) = dim E+ (B)

[1».

(or, equivalently, dim E(A) = dim E(B» (see Vaisbord Moreover, the conjugacy B. can be chosen in If(O, id) iff A As to non-hyperbolic linear vector fields, the question on topological conjugacy can

=

be answered as follows. By n+(A) (n-(A» denote the number of eigenvalues of the operator A with positive (recpectively, negative) real parts. The flows generated by

x = Ax

and

x = Bx

arc topologically equivalent iff n+(A)

= n+(B),

n-(A)

= n-(B)

and the restrictions of A and B to their invariant subspaces that correspond to the pure imaginary eigenvalues arc linearly equivalent (see Latiis

[1».

2.14. Remark. Our interest in hyperbolic singular points of vector fields is certainly motivated not only by the fact that there is a simple topological classification of these points. The main reason is that vector fields all of whose singular points are hyperbolic form a generic subset. To be more precise, the following assertions hold. ~ e rl(E). Then

(1) Let 0 E E be a hyperbolic singular point of the vector field there exist a neighbourhood every vector field hyperbolic.

'"

U(O) c E and a neighbourhood

E H(~)

(2) Given a vector field '"

vector

field

IID~(x)

- D1I(x)1I

~ e rl(E)


0,

>

0, a.

>

0 such that

V E ~),

(3.5) II Trp-t(p)VII

:is

c IIvll exp (- a.t)

(p

E

S, t

> 0,

V E ~).

3.11. Remark. The trajectory S is hyperbolic iff the spectrum of the operator A entering the normal form (3.4) has no points on the imaginary axis (i.e., Re ;>'t ¢ 0 for all eigenvalues ;>'1I"',;>'n of A). 3.12. Remark. Even in the case when S is untwisted and the vector bundles TM[S] and

NS are trivial the subbundles ~ and N' may fail to be trivial. This can be confirmed by the following simple example. Let SI

= IR

I 21tZ,

M

= SI

x 1R2 and let P(e): 1R2 -+ 1R2

denote the rotation through an angle e. Consider the following ~ smooth flow 1R3

= 1R2 x

IR;

It

on

20

.'ex, y, 0) Remark that xt(x, y, e

sl,

+

(1'(012) •

2x)

1i:). The set S

II

[~'

:.,l·

1"(012)

~l'

0

+ I).

xt<x, y, e), so the flow (1R1 x IR, IR, x) induces a flow

= {(O,

0, e): e E Sl} is a 2x-periodic trajectory of the latter flow. It is easy to see that S is hyperbolic and (1R1 x

IR,

~ = {(p(e/2) ~}

e): e

E

Sl, Y E R},

~

e): e

E

Sl, x

= {(p(e/2)

[~}

E

IR}.

Clearly, ~ and ~ are Mobius bands (see Figure 3.1)

NS

Figure 3.1.

3.13. Grobman-Hartman Linearization Theorem for Periodic Orbits. Let M be a smooth

manifold, (E rl(TM) and S be a hyperbolic closed trajectory of (. Then the flow Ip( is topologically linearizable in a sUfficiently small neighbourhood U of S, i.e., there exists a homeomorphism h: U -+ NS transfering the motions of Ip( to motions of (NS, R, Nip).

21 • This follows from Theorem 1V.3.S. 3.14. Remark. Suppose that the vector field ~ near the hyperbolic periodic trajectory

S is reduced via some demiperiodic pseudochart to the Floquet-Lyapunov normal form ~l(e, x)

~2(e, x)

=1+

Q(e, x),

= Ax + R(e, x)

(e E IR, x E IR").

It follows from the preceding theorem that, in this pseudochart, the flow topologically equivalent to the flow ,,/(e, x)

= (e + t,

near S is

tpo(

exp (At)x).

§ 4. Hadamard-Bohl-Perron Theory

In this section, we shall be concerned with the existence of some smooth local

manifolds in the vicinity of a rest point and a periodic orbit, namely, the stable, unstable, center, center-stable, and center-unstable manifolds. These manifolds allow us to obtain more detailed local normal forms of a vector field. Similar results hold for cascades, as well. We leave to the reader to formulate the corresponding definitions and theorems. The case of cascades can be reduced to that of flows by means of the Smale suspension.

Rest points 4.1. Notation. Let E be a finite dimensional Banach space and ~: E -+ TE be a smooth vector field. Let 0 be a singular point of ~. Without loss of generality, assume that ~ generates a (global) flow (E, IR, tp). Recall that the seemingly more general setting of vector fields on manifolds in the vicinity of an equilibrium can be easily reduced to this case. Let us identify

~

E

rr (E)

with

the corresponding

function

~

E

C (E,

E)

(see

subsection 2.1). Denote A = D~(O). Decompose the space EasE = E'. Jt= .~, where

E', Jt= and

~ are A-invariant linear subspaces such that AS'

II

AI E' (Au

Ii

AI~) has

22 eigenvalues with negative (respectively, positive) real parts and imaginary eigenValues.

Ac!! A I Jt=

has pure

4.2. Definition. A smooth submanifold W c: E is said to be locally invariant with respect to the vector field ~ if for each point x E W the vector ~(x) belongs to the tangent space Tx W. In other words, for each point x that " (x, t)

E

W for all t

E

W there is a number

C vector field E. Then in some neighbourhood U of 0 there exist C manifolds W, W", we, locally invariant with respect to

~,

t > 0 and Ip(x, [0, t]) c:

~

defined on

we·, and we'"

each containing the point 0 and such that:

(2) there exists a ~-invariant C smooth foliation

= Wo

> 0 such

E (- £, E).

4.3. Theorem. Assume that 0 is a singular point of the

such a way that W

E:

we· = u [ W;:

x

E

we]

defined in

and there are numbers C > 0 and a: > 0 so that if

we

YEW;,

then d(c/(X) , "tey» :s C d(x, y) exp (- a:t);

C smooth foliation we'" = u [ ~: x E we] such and there exist numbers C > 0 and a: > 0 with the following property:

(3) similarly, there is a

~-invariant

that W" = ~ ify E W;, t > 0 and Ip(x, [- t, 0]) c: ~

we then

d(lp-t(x) , Ip-tey» :s C d(x, y) exp (- a:t).

The proof is given in Chapter V (see subsection 4.9).

4.4. Remark. The manifolds W (W", we, we·, we"') are called the stable (unstable, center, center-stable and center-unstable, respectively) local manifolds of the singular point O. The stable and unstable manifolds are completely determined by the property (1) of the Theorem 4.3 and, according to the properties (2) and (3), they coincide with the sets of points x E U that exponentially approach the point 0 as t -+ + 00 and t -+ - 00, respectively. As to the center, center-stable and center-unstable manifolds, they are not, in general, unique. This is manifested by the following example. 4.5. Example (see Anosov [2]). Define a vector field ~ on ~2 by ~(x, y) Integrating the equation dx / dy

= - x / l,

we obtain x

= c e"P

= (- x,

y2).

(1 / y) ey '" 0, c is

23 the constant of integration), y 4.1:

= O.

The phase portrait of

~

is presented in Figure

x

/' Figure 4.1. Let

uCy, c)

M(c)

Since u(O, c)

= au ay

= {(x,

(0, c)

the singular point (0, 0). serve as

a center

={

=0

c exp (1 I y)

for y < 0,

o

for y

y): x

= uCy,

for all c

c), Y

E

E

of class

(c

IR}

E

IR).

IR, the curve M(c) is tangent to x

Hence it follows that for each

manifold

> 0,

c

E

IR

= 0 at

the curve M(c) can

C for (0, 0). Similarly, for the vector field

1)(x, y, z) = (- x, l, z), the center-stable singular point (0, 0, 0) are also nonunique.

and

center-unstable

manifolds

4.6. Remark. The assertions (2) and (3) of Theorem 4.3 allow us to defme

of the

C

maps

WC.. -+ WC and tu: WCU -+ WC as follows: t ..Cy) = x Cy E W!, X E WC), tuCy) = x Cy E ~, X E WC). These maps are called the stable (respectively, unstable) asymptotic t .. :

phase.

24 Theorem 4.3 permits also to introduce some special coordinates in the vicinity of 0 that straighten out the manifolds WC· and This fact is expressed in the following

WCu as well as the corresponding foliations.

c: smooth vector field (

4.7. Theorem. Let 0 be a singular point of the

there exists a c: diffeomorphism field 0:.( has the fonn: ~(x,

y, z)

= (A,.x

0::

on E. Then

E ... E such that the principal part of the vector

+ P(x, y, z), AuY + Q(x, y, z), AcZ + p(z) + R(x, y, z» (x

E

~,

Y E It', z E Jt=),

where P, Q, R and pare C /Unctions and P(O, y, z) ~

= 0,

Q(x, 0, z)

= 0,

R(O, y, z)

= R(x,

0, z)

= 0,

p(O)

= 0,

Dp(O)

= o.

The proof is given in Chapter V (see subsection 4.10).

vector field ( on E. The flow IP~ of ( near the point 0 is topologically conjugate with the flow of the following vector field (0: 4.8. Theorem. Let 0 be a singular point of the

(o(x, y, z) ~

= (A,.x,

AuY, AcZ

C

+ p(z»

(x

E

~, Y

E

It',

z

E

Jt=).

For a proof, see subsection V.4.11.

Periodic trajectories 4.9. Notation. Let M be a (n + 1)-dimensional compact manifold, (E rr('tM)' S be a closed orbit of (and III be its smallest positive period. Let (M, IR, IP) denote the flow generated by (. As before, let TM[S] = {v E TM: 'tM(V) E S}, and let NS = TM[S] I TS be the normal bundle. For every number I, the transformation

linear morphism NIPt: NS ... NS.

Pix a point b

E

TIPt: TM ... TM induces a

S and consider the linear operator

NbIPw: NbS ... NbS. The space N~ can be represented as a direct sum of NbIPw·invariant

25 linear subspaces N:" III:, and ~ such that the spectrum of N,:/pw1N:, (NbrpwlIII:,) lies inside (respectively, outside) the unit circle, and the eigenvalues of the operator Nbrpw I~ belong to this circle. Denote

U N:"

~ =

U 111:"

~ =

beS

~ =

beS

U ~. beS

It is easy to see that ~,~ and ~ are vector subbundles of the normal bundle NS invariant under the flow (NS, IR, Nrp).

4.10. Theorem. Let S be a periodic orbit of the vector field ~ E r""(TM). Then in a small enough neighbourhood U of S there exist locally invariant manifolds W,

WC·, WC

of class

U

(1) W[S] WU[S]

=

~

we,

C each containing S and satisfying the following conditions:

= TS$~,

TS ,.

WU,

,.

:rW'[S]

= TS,.~,

W[S]

= TS.~,

W·[S]

= TS,.~,.~,

~j

C smooth foliation WC· = U [ K/!: x e WC] such that W'" = U [ b E S] and there exist numbers C > 0 and« > 0 with the follOWing property: if x e WC, y e ~, t > 0 and rp(x, [0, t]) c WC then «(rpt(y), rpt(x» s Cd(x, y) exp(-«/)j (3) there exist a i;.-invariant C smooth foliation WC U = U [ ~: x E WC] with similar (2) there is a i;.-invariant

w::

properties. • The proof is presented in subsection V.4.9.

we

One can define the asymptotic phases I.: WC· .. WC and Iu: WC U .. for the periodic orbit S in the same manner as for singular points (see Remark 4.6). The mappings I. and lu are

C smooth.

4.11. Theorem. Let S be a closed orbit of the vector field i;. E r""(TM). Then there exists an w-demiperiodic C pseudochart such that the principal part of i;. with respect to this pseudorhan is of the form: ~(x,

y, Z, 9)

= (A,.x +

p(x, y, Z, 9), AuY

+

Q(x, y, Z, 9),

26 Ac:Z

+

+ R(x, y,

IPo(z, e)

+

Z, 9), 1

l/Io(Z, 9)

+

V(x, y, z, 9»

where the eigenvalues of As: IRk -+ IRk (Au: IRI -+ 1R1, Ac: IR m -+ IRm) satisfy the condition

°(

Re ~ < Re ~ > 0, Re ~ = 0, respectively). The functions P, Q, R, V, IPo, 1/10 are C smooth and 2w-periodic in 9. The following equalities hold:

= 0,

P(O, y, Z, 9) V(O, y, Z, 9)

=

= 0,

Q(x, 0, z, 9)

V(x, 0, Z, 9)

= 0,

R(O, y, z, 9)

IPo(O, 9)

= 0,

= R(x,

1/10(0, 9)

0, z, 9)

= 0,

= 0,

D 1IPo(0, 9)

= 0.

• The proof will be given in Chapter V, subsection 4.12. 4.12. Theorem. Let S be a closed orbit of the vector field ~

r 1 (T M ). Then there

E

exists an w-demiperiodic C pseudochart for S such that the flow of the vector field ~ described in Theorem 4.11 is topologically equivalent near S to the flow generated by the following vector field ~o: ~o(x,

y, Z, 9)

= (A-s-X,

AuY, Ac:Z

+

IPo(Z, 9), 1

+

l/Io(Z, 9»

~o

refered to in Theorems

Im ) (XEIRk ,YEIR,ZEIR ,9EIR,

• See V.4.13. 4.13. Example. Let us show that the vector fields

~

and

4.11 and 4.12 are not, in general, C conjugate with each other. For this purpose let us examine the following vector field: ~(X, y, 9)

= (x,

- y, 1

+ xy)

(x E IR, Y e IR, 9 e IR).

Clearly, it can be regarded as the Floquet-Lyapunov normal form of a vector field defined on 1R2 x SI, SI

=

IR I Z, and having S

=

{(O, 0, 9): 9

E

SI}

orbit. Our goal is to show that ~ is not C conjugate with the vector field ~o(X,

y, 9)

= (x,

- y, 1)

(x E IR, Y E IR, 9 E IR).

~

as a periodic

27 Suppose the contrary holds. Then there exists a carries the motions

C

mapping

«x, y, a) to periodic motions I//(a) or I

Set ,.,.(x, y, e)

= I(x,

=e

+

1 (a

E

Sl, 1

(xet , ye- t , a + 1 + xyt)

E

IR), that is,

= I(X,

I: 1R1 x Sl .... Sl

E

1R1 x Sl, t

I

0

tpt

E

= !/It

which

IR) 0

I

(I

E

IR),

y, e) + I.

y, a) - a. We obtain the following functional equation for ,.,.:

lJ.(xet , ye- t ,

e + t + xyt) + xyt

= J.I(x, y,

e).

(4.1)

Denote I

vex, y)

=

I

J.I(X, y, e) de.

o

Integrating (4.1) with respect to e 9, we get

E

Sl

and taking into account that

J.I

is periodic in (4.2)

Note that VEe, by our assumption. The general solution of the equation (4.2) is of the form vex, y) = - xy In Ix I + ~(xy), where ~ is an arbitrary function. Observe that ~ is differentiable at every point (X, y) with x ¢ 0, Y "* O. Therefore Bv

-By = - x In

Ixl

+ x D~(xy),

Bv

-Bx = - y In

Ixl

+ y + y D~(xy).

Let x be a fixed non-zero number, then lim Bv(x, y)

= -x

In Ixl

By

y-+O

+x

lim

D~(z),

z-+ 0

consequently, D~(z) (z '" 0) is bounded as that z .... O. On the other hand, if y is fixed, Y '" 0, then lim

x-+~

Bv(x, y) By

= -y

lim In Ixl x-+o

+ y + Y lim z-+O

D~(z),

28 hence, DI;(z) -+

C

CD

We get a contradiction. Thus, the vector field ~ is not

as z -+ O.

conjugate to 1;0. According to Theorem 4.12, there are continuous functions .: 1R1 x Sl -+ Sl satisfying

the condition •

0

rpt = I/I t

.(x, y,

9)

o.

=9

(t -

E

IR). For example, we can take

xy In Ixl for x

~

0, .(0, y, 9)

= 9.

Bibliographical Notes and Remarks to Chapter I Theorems 2.5 and 2.11 are due to Grobman [1] and Hartman [1]. The presentation of the Floquet-Lyapunov theory (see Floquet [1], Lyapunov [1]) is adapted from the books by Abraham and Robbin [1], Reizin5 [1]. Theorems 4.3 and 4.10 have a long history (see the historical comments by Anosov [2], but note that the works of Bohl [1] are not reflected there). The Hadamard-Bohl-Perron theory is presented in the books by Abraham and Robbin [1], Anosov [2], Nitecki [1], Hirsch, Pugh and Shub [1], Palis and de Melo [1].

29

CHAPTER II FINITELY SMOOTH NORMAL FORMS OF VECTOR FIELDS AND DIFFEOMORPHISMS

The Problem on Reducing a ex> Vector Field to in the Vicinity of a Hyperbolic Rest Point

§ 1.

CC

Normal Form

We consider the problem on reducing an infinitely smooth vector field at a hyperbolic rest point to normal form by the aid of coordinate changes of class ~, where k is a fixed positive integer. This section contains a short review of the results presented in Chapter II, together with some explanations and examples.

1.1. Preliminaries. Let E be a real Banach space of finite dimension n. Let ~ E r""(E) be an infinitely smooth vector field on the space E, and ~(O) = O. We assume that the spectrum of the operator A == D~(O) lies out of the imaginary axis. In order to simplify the subsequent discussion we suppose that the operator A is diagonal, i.e,

= diag[9 1,

where 9, are real numbers and restriction is not essential and will be dropped in the sequel.

A

... , 9 n ],

We shall use the following notation:

· co11ection

0f

== vi

+ ... +

vn

0 (i

= 1,

... , n). This

v = (vI, ... , v n ) denotes a multiindex,

. .10tegers v '('I = 1" , " n) ; xa- = non-negative

The number Ivl

9,"

a- I

X I '. , , ' X.,.n n

i.e., a

, a monomI'al , IS

is said to be the degree of the monomial xv.

1.2. Topological and smooth linearization. As it was shown in Chapter I (see Theorem 1.2.11), the topological normal form of (a germ of) a smooth vector field at a hyperbolic rest point is linear. Unfortunately, the linearizing homeomorphism may fail to be smooth. This fact can be confirmed by the following example (see Hartman [2]) of a polynomial vector field with a hyperbolic rest point at the origin which does not admit any

C

linearization,

30 1.3. Example. We consider two differential systems

x = 2x, and

Y = y + Xl, Z = - z;

u = 2u, v =

v,

W=

(1.1)

w.

-

(1.2)

The formulas x(t)

= xe2t ,

y(t)

= [y + tu]et ,

z(t)

= ze-t

(1.3)

and

= ue2t ,

U(t)

v(t)

= vet,

= we-t

w(t)

(1.4)

give the solutions of these systems. According to the Grobman-Hartman Theorem, the systems (1.1) and (1.2) are topologically equivalent in the vicinity of the origin, i.e., there exists a local homeomorphism transfering solutions (1.3) to solutions (1.4). Suppose this homeomorphism can be chosen differentiable. Then, without fail, there exists a conjugating diffeomorphism of the following structure:

x

= U,

Y

= v + rp(u,

w), z

= w.

(1.5)

°

In fact, the system (1.2) leaves the plane v = invariant. In virtue of our assumption, the system (1.1) has a locally invariant smooth manifold y = rp(x, z) tangent at the origin to the plane y = 0. Then (l.5) is a smooth conjugacy. Therefore, the identity

is valid, i.e., 2t

rp(ue ,

Let u

¢

0,

W

¢

we-t ) =

+ tuw]et •

[rp(u, w)

0. Take a sequence {tn} -+

+

00

(l.6)

and set (l.7)

Let rpn denote the solution of the system (l.2) beginning at the point (un' 0, w) t 0, i.e.,

=

rpn(t)

= (une2t , 0,

The identity (1.6), for t = tn and

-t

we )

Ii

(ue

u = Un' gives

2(t-t )

-t

n, 0, we ).

for

31

(1.8)

It is clear that 1p(0, 0) = O. The map (1.5) transfers the stable manifold of the point u == 0, v = 0, W = 0 for the system (1.2) to the stable manifold of the point x = 0, y == 0, z = 0 for the system (1.1). Consequently, 1p(0, w) = O. According to our assumption, the function Ip is differentiable. Therefore

Note that unwetn

= uwe- tn

by (1.7). Thus the equality (1.8) becomes (1.9)

Tending

11

to infinity, we get Ip(u, 0)

= O.

Consequently, (1.10)

Divide both parts of the equality (1.9) by tne- tn . Taking into account the relations (1. 7) and (1.10) and passing to the limit, we obtain uw = 0, contradicting the choice of u and w. This contradiction shows that the system (1.1) does not admit smooth linearization in the vicinity of the origin. 1.4. Polynomial resonant nonnal fonn. Consider the family of all smooth vector fields ~: E -+ TE each having the origin as a hyperbolic singular point. The group Dif~(E) of

O-germs Ip E

of (!t diffeomorphisms, k ~ 1, acts in a natural way on vector fields, namely,

Dif~(E) transfers ~ to Ip.~, where Ip.~

= Tip

0

~

0

Ip -I.

As it was shown above, the

orbit {Ip.~: Ip E Dif~(E)} may fail to contain a linear vector field. So we face the somewhat vague problem of finding an element of this orbit which has the most simple form. Such and element is referred to as (!t normal form of the vector .field ~.

e

Sternberg [2] and Chen [1] have shown that for a given vector field ~: E -+ TE with the origin being a hyperbolic equilibrium there exists a number Q = Q(k, 8 1"", 8 n ) such that ~ is locally (!t equivalent to its Taylor

expansion of order Q. In other

words, the terms of degree greater than Q can be deleted from the (!t normal form. Thus, when dealing with finitely smooth normal forms of C" vector fields at a hyperbolic rest

32 point, we can confine ourselves by considering only polynomial vector fields. As Poincare [l,2] and Dulac [1] have shown, by the aid of polynomial coordinate changes, the vector field ~ can be reduced to the following form Q

= 9,rj

Xj

+

L p/,x. Then there exists a polynomial h~ E P-.:(XI , ... , Xn; Xs) such that the coordinate change x = y brings the vector field (2.4) to the form

+

~y-':es

39 q

n

~(y)

= Ay +

L

L p;y'''et

+ l(y)

(y

E

(2.5)

E),

1... 1-2

t-I (t ....)-=(S.T)

where 1 E CJ(E, E) and n](y)1I

= O(lIyllq)

• After the indicated change

as lIyll -+ o.

of variables

is

accomplished,

the

vector

field

X = ~(x) takes the form n

LDJ[~yT]j~S

j +

= Ay + ~yTes + p~(y +

~yTes)Tes

J-I n

+

q

r r

+ h~yTes)"'et + fly + ~yTes).

p;(y

1... 1-2

t-I (t ....);t(s. T)

Try to choose a polynomial ~ in such a way that n

Ay

+

t-I

L

L p;y"'et + 1(J) + LDJ[~yT] AJYJ + 1... 1-2

p;"y'"

+ lJ(y)]es

1... 1-2

J-I

(t ....);t(S.T)

then we have

q

n

q

r

y = ~(y) be satisfied,

U .... )-=(S.T)

(2.6)

+

n

q

t-I

1... 1-2

r r

p;(y

+ ~yTes)"'et + fly + h~yTes)·

Equate the coefficients of yT in both sides of the s-th line and write the auxiliary equation for ~: n

-r

DJ[~yT]A;YJ

+

A~yT

+ p~yT = o.

(2.7)

J-I

Let Ss,T: PT(X1, ... , Xn; Xs) -+ PT(X1, .•. , Xn; Xs) be the linear operator defined as fOllows:

40

[21.,T9']yT

= A.9'y

"

T

-

LDjP(y)A..iYJ

J-I

According to the differentiation formula for a polynomial (see subsection A.7),

Here ~ denotes the 't'-linear map associated with the 't'-homogeneous polynomial 9'. Hence it follows that the real parts of the eigenvalues of the operator 2l.,T are equal to e. - 't'le l - ... - 't'"e" .. O. Consequently, the operator 2l.,T is invertible. Put (2.8)

Substitute the solution h! of equation (2.7) into (2.6) and observe that the coefficients of ya' coincide in both parts of the equation (2.6) for every For i III $, set

tr

(2 s Itr I s q).

q

1,(Y)

L IP!(Y + h!yTe.)a' - p!ya'] + ft(y

=

+ h!yTe.).

(2.9)

1a'I-l Finally, put

1.(Y)

= lid + D.h~(y)rl

q

"

{- L D ;h~(y) J-I Jill.

q

- D~(y)

L P;'ya'

+ 'J(y)]

1a'1-2

q

L p!ya' + L IP!(Y 1a'I-l cr_T

[

Icrl-l

+

~yT e.)a' _ p!ya']

(2.10)

41 It is not difficult to check that substituting the expressions (2.8),

(2.10) into (2.6) gives a true equality. lIyll ...

Moreover,

1 E CJ

and

IIl(y) II

(2.9) and

= o(lIyllq)

as

O.

2.9. Theorem. Let .f denote the Q-jet of a smooth vector field on E with 0 equilibrium point. There exist a vector field 'II

E

/2

and a

C"

E

E as

smooth coordinate system

t: E ... IRd such that ~, the principal part of 1.'11, is of the fonn ~(x)

= Ax +

and, moreover, p; - 0 implies at

(2.11)

= < 0',

a>.

The vector field (2.11) is called the resonant nonnal fonn of the jet jQ. ~ Let ~ belong to the jetjQ. By Lemma 2.8, every non-resonant monomial of degree

q :s Q entering the Taylor expansion of the vector field ~ can be deleted with the help of polynomial changes of variables. It should be noted that these transfonnations do not influence other monomials of degree not greater than q. Applying successively Lemma 2.8 to all non-resonant monomials of degree q = 2, ... , Q, we conclude that ~ can be reduced to the fonn Q

'II(y)

= Ay +

r

PaXtr

+ fly)

(y

E

(2.12)

E),

113'1-1 where

p; = 0

whenever

at - ""]es -

1P;y'" - p;(I/J(y»""]e,.

(2.24)

'-I Icrl-2 ('. cr)"(s. T)

It is easy to show that inserting the polynomial

~

and the expression (2.24) into

equation (2.21), we obtain a true equality. Besides, as lIyll -+ O.

1E

CJ(E, E) and n](y)1I = o(lIyllq)

2.17. Theorem. Let l

be the Q-jet of a CJ smooth diffeomorphism from E into itself

with the origin 0 being a fixed point. Then there exist a diffeomorphism G E j'Q, a neighbourhood U of 0 and an irifinitely smooth change of variables, t: (U, 0) -+ (E, 0), such that Q

t· 1

•

G

0

t(x)

= Lx +

L PaX

cr

(x

E

U),

(2.25)

Icrl-2 and, moreover, p; .. 0 implies at

. The expression (2.25) is called the

resonant normal form of the j'et j'Q . • This follows from the arguments used in the proof of Theorem 2.9. The only difference is that now we must use Lemma 2.16 instead of Lemma 2.8.

§ 3. Polynomial Nonnal Forms

In this section, we establish some sufficient conditions for the finitely smooth conjugacies of vector fields in the vicinity of a rest periodic orbit. These results allow to reduce the problem on normalization vector field to the same problem for the associated jet, which was discussed also consider normal forms of diffeomorphisms near a fixed point.

existence of point and a of a smooth in § 2. We

Vector fields in the neighbourhood of an eqUilibrium 3.1. Definition. Let Q, K, and k be positive integers, k:s min {Q, K}. Let I; and

11

48 be mappings from E to E of class

order (Q, k)

at

CC.

= O(IIXIIQ-P)

lIif(x) - if'll(x)1I as IIxll

~

The mappings ( and 'II are said to have COlllact of

the origin if

= Q :5 K, ( and if DP(O) = vP'II(O)

O. For k

(p

= 0,

1, ... , k)

'II have contact of order (Q, k)

II

(Q, Q) at the

(p = 0, 1, ... , Q). In this case, the mappings origin if and only ( and 'II are also said to have contact of order Q (see Definition 2.1).

y = Ay

3.2. Notation. Let

be

hyperbolic vector field;

let v .. ... ,

Vd

be

the

eigenvalUes of the operator A. As in the previous section, let 9.. ..., 9 n be all distinct values contained in the collection {Re Vt: i = 1, ... , d}. By the hyperbolicity assumption, 9 t '" 0 (i = 1, ... , n). Introduce new notation for the numbers 9 ..... , en

as follows: -

~l
min {k, QI}) at the origin, then ~ and conjugate in the vicinity of the origin .

71

are c;k

• Apply Theorem VI.1.3 to the case where A is a rest point.

3.S. Theorem. Let K and k be positive integers; ~ be a vector field on E of class c;k with the origin being a saddle rest point; A = D~(O). if K 2:: Qo(k) (see (3.1», then the vector field ~ can be reduced in the vicinity of the origin I7y the aid of a c;k transformation to the resonant polynomial normal form n

00

y = Ay +

(y

E

E) •

(3.3)

• Let ~ E r~(E) and K I: Qo(k). By Theorem 2.9, there exists an infinitely smooth coordinate change y = !pCx) reducing the Qo-jet of the vector field ~ to the resonant normal form (3.3). With respect to the new coordinates, the vector field ~ takes the form = ~(y) II !P.~(y), and since the vector field (3.3) belongs to the jet lo(~), the vector fields (3.3) and ( have contact of order Q at the origin. By Theorem 3.2, the

y

vector fields ( and (3.3) are locally c;k conjugate. Hence, ~ and (3.3) are also locally

r::c conjugate with

one another.

3.6. Theorem. Let k be a positive integer; ~ be a vector field of class c;k having the origin as a nodal rest point; A. D~(O). if k I: QI (see (3.2», then the vector

field ~ can be reduced in the vicinity of the origin I7y means of a c;k transformation to the resonant polynomial normal form n

01

Y = Ay + E E p;yf1'e,

'-I

(y

E

E).

(3.3)

1f1'1-2

• This statement follows from Theorems 2.9 and 3.3. 3.7. Remark. Theorems 3.5 and 3.6 give sufficient conditions for reducibility of a

50

vector field in the vicinity of a hyperbolic rest point to the resonant polynomial normal form via finitely smooth changes of variables. The condition K 2: Qo(k) of Theorem 3.5 involves the numbers K, the smoothness of the vector field, and k, the smoothness of the conjugation map, as well as the spectrum of the linear approximation operator. Let the number k be fixed. Because K 2: Qo(k) , formula (3.1) shows that the number K strongly depends on the value ~z Ilm ~z Ilm} max { ;;-. X-' X-';;- , "I

I

1"1

which might be called the spectral spread of the operator A. The situation in Theorem 3.6 is different: the smoothness of the normalizing mapping coincides with that of the vector field. The following reason can partly account for this difference: in the saddle case, resonances of arbitrarily high orders may occur, whereas in the nodal case resonances of order greater than QI are absent (see Remark 2.13(2». We note in passing that Qo(k) :s 2

[ (2k + 1) max {Az, {A It mln

Ilm} ] } III

+ 2 II

Q.(k).

(3.5)

3.S. Example. It should be pointed out that there exist smooth vector fields which cannot be reduced near a hyperbolic rest point to polynomial form by any transformation of class

d.

For instance, the following

C system

x = 2x, Y = y + xz(i + £-rl/3,

.z

= - .z

(3.6)

is not locally C conjugate with a polynomial vector field. Suppose the contrary holds. By some results presented below (see §§ 4-8), the polynomial normal form corresponding to the linear part of (3.6) with respect to the class of all C transformations is as follows: (3.7) = 2u, = \I + auw, W= - w, a = const.

u

v

By the assumption, there exists a local x

= j(u,

v, w),

y

conjugating (3.6) and (3.7). Clearly

C diffeomorphism

= g(u,

v, w),

111(0, 0, 0)

z

=

III

= h(u,

v, w)

(0, 0, 0). Since III transfers the

51

stable (unstable) manifold of the point (0, 0, 0) with respect to the system (3.7) to the corresponding manifold for the system (3.6), one has

=0,

j{O, 0, w)

h(u, v, 0) Because

~

=0;

g(O, 0, w)

(3.8)

O.

==

(3.9)

is a local homeomorphism, it follows from (3.8) that h(O, 0, w) '" 0

if w '"

o.

(3.10)

Write out the solutions of (3.6) and (3.7): t

[y

+ xz

I (ile

4t

+ le-2tr l13 dt] et ,

o z(t)

= ze-t ; u(t)

=

lie

U

v(t) = (v

,

+ auwt) e,t wet)

~

= we .

Let u and w be fixed sufficiently small non-zero numbers. The map solution

!p(t)

=

(ui t , auwttl, we-t)

of (3.7) with initial condition

~

!p(0)

transfers the

=

(u, 0, w)

to a solution of (3.6). Hence 2t

-t

t

j{1Ie , auwte , we )

= f(u,

g(ue2t , auwtet , we-t )

=

{

2t

0, w) e ;

g(u, 0, w) (3.11)

t

+ j{u, 0 ,w,'\h(u, 0 ,w,,\

I [g2(u,'0 w,e,\

4t

+ h2(U,

0, w'\e, 2t ]-1/3 dt} et .,

o (3.12) Let {tn } -+

+

III.

Denote

-2t

U"

= lie

(3.13)

n.

Then 2t

t

-t

(u"e , aU"wte , we )

= (ue 2(t-tn >,auwtet-2t ",

t

we-).

52 Replace, in the identity (3.12), t by tn' and U by Un' then

(3.14) tn

+ ft.u n'

0 , w'\h( , un' 0 ,w,~

J

[g2(Un"

0 w,e \ 4t

+

h2(Un , 0, w~e-2tl-\13 dt} etn • ,

o

Since g(O, 0, w) = 0 (see (3.8», one has (3.15) Further,

and by using the equalities (3.11) and (3.13) one obtains

By virtue of (3.15), the equality (3.14) becomes

+ ft.u,

-tn

auwtne

-tn

,we

tn -tnJ

)h(um 0, w)e

2 [g (U m

4t

0, w)e

o

The following estimates are valid: tn

In

==

J[g2(U o

m

0, w)e4t

+

h2(Uno 0, w)e-lt r\13 dt

(3.16)

53 tn

S

J[h(u

n , 0,

w)r2l3elt13 cit

const./tn13 ,

s

o since

Ii m h(uno 0, w)

= h(O,

0, w)

;I:

0

n-++oo

by (3.10) (recall that

W;l:

0). Hence it follows from (3.16) as n -+ g(u, 0, 0)

00

= 0,

that (3.17)

therefore

(3.18)

-t

Divide both parts of the equality (3.16) by tnt n, then by force of (3.18) one obtains

g~(u, 0, O)auw

+ g:"(u, 0,

O)wt~ 1

+

o(t~ I) (3.19)

The left-hand part of the last equality and the first term of the right-hand part are bounded as n -+ 00 (see (3.13». Let t E [/n/4, 3/n/4], then by (3.1S) and (3.13) one gets

Consequently, (n -+ .).

It therefore follows from (3.19) that J(u, 0, O)h(O, 0, w)

= o.

54 Since w '" 0, one has h(O, 0, w) '" 0 by (3.10), hence, j{u, 0, 0) = O. Using (3.9) and (3.17), one concludes that lII(u, 0, 0) = (0, 0, 0). Because U"# 0, q,(O, 0, 0)

=

(0, 0, 0)

and q,

is locally one-to-one, we get a contradiction.

This contradiction

shows that the vector field (3.6) cannot be reduced to polynomial form by a local C diffeomorphism. This example answers affirmatively the question raised by A.D.Myshkis. A similar example of a 4-dimensional vector field was proposed by Belitskii [1]. 3.9. Remark. Let us remind the reader that a vector field

~

is said to be

Q-delermined with respect to a group G of coordinate changes if every vector field 11, belonging to the Q-jet of the vector field ~, lies in the G-orbit of ~. From this point of view, Theorems 3.3 and 3.4 affirm that every C" smooth vector field with saddle (nodal) linear part at the rest point 0 is Qo-determined (Qcdetermined) with respect to the group Difta'(E).

Vector fields near an equilibrium of general type 3.10. Statement of the problem. If a rest point of a smooth vector field is not hyperbolic, then the phase flow in the vicinity of this point is not finitely determined even with respect to the group of homeomorphisms. For instance, an equilibrium of center type of a planar linear vector field can be transformed into a focus by polynomial perturbations of arbitrarily high order. Therefore, the problem on topological (and, moreover, smooth) normalization of a vector field near an equilibrium of general type cannot be reduced to the normalization of finite order jets. By Theorem 1.4.8, the dynamical behaviour in the vicinity of such a rest point is to a great extent determined by the properties of the dynamical system restricted to the center manifold. When investigating rest points with pure imaginary spectrum, it is reasonable to discuss either normalization of finite order jets or topological normalization of vector field families of finite co-dimension (see Arnold and n'yashenko [1]). These problems are outside the scope of this book. Our aim is to investigate (up to finitely smooth changes of variables) dynamical systems in the vicinity of the center manifold. Therefore, in the sequel, we shall assume that all changes of coordinates are of the form y = x + h(x) , where h vanishes along the center manifold. We shall obtain normal

55 forms expressed as polynomials in the hyperbolic variables with coefficients being functions defined on the center manifold. 3.11. Preliminaries. Let t; e r~(E), A

linear subspaces such that E

= F!' III E,

= Dt;(O).

Let

F!'

and E be the A-invariant

the eigenvalues of the operator Ah

= A IF!'

lie

out of the imaginary axis and the eigenvalues of the operator Ac = A IIf lie on the imaginary axis. Let 9\1 ••• , 9 n denote all distinct values of the real parts of the eigenvalues of the operator

Ah ;

let

XI' ... , Xn

be the corresponding Ah-invariant

linear subspaces. By Theorem 1.4.7, there exist coordinates of class principal part ~ of the vector field to the form ~(x, z)

=

(A,.x

+

F(x, z), AcZ

+

+

p(z)

R(x, z»

where the mappings F, R and p are of class

CJ,

(x e

F(O, z)

F!',

Z e

= R(O,

CJ

bringing the

E), z)

= 0,

(3.20) p(O)

= 0,

=

O. Without loss of generality assume that the supports of the functions F, R and II (X, z)1I :s c, where c is a small positive number to be specified later.

Dp(O)

p are contained in the ball

3.12. Definition. Let / subspace of E; /I W = g I W. order Q along W, if

and g be

CJ

smooth mappings from E to E; W be a linear

The mappings / and g are said to have venical contact l!f(y) - g(y)1I

=

0/

0([P(Y, W)]Q)

uniformly on Easy tends to W (here p is a metric on E). This is an equivalence relation. We define venical Q-jets of mappings from E into E with respect to W to be the corresponding equivalence classes. 3.13. Example. In order to illustrate the significance of the above notion, let us return to Example 1.3. The system (1.1) contains only one resonant monomial yz. Since the linear part of (1.1) is diagonal, it is clear that this system admits no

C

linearization. Moreover, we know that there is no C linearization. The fact that the 2jet at the origin (or, in other words, the quadratic term yz) turns out to be an obstacle to

C

linearization seems to be a surprise. We shall see in a moment that the

56 problem on linearization of (1.1) near the origin is equivalent to the same problem in the vicinity of the invariant manifold z = O. Therefore the monomial yz, the obstacle to

C

z

= O.

linearization,

is

best

to regard as vertical

I-jet with respect to the plane

Next we shall give another proof of the fact that

C

(1.1) is not

linearizable

(based on the notion of vertical jet).

Suppose, to obtain a contradiction, that (1.1) and (1.2) are locally Then, as it was shown in subsection 1.3, there exists a x

and .,(0, w)

= !p(u,

invariant manifold

=

u, y

=v+

= O. The vertical = 0 is of the fonn

0)

w

C conjugacy

!p(u, w), z

C

conjugate.

of the fonn

=w

I-jet of the mapping ., with respect to the 1/I(u)w, where 1/1 is a continuous function. In

other words, !p(u, w) = 1/I(u)w + A(u, w), where I A(u, w) I to determine the functions 1/1 and A, consider the equation

= o( I wi)

as w -+ O. In order

(I

In what follows, we shall deal with the domain u

E

IR).

> 0, w > O. The last equality implies (3.21)

A(U, wJ~

The equation (3.21) has

-t) -t = A(2t Ill! ,we e •

a particular solution 1/Io(u)

=i

(3.22) u In u. The general solution of

the homogeneous equation

on the half-line u > 0 is .,(u) = CU, where C is a constant. Hence it follows that the general solution of the equation (3.21) in the domain u > 0 can be written as

_ 1

1/I(u) - 2: u In u

+

cu.

By continuity, set 1/1(0) = O. In order to find A, let us first note that Ao(u, w) = UW is a particular solution of (3.22). The general solution of (3.22) will be sought in the fann A(U, w)

= UWJ.L(u,

w) (u

> 0,

W

> 0). Then J.I(u, w) = J.I(uWZ, 1). Thus, the general

57 solution of equation (3.22) in the domain u > 0, A(U, w)

> 0

W

can be written as

= uwt(uwl)

where I is an arbitrary function. Since A(u, w) = o(w) as w -+ 0, we get ~ -+ O. Therefore, we may put 1(0) = O. So we have

x

=

u, y

l

=w

=

+

v

r

In u

+

Cuw

+

UWl(uWZ)

= "'(u,

I(~) -+

0 as

v, w),

(3.23) (u

> 0, w > 0).

Let us show that the function y = ",(u, v, w) is not differentiable with respect to u at u = O. In fact, if u > 0 and u -+ 0, then I(uwl) -+ O. Hence, the map

u ...... UWI(uWZ) is differentiable at the point (0, w), the derivative being equal to O. On the other hand, the function ./(u) = u In I u I (u'" 0), differentiable at u = O. Consequently, the partial derivative exist. We have reached a contradiction.

./(0) = 0, is not a",/au I u-o does not

3.14. Vertical jets. Let us dwell on the notion of vertical Q-jet of the vector field (3.20) with respect to the center manifold W'. Let have one and the same center manifold and

11

MP, and

1;, 11

I; I W

r~(E). Suppose that I; and

E

= 11 I W'.

11

It is easy to verify that I;

have vertical contact of order Q along W if and only if there is a coordinate

system of class CJ such that the function ~ is given by formula (3.20) and the principal part ~ of the vector field 11 is given by the equality ~(x, z)

=

(A,.x

+

G(x, z), A.,.z

where lIG(x, z) - F(x, l)1I IIxli -+ O. For

I;

E

=

r~(E),

+

p(z)

O(IIXllo),

+

ex If',

Sex, z»

E

IIS(x, l) - R(x, Z)II

=

ff),

(3.24)

O(IIXllo), uniformly on

ff as

z

E

let )'$(1;) denote the vertical jet of the vector field I; along

the center manifold W. The vector fields I; and

11

have vertical contact of order Q with

respect to W if and only if D~G(O,

(p

=

I, ... , Q; z e ff,

z)

= D,(O,

z),

D~(O,

z)

= D,(O,

z)

IIzlI:s E). Therefore the vertical jet J~(I;) contains one and

58 only one vector field 11 with Q

~(x.

z)

= (A~ +

L J,r D~F(O. z)?

AcZ

+

p(z)

p-I

(3.25) Q

+

L ftr D~(O. z)?)

(x

E

E'. z E

F)

p-I

3.1S. Theorem. Let l,(f.) be the vertical Q-jet of a CJ+k. smooth vector field f. with respect to the center manifold at the origin. There exist a vector field

11 E

J'?,(f.)

and

~ smooth coordinates bringing the principal part ~ of 11 to the form Q

Q

(3.26)

where Ah

E

CJ+k.·I; t

E

CJ+q; Ptr' qtr E CJ+k.-ltrl; Ah(O) = Ah ; teO) = 0; Dt(O) = Ac;

P;' ;! 0 implies at = (i = 1•...• n); qtr II! 0 implies = O. The vector field (3.26) is said to be the generalized polynomial resonant normal form of the vertical jet J'?,. • The validity of this theorem follows from the lemma below.

3.16. Lenuna. Let K and q be positive integers. K

2:

q. ~

E

r~(E) be

a vector field

given by Q

X = Ah(z)x +

L

P...(z)x'" Itr,1-2

+

rp(x. z).

(3.27)

z=

Q

t(z)

+

L qtr(z)xtr + !/I(X. z)

(x

E

E'.

Z

E

F).

I tr 1-\

y

= Ah(O)y

be a hyperbolic linear vector field;

9\ ••••• 9 n

the real parts of the eigenvalues of the operator Ah(O). Let

E'

be all distinct values = X\

El ••• El

0/

Xn be the

59 corresponding direct sum decomposition into Ah(O)-invariant subspaces;

=

t(O)

0;

be a linear operator with pure imaginary eigenvalues; t E c"+l; Ah, PO', qO", rp, I/J E c" (IT E z~, 1:s IITI :s q); IIrp(x, z)1I = o(lIxll q); III/J(x, z)1I = o(lIxll q) Dt(O): g= -+ g=

uniformly on g= as IIXIl -+ (1)

~

If

s

o.

{1, ••• , n}, or

E

E

z~,

lorl

=q

and 9. ""

+ g(W)yT)

- G(w)

;I: II

O. Let us note that if G and g are G1(y, w) is also

c< smooth mappings, then

c< smooth and, besides, IIG1(y, w)1I = O(lIyIlITI-I)

as lIyll -+ O. Taking into account this remark, we easily find the functions ~ and ~ with the required properties. Applying successively this lemma for q

= 1,

... , Q, we get (3.26).

3.17. Definitions. Let Q, K and k be positive integers, k:s min {Q, K}. The vector fields (3.20) and (3.24) are said to have venical contact of order (Q, k) along their common center manifold at the origin if there exists a positive number c such that

IIDS[R(x, z) - Sex, z)] II :s C IIXIIQ-.S' (s

= 0,

1, ... , k;

II (x, z)1I :s E).

.{3.34)

63 If k = Q :s K. then the notion of vertical contact of order Q is equivalent to the notion of vertical contact of order (Q. k) Ii (Q. Q). In some cases it is useful to sharpen the definition of vertical contact. Namely. replace the inequalities (3.34) by the conditions IID~~[F(x. z) - G(x. z)]11 :s c IID~~[R(x. z) - Sex. z)]11 :s c

(p

If K

= Q + k.

E Z+.

q

E Z+.

P

IIXllmax{Q-q,O}.

(3.35)

IIXllmax{Q-q,O}

+ q = O.....

k;

II(X. Z)II

:s E:).

then (3.34) and (3.35) are equivalent.

3.1S. Theorem Let k be a positive integer. ~ and 11 be c!' smooth vector fields with principal parts represented in the form (3.20) and (3.24). Define the numbers >-lo >-1. "'I. and Qo(k) for the operator A Ii A" as it was done in 3.2 and (3.1). U ~ and 11 have vertical contact of order (Qo(k). k) along the center manifold at the origin. then

"'m

they are c!'conjugate in the vicinity of the origin .

• The proof will be given in § 2 of Chapter VI (see VI.2.8). 3.19. Theorem. Let k be a positive integer. ~ and 11 be c!' smooth vector fields with principal parts of the form (3.20) and (3.24). Suppose that all the eigenvalues of the operator Ah lie in one and the same side of the imaginary axis (for definiteness. in the left side). Define the number QI by formula (3.3). U ~ and 11 have vertical contact (in the strong sense) of order (QI' k) along the center manifold at the origin. then they are

c!'conjugate

with one another near the origin .

• The proof will be presented in § 1 of Chapter VI (see VI.1.6). 3.20. Theorem. Let k 2: 1. ~ E r~(E) and K 2: Qo(k) + k. There exists a c!' smooth coordinate system near the origin bringing the principal part ~ of the vector field ~ to the generalized resonant polynomial normal form

64

(3.36)

• Let Q = Qo(k). By Theorem 3.15, there exist a ~ smooth coordinate system and a vector field 11 E J~(I:.) such that the principal part ~ of the vector field 11 with respect to this coordinate system is of the form (3.26). In virtue of Theorem 3.18, the vector fields I:. and

are locally ~ conjugate.

11

3.21. Theorem. Let K and k be positive illlegers and I:.

the numbers

9 ..... , 9 n

have the same sign (for definiteness,

E

9,

r~(E). Suppose thal all

< 0 (i

=

1, ... , n».

Define the number Q1 by formula (3.3). Assume that K!! Q1 + k. Then there exists a ~ smooth coordinate system near the origin which brings t:. to the generalized resona1ll polynomial normal form

°1

L Pa-(z)ra-,

X = Ah(z)r +

.i:

= tCz)

Cx

E

E', z E F)

C3.37)

la-I =2 • The validity of this theorem follows from Theorems 3.15 and 3.19 since

~

0

1.

Vector fields near a hyperbolic periodic orbit 3.22. Statement of the problem. Let M be a smooth manifold of dimension (d be a positive integer and

I:.

E

+

1), K

rKCTM)' Let S be a closed trajectory of the vector field

1:., with prime period w. By Theorem 1.3.9, there exists a w-demiperiodic

c
'l

... , l), Yj

= ""jYj

> ... >

> 0,

;>'1

(j

=

1, ... , m),

""m > '" >

""1

> 0;

9

= - «l~l

68 I I m « • ./3 _ «l «1 • ./3 1 13 m - ... - a. ~I + 13 JlI + ... + 13 Jlm & - , I.e., x / - Xl ... Xl/I'" Ym is a resonant monomial. It is easy to show that system (4.1) cannot be linearized by any

cI «I + 1131

smooth transformation. On the other hand, according to the Grobman-Hartman

Theorem, (4.1) is topologically linearizable. We would like to obtain an estimate from below for the highest smoothness class of linearization maps.

x«1 satisfies the condition S(k) (see Samovol [1-3]) if at

We say that the monomial

least one of the following inequalities r k a.I ~I + .. . + a. ~r > ;>'r

(1

:II

r

I),

:II

(4.2) 13 I JlI

+ ... +

13• Jl. >

kJl.

(1

holds. Let us show that the condition S(k) guarantees near the origin,

:II

s

c!

:II

m)

linearizability of system (4.1)

i.e., the existence of a c! conjugacy between (4.1) and the linear

system

w

= aw,

Ut

= - ~tUt

(i

= 1,

... , I),

vJ

= JlJvJ

= 1,

(j

... , m).

(4.3)

For definiteness, assume that the inequality

holds. Select a number M such that M

Z

= W + ~ u«"

>

k~r'

We shall check that the change of variables

r

In (

L1

M/~t

Ut 1

),

Ut

= Xt

(i

= 1,

... , I),

t-I

VJ

is of class

c!

Denote v

= YJ

(4.4) (j

= 1,

... , m)

and conjugates (4.3) and (4.1).

= a.I ~I +

•••

+

a.r ~r and consider the auxiliary differential system

(I

Since v

>

= 1,

... , r).

(4.5)

k;>'r and by Theorem A.33, the extension (4.5) has a uniquely determined local

69 invariant section function

Ip

=

Ip(UI'

Ip

•••

,ur ) of class c! . It is easy to check that the

= ~ "~I

... "~rln (

rl"tIM/~t r

)

t-I

satisfies the system (4.5). Hence it follows that (4.4) is of class c! (one can also refer to Proposition 5.13 below). A straightforward calculation shows that (4.4) conjugates (4.3) and (4.1). Let us note that the transformation (4.4) agrees with the identity map when

"r

M/~I

•

M/~r

restricted to the surface I"II + ... + I I = 1. This property uniquely determines the change (4.4) because almost all trajectories of system (4.3) (more exactly, all trajectories except those in the subspace this surface exactly once.

"I

= ... =

u,.,

=

0) intersect

Samovol [3] has shown that the condition S(k) is sufficient for c! linearizability in a much more general setting than (4.1). If I = 0 or m = 0 in (4.1), then the estimate (4.2) of the smoothness class of a linearization cannot be improved. 4.2. Example. Consider the system

(4.6) The monomial

x8y~y~

is resonant and satisfies the condition

S(7).

By the above

arguments, the system admits a C7 smooth linearizing transformation, namely

4.3. Example. Let 11 be a positive integer. Let us show that the system

x = x, y = ny + x" is C'.I linearizable but does not admit C' linearizations. A conjugation between the initial system and the linear system

u

=",

v

= nv

70 can be chosen, without loss of generality, in the form x

Clearly,

= u,

Y

=v+

rp(u).

satisfies the equation

rp

nv + ntJI + un

= nv +

T

dcp u

du

'

or

u dcp

du

Hence, rp(u) =

un

In lui

+

= nrp +

un.

cun , c = const, completing the demonstration.

4.4. Example. Let us return to Example 4.2. The monomial

x8y~y~

satisfies the

condition S(7) but does not satisfy the condition S(8). Nevertheless, we can C linearize the system (4.6), In order to do this, introduce an additional variable

Yo = y~. Then

i=

15z + Y~M, x

= - x,

Yo

= Yo,

YI

= Yh

Y2

= 3Y2'

The monomial y~~y~ satisfies the condition S(8) for r = 1, namely, 4· 1 In accordance with subsection 4.1, the system (4.7) admits a Z

where

Vo

=W + =

V2U 2 •

110

4J.i

I~ ,

= x,

C

(4.7)

+

5· 1

>

linearization

= YJ'

= 0,

10 (vo

+

VI

Hence, we get a

C

smooth linearizing transformation for (4.6):

Vo 1 2

In

U

vJ

8· 1.

(j•

1, 2),

Thus, by introducing one extra monomial variable, we produced a new linearization map (as compared with (4.4» and thereby improved the smoothness class. This motivates the following 4.5. Dermition. A monomial xOl.y'l is said to satisfy the condition MS(k) with respect to system (4.1) if there exist multiindices

C( E

z~,

Kt

E

Z':'

(i = 1, ... , p)

and

71

positive integers

(2) letting



= (u4 +~)

along

the orbits of system (4.13).

0, and let to

surface F(u, v)

= 1,

= - ~ In

+ ~).

to

u4 + ~

(u4

= 1,

suggested by formula (4.4). Note that the equality

= to(u,

be a point with

v, w) denote the moment when (u, v, w) reaches the

vt~

F(uto ,

i.e.,

Let (u, v, w)

II

e4to (u4

+ ~) =

1. Hence it follows that

Assume that the conjugation, when restricted to the surface

agrees with the identity map. Note that U

(u

4

+

2 1/4'

v)

Yo

2to

II

ve

=

V

(u 4

+v

2)

1/2'

Zo

5

we

The solution of system (4.12) is x(t)

= xet,

y(t)

= yeCJ.+XZ)t,

z(t)

= ze-t .

-to

4 2 1/4

= w(u + v)

.

75 Substituting x = xo, y = Yo, z = Zo, 1 = - 10 into the last formula, we conclude that the conjugation map sends (u, v, w) to the point (x, y, z), where

X

If

u

= 0,

= u, v

y

= 0,

1 4 2 (2+uw)!ln(u +v) _

V

= -(u-:4-+-';"-:-)"-I/:-::"Z e then

x

= 0,

y

-

4

v(u

uw

+

_.2 """'1"

v)

,

z = w.

= O.

4.11. Example. Consider the following systems of differential equations:

x = x + rye, u = u,

y = 2y, i = - 2%;

(4.16)

V = 2v, W = - 2w.

(4.17)

Write down the solutions of these systems: (4.18)

U(/)

= uet ,

v(1)

= ve2t ,

= we"2t.

wet)

(4.19) ()

It is easy to check that the system (4.16) satisfies the condition S(2) and does not satisfy the condition MS(3). Nevertheless, the system (4.16) admits a c! linearization. In order to prove this, we use the second method described in Example 4.10 above. We set F(u, v, w) = u20 + U2ZWI2 • Let the point (u, v, w) satisfy F(u, v, w) '" O. Note that F(uet , vi t , we"2t)

= eZOtF(u,

v, w). Therefore 10

= - ~O

In (u zo

+

';ZWI2).

By formula

(4.18) and (4.19), we get x

= u(1 - !.. ;wl' In 10

(u20

+ y2ZwI2»"II2,

y = v, z = w.

It is not difficult to verify that (4.20) is a local diffeomorphism of class can be deduced from Proposition 5.13 below). Using the formula (1 -

0:)"112

= 1 + ~ _!.:1 0:2 + 2

2.4

1· 3 ·5

2.4.6

0:3 _ •••

we conclude that the transformation (4.20) can be written as

(I I < 1)

0:,

(4.20)

c!

(this

76 x

= u + ~o u3 w/- In + _S_ uVw6 16000

(uzo + ~2W11

[In (u20

+ ~2W12)]3

-

8~ uVw· - ... ,

Y

= lI,

Observe that the first two summands in the last expression for (4.11).

c!

§ S. A General Condition for

This section is technical in nature.

[In (u zo

z;

+ ~WI2)]2

= w.

x correspond to formula

Linearizability For a multiindex

"t' E

z~,

we introduce three

pairwise equivalent conditions I!l(k) , I!lo(k) and !!I1(k) such that if "t' satisfies these conditions and L: E ... E is a hyperbolic linear operator, then the polynomial map

x

1-+

Lx

+ p~TeJ

x = Ax + p~TeJ'

is

locally

c!

linearizable (the same is true for the vector field

if the linear vector field

x = Ax

is hyperbolic). We also show that

in the particular case when "t' J = 0 there is a large class of linearizing transformations whose smoothness class can be completely characterized in terms of I!l(k). The proof of the linearization theorems is postponed until § 6.

Condition I!l(k) and a

c!

linearization theorem

We introduce here a condition, I!l(k), imposed on a resonant multiindex

"t'

under which a

polynomial diffeomorphism (vector field) which contains the monomial xT non-linear term admits a local

c!

linearization.

5.1. Preliminaries. For x, Y ERn, we shall write x

for every j

~ = 0 Let it was = Ax

x

E

as the only

l!:

y (and y:s x) if ~

{1, ..• , n}. Given a positive integer k, denote 1R~(k)

=

{x

E

l!:

r

R~: either

or ~ l!: k (j = 1, ... , n)}. L: E ... E be a hyperbolic linear operator. Define the numbers 9 1, ••• , 9~. as done in subsection 2.1S. Similarly, given a hyperbolic linear vector field (x E 8), let 9 1, ••• , 9 n denote the numbers defined in subsection 2.3.

77

. A collection

t1'

=

0f

(t1'\, ... , t1'p)

n-vectors crl

) =( crll\..." , crl

" 9-collection if all the quantities

Lcr{eJ

I = M, and cr is a a-collection. Let U = (u t , ... , un) be a vertex of the domain D'. Then v = ~ U is a vertex of the domain D defined by the relations (5.3). Since or E S(k, v), we have Then cr{ = Mv{ = I <Mv" a> I

i!:

Conversely, let cr = (cr\> ... , crp ), crt E 1R~(k), be a a-collection inequality (5.2) holds for every vertex u of the domain D'. Put

=

It,

1 crt (i I i!: 1 can be rewritten as i!: I I and, consequently, the domains D and D' coincide. Therefore inequality (5.2) holds for every vertex of the domain

u

Uj i!:

Thus, or

E

0 (j

=

1, ... , n),

I)

I (t

k. For every j

E

r

= o.

In fact. if O'~

solution of (5.6), then without fail Us

(j

E

7-1 = yl

{I .... , n}. put

E {I ..... ra} denote Is S

=0

I). UJ

E

J).

if

= {tEl: O'~ > OJ. = 0 for every tEl

= O.

(5.6)

uJ

u is

and

r = O.

hence

the unique

S

r

S

p(

=1

(r. p)

E

p

lact. if 'I-s I:.

S

~ 't" •

then

't"

s

+ Lrp' 0'( - 'I-s

s

~ 't"

-

-s

7

2:

0• Let -s s 7 > 't" • then

(=1

p

_s

•

Since 't

E

+ r ' s L p 0', - 'I

Ao(k. 0') and

(r,

~ T

s

+

ts

s

-s

p O't s - 7

2:

'ts + k -

'I

s

2:

p) e rl«O'), we have p

p

o < - Ep'I 't"s. and p' = 0 otherwise. Let us show that Let

> 0, and 7-1

O.

if i

= ts

rk(O'). In

81

=

0;

(5.8)

otherwise.

The reader can directly check that the transformation z

=w+

C t(x)

= _1_

(5.9)

for every i e {I, ... , pl. Ce Tl We note in passing that most of the linearizing coordinate changes used in § 4 are of the form (5.9) for an appropriate choice of the collection a' = (f1'I' .•. , f1'p). The following proposition establishes the exact value of the smoothness class ,of the

linearizes the model map iff O. Then there exists a number L > 0 such

UO II

that the point (L, ... , L) belongs to the domain D. Denote Do (j = 1, ... , n)}. Clearly, Do c: D.

= {u

E

IRn :

UJ it

L

Fix a number c, 0 < c < 1, and put c5 = c L • For every point x E IRn , which fulfills the inequalities 0 < xJ < c5 (j = 1, ... , n), we can choose a vector ;:; E Do in such a way that ;:;

it

Uo and x J

= c V. J

(j

= 1,

... , n). Furthermore, there exist numbers s

and 1 E {I, ... , p} such that the vector u

= sol;:;

it

1

belongs to the domain D and,

besides, Denote t

= CS

O.

(5.13)

From (5.13) and (5.1) we deduce the existence of a number II' > 0 such that the minimum of S over all ('1, p) E r ,,(rr) and U E D (denoted by SmiJ is greater than 211'. In order to prove CC smoothness of ., it suffices to show that sup ~VI

+ ... + xVPfv

D'I'[x"C'

In ~VI

+ '" +




+

xVp

2: X Vl ,

equality

(5.11),

0, we get

III.

't' II!

A(k, rr) implies •

II!

CC(Rn , R). If

't' II!

A(k, rr),

then there exist a vertex U of the domain D and multiindices 'I E z~, P E z~ such that ('I, p) E r ,,(rr) and the value S of the left-hand side of the formula (5.13) is nonpositive. Choose u, 'I and p in such a way that S = Smin' Consider first the case SmiD < O. Let xJ(t) = t J (j = I, ... , n) and G be the curve I 1-+ ~I(t), ... , xn(t» (0 < t < 1). Let us examine the behaviour of the derivative

D'I'.

along the curve

G.

There exists a number I such that (5.1l) is valid.

85

x

Therefore. the inequality C

= p-(pI+ ... +pPj.

+

(XVI

CI

+... +

Vp X It

Vl

px

is fulfilled

G.

along

Denote

In virtue of (5.10). we get

xT

It

VI

P IVI

+

. .. + pPvp -

+ ... +

'I"

XVp)pl+ ... +pP

O.

Therefore the system (5.15) is compatible iff the following system of relations Vt

l!!

- vt -

uJ

l!!

wi I

p

(j = 1, ... , n),

0

=0

(i

W

> O.

= 1•...• pl.

p

- wL/I1:s 0 has no solutions or. equivalently. the conditions UJ

l!!

0

(j

= 1•...•

n).

W

> 0, (5.17)

- wi I

l!!

0

(i = 1, ... , p)

imply the inequality p

p

- wLptl

1 (i

I:

=

1, ... , p)

and D·: uJ

I:

0

(j

=

1, ... , n),

.

IT I ~ 2. The polynomial (6.1) is called T-divisible if p~ - 0 implies

T:S

w.

Denote J(T)

= {w

E

z~:

3j

E

{1 ..... n}: a J

= <w,

a>;

1.01

~ T.

11.011 :S

Qo(k)

+

I"r!

+

1}

(for the definition of Qo(k) see formula (3.1».

6.2.

eft

multiindex.

T E z~ be a 2, and p: E .... E be a T-divisible resonant polynomial of degree

Linearization Theorem for a Map. Let k be a positive integer. IT I

~

Qo = Qo(k). If T satisfies the condition 5!l(k). then the map x conjugate to the linear map y H Ly near the origin.

H

Lx

~ The proof will be divided into several parts. First. we shall find a

+ p(x) is eft

c!

map H such

that the local diffeomorphisms HI • (L + p) • H and L have contact of order (Qo, k) at the origin. This will be done by solving some triangular system of affine functional equations in certain special functional spaces. Then. in order to prove that HI • (L

+ p)

• H and L are

c!

conjugate, we shall apply Theorem 3.3.

Analysis of the linearization problem Let T E 5!l(k) = SI(k) and rr corresponding a-collection. i.e.. follows. we shall assume that

= (rrl .... , T E

rr p ). rrt E 1R~(k) (i = 1 • ....p). be the Without loss of generality, in what

A(k. rr).

91

"

erie J = - 1

sign ~

(i

= 1,

pl.

... ,

J-l

We try to find a II(L

CC

+ p)

diffeomorphism H: E .... E meeting the requirement 0

H(x) - H(Lx)1I

= o(IIXIl Qo+ h:1 +1 )

as

IIXIl ....

0

and having the following form:

H(x)

= x + hex)

:; x

~ hw(lxla)xw,

+

(6.2)

weJ(T)

where the mappings hw: IRP .... Pw(E, 8), hw = (h!, ... , h:), are such that implies 9 J = <W, 9>. We have (L

+ p)

0

H(x)

= (L + p)

0

(id

+

h)(x)

= Lx + Lh(x) + p(x +

h(x» ,

where

" ",eJ(T)

t-l

wel~

weJ(T)

(j = 1, ... , n).

6.3. Lemma. if q~ • 0, then 9 J = <W, 9> and W2: T-divisible resonant polynomial with variable coefficients.

T,

i.e.,

p(x

+

hex»~ is a

• In virtue of the binomial theorem, we get

(6.3)

Where s =

"

~

t-l

St,

CIt

( {3 11 ,

... , {3

l,sl

" 1

, ... , {3 , , ... , (3

",Sn

)

92

s"

SI

E

Z:;

A= (

L 13 t-I

o ~ A ~ IX;

L 13"t);

1t , ••• ,

wtt

E

J(T)

1, ... , n; t

(i

= 1, ... , St);

t_1

(j, IX) and (i,

Wtt)

are resonant pairs, i.e., 9J

=

, 9 t

=

<Wtt, 9>

(i = 1, ... , n; t = 1, ... , St); c~ E Pa.-Il.(3(E, The sum in (6.3) is taken over all IX and f3 such that Sr

wr

=

fir -

"S(

L f3M

+

t-I

L L W~t/3(t

(r = 1, ... , n),

(-I t-I

Sr

L 13

M

~ IXr

(r = 1, •. , n).

t-I

Let us show that the pair (i, w) is resonant. In fact,

<W,

9>

=



"St

=



L L /3 tt

(-I t-I

" (9( -

L W~t9r)

=

< IX, 9 >

r-I

since both (i, IX) and (i, Wtt) are resonant. It is easily seen that W ~ T. p(x + h(x» is a T-divisible resonant polynomial with variable coefficients.

Thus,

6.4. Notation. Let q denote the sum of all terms entering the expression p(x + h(x» and having degree less than Qo + I T I + 1, i. e. , q is of the form q(x) = (ql (x) , ... , q"(x» , where - p(x)

93 rr(x)

r q;!(lxl")xW

=

(j

=

1..... n).

wEJ(or)

Put r(x)

= p(x + h(x»

- p(x) - q(x).

6.5. An auxiliary system of functional equations. Set down the functional equation h(Lx)

= Lh(x) + p(x) + q(x)

(x

E

E).

(6.4)

or. in the coordinate form.

L h~(lLxI")(Lx)w = L LJh~(lxl")XW wEJ(or)

wEJ(or)

L p-!;cW + L q;!(lxl")xw

+

wEJ(T)

=

(j

1..... n).

wEJ(T)

Consider the related system of functional equations

(6.5) (j

= 1.....

n; WE J(T);

9J

= <W. 9».

Clearly. substituting the solutions h~ of system (6.5) into formula (6.2). we get the needed solution of equation (6.4). Consider first the case when the operator L is diagonal (this somewhat simplifies the further arguments; the general case will be considered separately in the final part of this section). Let few: Pw(E. XJ ) -+ Pw(E. XJ ) be the linear operator defined by

Let t.: IR P -+ IR P

be the linear operator having the following matrix form n

n

94

Rewrite system (6.5) as follows: (6.6) Along with (6.6), consider the following system of functional equations:

(u e IR P ;

x e E; j

=

1, ... , n; we J('r); BJ

=

<w, B»,

or, equivalently,

(6.7) (j

=

1, ... , n; w e J(T); BJ

=

<w, B».

Clearly, every solution of (6.7) satisfies (6.5). Note that the functions q~ depend only on h~ with 1cr.1 < 1w 1 (see formula (6.3». Therefore the system of equations (6.7) has block triangular structure with respect to the blocks {h~: 1w 1 = r} (r = 1T I, ... , Qo + 1T 1 + 1) and, besides, the equations within every fixed block are independent. Consequently, system (6.7) can be solved inductively on the number

=

1w 1

Consider first the case r in (6.7) takes the form

=

r.

1T I. Since q~;;; 0, w

(B J

=

= T,

the corresponding equation

(6.8)

Tie, then D"""~ ;;; O. Finally,note that D""": P"C'(E, Xj) -+ L:_a(E, P"C'_,,+a(E, XJ» is a bounded operator. For I a I ~ 1, by the composite mapping formula (see subsection A. 6), the following equality holds:

Da[cp(lxl a )]

=

r

P

e(p, ~)[ifcp](lxla\ ... , Ixi"'P) IT

(i = 1, ... , p; j = 1, ... , pt), e(p,~) = const P

sum is taken over all p and

~ = (~tj)

Therefore

p

Hence it follows that

such that

1 ~ Ipl ~ 1151,

and the

p'

rr

~tj = a.

97 (6.10) P

= const and the summation extends on all

where C(p)

+

I

T+pcrl

p

E

z~ with 0 sip I s lor I and

p

.. ·+pcrpit:r.

Let IIxll s 1, [x] ~ O. In accordance with formulas (6.9) and (6.10), if rp 7 E z~,

then for every

E

fi~,.,

0 s Irl s k, we have (6.11)

Thus, the map

X t-+

CC

rp(lxl")x'" can be extended

1P(lxlcr>x... ;;;; 0 (IIXII s 1,

smoothly to B{O, 1) by putting

[x] = 0).

6.10. Lemma. Let T E AI{k, IT). Then/or every (r, p) E rJ«cr), a vector s = S(7, p)

p( Ix I cr)

with

IR P -+ P-.:(E, X J ),

i.e.,

can be chosen in such a way that there exists a local representative p; = -

P

E

-J
, q iii 0 implies = 0, Ah(O) = A h , 11'(0) = 0 and Drp(O) = Ac. Suppose the multiindices T and ~ satisfy the condition i!l(k) with respect to the operator A h • Then the vector field E is locally

c!
: supp II> C {(x, z): IIxll:S I}, endowed with the norm (IRP \

r

E

z~,

P E Z!, 11 E Z+o X E

E',

[x] .. 0, Z

E

Jt=, Irl +

11 :S k, (r, p)

E

rJc(cr)}.

Since the eigenvalues of the operator Drp(O) lie on the imaginary axis, all the methods and estimates used in the proof of Theorem 6.2 continue to be efficient in this situation. 6.20. Remark. As it was already pointed out (see Introduction), the problem on conjugation of diffeomorphisms (vector fields) is closely tied with the problem on the existence of invariant manifolds. For example, in order to conjugate the mappings x t-+ Lx + ~(x) and y t-+ Ly + ~Cy) with one another, it suffices to find an invariant section of the following extension:

[x] Y

In fact, let x

= hCy)

~ rLx + ~(x + lLY + IIICy)

y) - IIICy)]

be an invariant section then

= h(Ly + IIICy»,

LhCy)

+

ICy

+

hCy» - IIICy)

LhCy)

+

ICy

+

hCy»

= Ly +

+

= (id + h)

hence

Ly

+

IIICy)

+

h(Ly

(L

+

+

i.e., (L

+

I)

Thus the coordinate change x

0

(id

=y +

h)

0

hCy) is a conjugation.

III).

'IIcy»,

107 The converse statement does not, in general hold, i.e., not every problem concerning the existence of an invariant section of a non-linear extension can be reduced to a conjugacy problem. Note that Theorem 6.2 can be rephrased as follows: if a multiindex 1: satisfies the condition l!I(k) and p: E -+ E is a 1:-divisible resonant polynomial, then the extension

has a

C


(i

aPs; > k s;.

= 1,

tl't 2:

e 1R~(k)

0 (i

=

Let us show that

Te AI(k, tI'), where

... , p) and R is a large enough positive number.

(i = 1, ... , p) and

sufficiently large, then there exist numbers t;t

1, ... , p) be the corresponding elements. Denote

Without loss of generality, we may suppose that

sp• and a I SI• + ... + RCt

=

=

(i

1,

s· > 0, ... , pl.

hence, Let

tI'

is a e-collection. If R is

(" p) e rlc(tI').

Show that

1, ... , p) such that

(7.2)

t'

..

:5 '" .t

+ p'....!i... (.r = 1, ... , p.)

Consider the convex domain Q c

s;

RP

defined by the inequalities (7.3)

The set of vertices (angular points) of this domain is contained in the set of nonnegative solutions of all subsystems

t;'

=0 (7.4)

P

t' ttc J, = ,J l....

(j e J '"

UI'

'}) (s ... J",

1, ... , min {P,

nIl,

110

each of which has a unique solution. Let (0

= «(~, ... , (~)

p

be an angular point. Then

p

LrJ = L L~~ e{ = L(~ Le{. JEJ

Besides, if

Le{ = 0

JEJ t-1

(j = 0

then

(since the point (0 is a vertex). and

JEJ

Le{ - 0 JEJ

Le{ ~ 1. Therefore,

implies

JEJ p

p

L(~ Le{ ~ L(~. t-1

JEJ

t-1

p

Hence. 1(0 1

=

L(~~ LrJ ~ 1r I. t-1

Thus. the origin and all the angular points of the

JEJ

= 1r I.

The condition

+ pP ~)

belongs to the

domain Q lie in one and the same side of the hyperplane 1(I (r. p) E rk(er) implies that the point A

= (.s1 + pi

~ ..... .sP al

ap

domain Q. Consequently. the point B. the intersection of the straight line OA with the boundary of the domain Q. satisfies (7.2). Set At = gla; (.st - (t) p

+

pt ~ 0 (i = 1.... ,

pl.

Then

p

LAtert = L.stet + t-I

'-I

Further. p

L(A, - p')<ert. a>

=:

•

p

La;(.s' - (t) ~:

•

p

(

L.sta; - k a;)

> O.

7.4. Definition. We say that a multiindex 't E z~. 't = (a.. 13). satisfies the condition S(k) (and write 't E S(k» if at least one of the following n inequalities holds:

111

f3 I

+ ... + f3II"'II >

"'I

k

"'II (1

S

ssm).

7.5 Theorem. The condition S(k) implies MS(k) .

< 't

°

• Rearrange the numbers 9 1, ••• , 9" in such a way that 9 1 < ... < 9 t < < 9 t + 1 ... < 9" and change accordingly the numeration of components of multiindices. Let

E

Suppose, for definiteness, that the inequality Ch'l

S(k).

fulfilled. Set £~

£~

=

°

= 1,

(s '" 1 - r

9 *t

=

1* a 91

Hence, 't

E

+

£~

1), 15 1



+ ... +

=

°

(s '" I), £~.I

= a. 1,

= - 7I. t


k 7I.f" is

°

(s '" 1 - 1), ... , £~""+I == 1,

7I.f"

= k* 9f'"

... , r);

f" - ... - a. 7I.f"

< -k

MS(k).

7.6. Remark. Thus, the conditions S(k), MS(k) , and 2I(k) introduced above satisfy the

relations S(k) .. MS(k) .. 2I(k).

Therefore, each of these conditions ensures the possibility of CC smooth linearization of the diffeomorphism X 1-+ Lx + p(X) , where p(x) is a 't-divisible resonant polynomial. It is easy to show that the converse implications do not take place. In fact, the multiindex (5, 6, 0, 8) satisfies the condition MS(8) (see examples 4.2, 4.4 and remark 4.6) but does not satisfy the condition S(8). Hence MS(k) does not imply S(k). Let us show that 2I(k) is, indeed, weaker than MS(k). 7.7. Example. Consider the multiindex

't

= (0,

resonant monomial ;y:Oy~ (see Example 4.7). Here 9.

= 200.

Show that 't

E

21(13).

Set

IT1

= (0,

5, 10, 4) 91

that corresponds to the

== - 690,

0, 1, 0),

IT2

92

= - 300,

= (0, 1~'

0,

~).

93

= 1,

Then

112

1300 = 13 . 100 = k max A J 83 83 83 J-l,3,4 ' 4 . -4

5

= -66 > -65 5

=

13

5

=

k max BJe J-2,3,4

Thus, T e A(13; cr), cr:J and, consequently, T e 21(13). Nevertheless, the maximal value of k such that T e MS(k) is equal to 9. Therefore l!I(k) does not imply MS(k). 7.S. Theorem. If the operator L is contracting or expanding, then the conditions l!I(k) and S(k) are equivalent. ~

By Remark 7.6, it suffices to prove that in this case l!I(k) .. S(k). For definiteness, let L be an expanding operator. If T e l!I(k) , then T e A(k, cr) for some normalized 9-collection cr = (cr), ... , crp )' Since L is expanding, we have m = n. Rearrange the invariant subspaces XIo ... , Xn in such a way that 0 < 9) < ... < 9 n • By s denote the maximal number j e {I, .... n} such that e {I, ... , pl. Let us verify the inequality

Since

=

1 (i

=

cr{ > 0

1, .... p), the point

U=

(9) •.•.• 9 s ,

0, ... , 0)

is a vertex of the domain D defined by (5.3). Hence it follows that

for some i = io

113

= Tiel + ... + TSes > k max e, = k es ' ,- 1 ,s

Therefore,

T E

S(k).

7.9. Derlnition. The multiindex T = (ex, 13) satisfies the condition i!'1(k) if at least one of the following four properties is valid: (1) there are a number r, 1 s r s I - 1, and positive real numbers I:r+1 S ... S I:l such that (r

+

1s i

:5

I)},

(7.S)

(p

(2) there are a number s, such that

1:5

= r + 2,

ssm -

1,

.. " I);

and POsitive real numbers I:s +I:5 ...

S

I:m

(7.6)

(p

=s +

2, .. " Ill);

114

m

(4)

L

(fJJJ.lJ

>

k J.lm •

J-=I

7.10. Definition. The multiindex T = (a., (fJ) satisfies the condition Irl(k) if at least one of the following properties is fulfilled: (1) there exist a number r, 1 ~ r ~ I - 1, and positive real numbers C 1 ~ ... ~ Cm such that J

r

m

t-I

J-I

La.t~t + L ~ cllJ

> k max {~r;

Il J

(1

~j ~

m)},

cJ

J

(7.7)

= 2,

(q

(2) there exist a number s, such that s

L

(fJJJ.lJ

1

+

~

s

m - 1, and positive real numbers

t

>

k max

~t {J.l s ; -

(1

~ j ~

CI ~ .. , ~ C1

l)},

Ct

Ct

m

L~J"'J + L 1~J+ "'J

J-I

~

L a. ~t t-I

J-I s

1

... , m);

J-s+I

CI

>

k max

{"'s;

1

"'m

+

}, CI

(7.8)

115 s

m

(q =

2, ... , I);

m

(4) [(3JftJ

> k

ftm .

J~I

7.11. Tbeorem. The condition !!'I(k) implies fI(k). Fix a number r, 1:s r:s 1- 1. Let EtJ denote some positive numbers (i = r + 1, and p = m(1 - r) + r. By 11' = (cr l , •.. , I1'p) denote the normalized a-collection such that ~

..• , I; j = 1, ... , m)

... ,

The condition A(k, 0") can be rephrased in the following way: if •.. , vm ) is a vertex of the polyhedral convex domain D determined by U, 2:

0 (i

=

r

+

vJ ~

1, ... , I),

0 (j

(Ur+I' ••. ,

uz ,

VI'

1, ... , m), (7.9)

u(

1

+

c'J

~t

+

EtJ

vJ -

2:

1

(i

=

r

+

1, ... , /; j= 1, ... , m),

IlJ

then r

Lcx(~( (=1

m

+

(

L a. U, t-r+1

+

L{3JvJ

>

k max {~r; ur + l , •.. , uz ,

J=I

Let the property (1) of Definition 7.9 be fulfilled. Set

VI' ••• , v m }.

(7.10)

116

=

Ctj

c,

(i = r

+

1, ... , 1; j

= 1,

... , m).

Then system (7.9) takes the form

r + 1, .... I),

U,

a: 0 (i =

VJ

a: 0 (j = 1.... , m),

1

+

U,

c,

(7.11)

E, a: 1

+

vJ -

(i

I'J

A,

=r +

1, ... , 1; j

= 1,

... , m).

Let us show that the convex domain D c: IRn -r defined by (7.11) has no more than I - r

+

1

vertices of the following structure:

Ao = (

Ar+l =

A r+l

l+c r +l

AI

..... 1

)

' 0, ... , 0 ,

+£1

(0.... , 0, ~, .... Cr + 1

= (Ar+ 1 (

1

Er+ 1) •

CP -

+

I'm ), Er+ 1

£r+1

(p = r

+ 2.... ,

I).

A = (U~+h ... , U~, v~,

... , v!) be a vertex. Suppose there exists a number t E {I, .... m} such that v~ = O. Then, by (7.ll). u~ = A,/(1 + E,) (i = r + 1, .... I). hence v~ = ... = v! = 0 (thus, we get the vertex Ao). Let v~ > 0 (j = 1,

In fact. let

.... m). Assume u! (i = r B

+

such that

P

E

{r + 1..... I}. Show that p < I implies u~ = 0 in virtue of (7 .ll). hence,

R.ea1lY,

1.... , I).

= (u~+lt

u!+1 = .. ,

= 0 for some

... , U~h 0, ,,_, 0, v~, ... , v!)

= u~ = O.

U!.I

$

If U~+1

0, P VJ

it

r .f-

a: ~ f:p

also belongs to D.

= 0, we get the vertex

the point

Since A. is a vertex

A r + 1• Letp be the maximal number

2. Then system (7.11) is equivalent to the following one: (j

=

1.... , m),

up

= ... = Ul

= O.

117

Because these relations are independent. we get the vertex Ap. Let us show that there are no other vertices. In fact. let A be a vertex and u~

=r

= 1•...•

1•...• I). v~ > 0 (j of some subsystem of the system

(i

+

(i

> 0

m). Then this vertex is the unique solution

=r +

1•...• I;

J = 1•... ,

(7.12)

m).

But (7.12) has the solution A o, a contradiction. Substituting the coordinates of the vertices Ao. A,.+1o •..• Ap into (7.10) and applying inequality (7.S), we establish that 't' E A(k, 0'). It remains to note that the property (2) of Definition 7.9 is equivalent to property (1) for the operator L- I , and, besides, (3) ~ S(k), (4) ~ S(k). 7.12. Theorem. The condition I!:l(k) implies S(k). ~

E'J

Let

= EJ

't' E

(i

=

Suppose the property (1) of Definition 7.10 is fulfilled. Set r + 1, ... , I; J = 1, ... , m). Then system (7.9) takes the form

I!:l(k).

U, it

0 (i

=r +

VJ it

0 (j

= 1,

u, 1 +

EJ

~t

+ vJ

1, ... , I),

... , m). EJ -

it

1

"'J

(7.13) (i

= r + 1,

... , I;

J = 1,

... , m).

Using the same arguments as in the proof of Theorem 7.11, we can check that the convex polyhedral domain Dc: the following structure:

Bo

=

(0, ... , 0,

IRn-r

determined by (7.13) has no more than m

"'I. ... , "'m ), EI

Em

+

1 vertices of

118

(q

= 2,

... , m).

Substituting the coordinates of the vertices Bo, BI , inequality (7.7), we establish that 't E A(k, 0').

... ,

Bm

into (7.10) and applying

7.13. Dermitions. We say that the multiindex 't = (a., /3) satisfies the condition It'(k) if T E It'1(k) or 't E It'2(k). The multiindex 't is said to satisfy the condition C(k) if at least one of the following properties holds: (1) there are numbers I: > 0 and r, 1 :s r :s I - 1, such that ,.

l

( ( + -1- L 0::>', L0::>', 1+1:

(-I

:>'z > k max {:>',., - },

1+1:

(-,.+1

(7.14) ,.

m

Lo:'i\, +! L/3JjJ.J

> k max {:>',.,

jJ.m };

I:

(2) there are numbers

I:

J-I

(-I

> 0

I:

and s, 1 :s

S

:s m - 1, such that

(7.15) IS

L/3 JjJ.J

I:

L0:':>., (-I

> k max

> k:>.z or

L/3JjJ.J

> k

{jJ.s, i\z }; I:

(-I

m

Z

(3)

La.'i\,

+!

J-I

jJ.m'

J-I

7.14. Theorem. The condition C(k) implies Il:'(k).

119

• It suffices to note that putting £( = £ (i (7.14) (and similarly, (7.6) goes over into (7.1S».

=r+

1, ... , I),

(7.S) turns

into

7.1S. Theorem. The condition S(k) implies C(k).

• For definiteness, suppose that the following inequality holds: a: 1~I

Put

£

= max { ~l

~r

-

+ ". +

a:r ~r

> k

~r'

1, /-1m }. Then the condition C(k) is fulfllied. ~r

7.16. Notation. Fix an integer r, 1::s r ::s I. Set

For

£

Let

£1

> 0 put

be the solution of the equation

that belongs to the segment [(;\l - ~r) I ~r' /-1m I ~r] and

7.17. Theorem. Let T

=

leo

£'1.

be the solution of

be defined as indicated in Table 7.1. Then the multiindex (a:, (3) satisfies the condition C(k) for every integer k < leo.

120

Conditions

1\ I" - + -

?r?,

leo ?, -

:5

?r

1\ I" -+-

13

1\ - - + ;>'r f.Lm

f.L m

I'].

1 :5 1 ?r

+

?r

I" +

£

?r

;>.,

13 1\ 13 >- -+-

f.Lm

f.L m

?r

f.L m

?, -

?r

-;>'rf.Lm

-

;>'r

?, - ?r 1\ 13 I" -+- >1\--+?r?'

?r f.Lm

?r

1\

f.Lm

+

?r

f.Lm

1> 1 J\

+

?r

f.Lm

?,

K\(£\)

£

\

f.Lm

?r

f.Lm

1\ 13 -+?r

f.Lm

13 I" -., - ?r

-;>'r

1\ 13 I" +-

K2 (

~m ), i.e., n. ,.

122

and let

(b) Let 1

£1

be the solution of the equation

> 1. Then

Observe that the function Kl is increasing and the function K2 is decreasing on the segment [IJ.m, ~t ~r

Let

12 _ ~

~t

-

~r

].

~r

13

~t - ~r '

and let

£

2

denote the solution of the equation

123

7.1S. Dermition (see Samovol [5]). Let 1:s r :s I, duce the following notation: r

1:s s :s m,

= ( Ia,.l. Then J I = k A,., J2 > 0, J3 > O. By Theorem 7.17, the relation ko > k holds, i.e., T E C(k). Suppose now that the property (2) is valid. Without loss of generality, we may assume that lal > k,

alAI

+ ... + alAl = k Al,

f31J.1.1

besides,

ri

< k, J I =

L«tAt,

+ ... + f3mJ.l.m

= k J.l.m,

al

'"

0

and,

m

l-I

J 2 = alAl, J 3 =

t-I

we have J I = (k - «l)A 1• J3 = k"m. Therefore,

L{3JJ.I.J '

J-I

Since

p~(T) = pic(T)

= 0,

124

-

JI

>"-1

-

J" >.,

+ - -k =

JI

>'1-1

J ("

>'1-1

+

(k - « )>'1

Jl.m

>"-1

+

cr. - k

= (k - «, )(>',

- >"-1)

> 0,

,

J3

+ - -k =

I ", - I

Jl.m

Al_ I A,

By Theorem 7.17,

(k - cr.')>."

ko >

) +....! J _k = A,

>

0,

(k _ «')

Jl.m

> O.

A'_I

k and, consequently,

T E

C(k).

7.20. Remark. Thus, we have obtained several relatively simple conditions each implying the condition 21(k) and, consequently, sufficient for c! linearizability of the corresponding diffeomorphism (vector field). The main logical hierarchy between these notions is indicated in Table 7.2 (where arrows denote, as usual, implications).

SoCk) -

C(k) -

~ S(k) _

It' (k)

~ MS(k) _

21(k)

Table 7.2 7.21. Example. Let us apply Theorem 7.17 to the vector fields (4.6) and (4.9).

(1) Reversing the time direction in (4.6), we get «I = 5, «" = 6, 13 1 = 8, AI = 1, >.,. = 3, Jl.1 =1. Put r = 1, I = 2. Then J I = 5, J" = 18, J3 = 8, 1 = 2 > 1. Hence

+ Jl.1) J" 10 18 S JI J" J" J3 -......;;....-......;...-+-=-+-4=-~A I >." A" 3 3 3 >." Jl.1' >." A" - AI

JI(>'I

According to Table 7.1,

125

Thus, the condition C(9) is fulfilled. In fact, for

• «:>..

£

= 1

1 2 +- «~= 5 + 9 > -272 = 9 max {I, -32 } = k 1+£"2 1 «:>'1

+ -£1 (3 1J.il = 5 +

= 13 >

8

9

=9

max {I, I}

:>. 2 max {:>.. - - } '1+£'

=k

max

J.il {;\I' }. £

The coordinate change

is

f!

smooth and conjugates (4.6) with the linear system

(2) Reversing the time direction in (4.9), we get ~

= 200,

J.il

1 = 199/300

= 300,

r

= 1,

I

= 2.

Therefore,

= 10, «2 = 4, (31 = 5, ;\1 = I, II = 10, 12 = 800, 13 = 1500,

«I

< 1. Hence it follows

Thus, leo = K 1(£\ where £1 is the solution of the equation £2 - 149£ - 24150 = 0 that lies between 199 and 300. Take the approximate solution £ = 246. Then I 1 2 «:>.. + ~ «~

=

800 10 + 247 > 13

i.e., the condition C(13) holds.

=

200 13 max {I, 247 }

=k

:>. 2 max {:>." ~ },

126

7.22. Example. Let us show that t!(k) does not imply C(k). Consider the vector field

= 3, cr.3 = 3, ~I = 6, ~I = 1, J 2 = 75, J 3 = 36, 1 = 1~ > 1.

= 3, = 3,

cr.2

=2

then J 1

Here cr. 1 then J 1

~

= 5,

~3

= 20,

fl.1

= 1.

If r

=1

Besides,

hence,

If r

= 18,

J

J

~2

~3

= (£2'

£3)

J2

leo = ....!. + ..! =

Consequently,

= 60,

J3

= 36,

1

= 1~ >

1, and

6.6. Hence, the condition C(6) is fulfilled (but C(7)

does not hold). Let r

1

cr. ~I

+ -~

= 1,

£

= (4,

19). Show that

1

fl.1 £2

= 3 + 9 = 12 >

10.5

=7

3

• -2

't E

= k max

t!(7). In fact,

{~I'

fl.1

-

},

£2

Thus, the condition t!(k) does not imply, in general, the condition C(k). The change of variables

127

conjugates the vector field under consideration to its linear part and is of class C7 •

The condition S(k) for n

= 2, 3

Samovol [3] has shown that in the nodal (diagonal) case the condition S(k) gives the best estimate of smoothness of a linearizing map. For such equilibria, S(k) is equivalent to S(k) (see Theorem 7.8). Our next goal is to examine the condition S(k) for n = 2, 3, assuming that the equilibrium is of saddle type.

7.23. Lemma.

1fT E

= 1,

for every i

A(k, cr) and there is a number j

... , P then k
l.

'-I Put '1'

=0 T

SI

(i

+

It}), lSI

p crl

r!

+ ... +

In fact, for s. j, We get because crT

it

= k,

TSI

p pSI

p cr p

+

k. Hence, ('1, p)

= (1,

E

SI

=

T

'1S1

=

TS

- '1

cr~ -

0, ... ,0). Then SI

+

cr l - '1

+

cr~

SI

it

rk(cr) and therefore

SI

it

0

(s = 1, ... , n).

O. If s = j then

TJ

+ crT - k it 0

128

Consequently,

'tJ

> k.

7.24. Theorem. 't2

= 2,

Let n


O. ,- \ ,2,3

i.e .• \

T ).\

Thus.

T E

A(k;

+ T2~ >

k).2'

IC\I IC~.

7.27. Lemma. If n = 3. T E A(k. cr). crt = (1/).\. o. 0). cr2 = (0. and E > 1/).2' then there exists a number IS such that T E A(k; IC\. 1C4)'

E.

(E~ - 1)/Jt)

• Denote

If there exists a number i E {I. .. .• p} such that ~,.- 0 and «,).\ + ~').2 = I. then the assertion holds by virtue of Lemma 7.26 for IS = 1/).2' Suppose now that ~,.- 0 implies «,).\ + ~').2 > 1. Consider separately two cases: (a) «,E).\ + ~, ~ E (I = 3..... p). Then the domain D determined by

(7.16) U\«,

+ ~, + ~(

1-

a.Ji\.

it I:

1-

fl3

(1 - a.3i\.)

= "Ji\. + {3J

BJ> =

1;

= 1;

{33

1-

{3t

"Ji\.

+

it a.ti\J

{33

=0

then

i\.a.,

l!:

1-

"ti\J

f3, =

1

{3,

1.

For B2 :

(1 - "si\ J)(£i\2 - 1) l!:

+

Q:~.

{3~2 -

> f3~2 + a:~. «,5'i\. + f3 s i\2

{3,

=- 0, i

Put a =

= (1

B2 >

= a:ti\J +

a:ti\.

f33 -~-. 1 - «3i\J

or e A(k, 17') yields

pl.

If

(3,

s i\J)£i\2

' a.Ji\J

{4, ... ,

Q:

+f3 s i\2 - 1

= 1.

1 1

= "Ji\J +

E

-

a.si\J



.

Now let us tum to the proof of Theorem 7.25. Let number i E {I •...• p} such that cr~ 1 - otA. - - - • 0). where 0 s ot < I/A 1• and

= O.

't E

A(k. cr).

Suppose there is a

In this case. if cr~;I: 0 then crt

= (ot.

•

't

A'}.

e A(k; " •• ":z} accordmg to Lemma 7.26. If

cr~ = O. then crt = (lIA •• O. 0). Consider the following two possibilities: (1) cr!;I: 0 for every

S E

{I •. , .•

pl.

Then k


"',

~ -~,

~2

the value of ~,T' three cases, fact, ~2 - ~,

+ "')

",T3 it:

> '"

+

2

+T >

~2T2

- ",T3

T

,

3

+ T.

However, by virtue of the resonance condition,

belongs to

the set {~" ~2'

- ",}.

In each of these

~,T' + ~2T2 - ~ it: ~2(T2 - 1). Show that the domain D" is empty. In implies ~/", > 1 and (~, + "')/~2 < 1. Therefore, the relation

T'(~,

+ "')/~2 + T2 > T' + T3 gives T2 > T3. Besides, from T2 > 1 we get T3 it: ;>'2(T2 - 1)/", > T2 - 1. Thus, if T2 > 1, then T2 > T3 > T2 - 1, contradicting the fact that T2 and T3 are integers. Consider now the case T 2 !iS 1. Since T2 > T 3, we have T2

= 1,

= O. The resonance condition implies

T3

T'

= O. Therefore,

= 1, contradicting the assumption I T I it: 2. Consequently, D" = Table 7.3 establishes examples showing that all the sets D, - DIO finish the proof, we refer to Theorems 7.17 and 7.25.

IT I

= T' + T2 + T3

We point out that are non-empty. To

Ill.

7.30. Example. Let us consider in detail. one of the examples given in Table 7.3, namely, the following vector field: _.3 2' ' = 2y. x,• = - x, + x,xV' , X2 = - 22'x Y

Here

= "';

;>.,

= 1,

T'

+

T2

i\2

= 2,

=4 >

(7.18)

= 2, T' = 3, T2 = 1, T3 = 2. Therefore ;>'2 - ~, = 1 < 2 = T'(i\2 - ;>.,)/", + T3; ~T2/(;>., + "') = 2/3 < 2 = T3. Consider

'"

7/2

the equation

3

+ _2_ = 3£ + 2 1

Let

= [1;

Thus,

£'

be

the solution

2]. Clearly, 7/6

(;>."


'2 - l)/J.I) = (0, 6/5, 7/10). The corresponding domain D (see (5.3» is defined by UJ 2:

0 (j

= 1,

2, 3),

and has the following two vertices: A. ditions (5.2) take the form

3 Thus,

+

-r II (-rl, -r2, -r3) E A(3, "I'

= (1,

>k

20 7

ul

"J.

2:

~ ~ + .1 ~

1,

5

0, 1017) and B

10 3 7 '

+

5 -6

Consequently,

2:

10

1

= (1,

5/6, 0). The con-

> k. system

(7.18) is locally

C

linearizable. 7.31. Remarks. (1) We conjecture that Theorem 7.29 gives the best possible value of the smoothness class of linearizing coordinate changes (in the diagonal case).

(2) Recently we have obtained some results analogous to Theorem 7.25 for n = 4, but we have not included them because they are only fragmentary. Investigation of this case is in progress.

f 8. Theorems on

ck

Normal Fonns

In this section, the results obtained thus far are applied to prove several theorems concerning ck normal forms of finitely smooth vector fields (diffeomorphisms) near an equilibrium (fixed point, respectively).

8.1. Lemma. Let L: E ... E be a hyperbolic linear operator; q: E ... E be a resonant polynomial of order "(q) not less than 2; -r be a multiindex satisfying the condition S(k); p: E ... E be a -r-divisible resonant polynomial. Then the mappings F(x) = Lx + q(x) and G(y)

= Ly + q(y) + p(y)

are

ck

conjugate near the origin.

~ Define the number Qo = Qo(k) by formula (3.1). When proving the lemma, it will be useful to return to the scalar variables %1' ••• , Zct (see subsection 2.3). Denote

137

E(w)

Let

VI'

=

{r

E

Z!:

==

{r

E

z~: er(r) = er(w)}

••• , I'd

rl.1

+ ... +

rt.ml

= Wt.1 + ... + Wt.mt

be the eigenvalues of the operator L.

i - j if I V t I = IvJ I. For every multiindex for at least one number i E {I, ... , n},

For i, j

(i

E

=

1, ... , n)}

{I, ... , d}, write

which satisfies the condition q~;e 0 introduce additional monomial variables

W

= i~

°

(r E E(w» and form the vector ul == fil(Z) = {u!: r E E(w)}. Let 1 be the dimension of this vector and BI denote the d x 1 matrix composed of the elements

u!

btJ =

°

q! (.,.

E

E(w),

i

= 1,

... , d).

Describe the transformation of the vector ul induced by the map F. Since the subspaces E I , .... En are invariant under the operator L = {IU}l.J-I .....d we have

Zt 1-+

r

Qo

It;ZJ

+

r

q!zw

(i = 1, ... , d),

I W I =1C(q)

t- J

and, consequently, d

u!(Lz + q(z» I W 1-IC(q)

Hence it follows that the induced transformation is of the form

where CI is a linear operator such that the moduli of its eigenvalues take values of the fiorm

..,1

d

11"11 ... IVdl"', and ql(z) is a resonant polynomial. Moreover. since q is resonant and 1.,.1 2: 2. we conclude that these moduli belong to the set {lvII, ... , 1",,1}, and K(ql) 2: rc:(q) + 1. Thus. we get the mapping

Continue the process of introducing additional monomial variables untill the least degree of monomial terms that are not T-divisible exceeds the number Qo(k). As a result.

138

we get a mapping of the form

where u' are gcvectors (i

= 1,

... , r); the moduli of the eigenvalues of the operator

(where the blank entries are zeros) belong to the set {exp 9 1, ... , exp 9 n }; qr is a resonant polynomial having contact of order (Qo(k) , k) with the zero map. Besides, it is easily seen that the

c! smooth manifold

is invariant under t r • It follows from Theorem 3.3 (reformulated for diffeomorphisms) that conjugate with the map

~r(~' v)

= (L~ + Bv,

Cv)

(~E E, v

= (VI,

tr

c!

is locally

... , v».

Let (z, u) = he!;, v) II (hl(~' v), ~(~, v» be the conjugating map. Then the map has an invariant manifold v = ~(~), where graph tp = h(graph ~), t.e.,

~r

(8.1)

Put H(F.) = hl(F., ~(F.», R = HI 0 F 0 H. Since h has contact of order (Qo, k) with the identity map, the same is true for the mapping H. Therefore, R(F.) = ~ + q(F.) + 1(F.) , where I is a c! smooth function having contact of order (Qo, k) with the zero map. Since h = (hI' ~ conjugates ~r and t r , we have

(8.2)

139

+

C~(~, v)

Letting v

= ~(I;),

q,.(hl(~' v»

=

h,,(1.(

+

Bv, Cv).

(8.3)

we deduce from (8.2) that

Taking into account (8.1), we get

or, equivalently, (8.4)

That F and R are conjugate means

Comparing (8.4) and 8.5) yields h,(Lt;

+

B~(I;), cq;(~»

The equality (8.3) for v

whence it follows U

= "(l»

=

= ~(I;)

hl(Lt;

+

+

q(l;)

1(1;), ~(LI;

+

q(l;)

+

1(1;))).

(8.6)

gives

(because of (8.1)

and

the invariancy condition for the manifold

that

In virtue of (8.5), h,,(Lt;

+

B~(I;), C~(I;»

= ,,(hl(LI; + q(l;) + 1(1;),

~(Lt;

+

q(l;)

+

1(1;»))).

Applying once more equality (8.1), we get h1(Lt;

+

B~(I;), C~(t;»

= h,,(Lt; +

q(~)

+

I(t;) , ~(I.(

+

q(l;)

+

Since h is a local diffeomorphism, it follows from (8.7) and (8.6) that

1(1;»).

(8.7)

140 B~(~) = q(t;)

+

1(t;) ,

(8.8) C~(~) = fi(Lt;

+

+

q(t;)

Let us show that the mappings G and R are locally

1(t;».

eft conjugate with

one another. By

Remark 6.20, it suffices to find an invariant section of the extension

[

Y ]1---+

[LY

t;

+ q(y + t;) + p(y + + q(t;) + l(~)

L(

Since I has contact of order

t;) - q(t;) - 1(t;) ].

with the zero map,

(Qo, k)

(8.9)

the extension (8.9) is

eft

conjugate to the extension

1---+ [Lx + q(x + t;) + p(x + t;)

x]

[

t;

L(

+

Consider the additional variable v, (S.lO) to the extension

q(t;)

+

- q(t;) ].

(8.10)

l(t;)

v = ~(t;).

Taking into account (8.S), pass from

Lx + [q(x + t;) - q(t;) + p(x + ~) - p(t;)] + p(t;)

1---+ [ Lt;+Bv

(S.l1)

Cv The expression in square brackets is a resonant polynomial vanishing for x = 0, and p is a or-divisible resonant polynomial. In virtue of Theorem 6.21, (S.ll) has a invariant section x = g(t;, v). invariant section x = g(t;, f(t;».

Therefore, Thus,

the extension

(8.10)

has a

the extension (S.9) also has a

invariant section and, consequently, the mappings G and R are locally Because F = H origin.

0

R

0

8.2. Lemma. Let ~

HI, we conclude that F and G are also

= At;

eft

eft smooth eft smooth eft smooth conjugate.

eft conjugate near the

be a hyperbolic linear vector field, q: E -+ E be a resonant polynomial with K(q) ~ 2, or e l!I(k) , and p: E -+ E be a or-divisible resonant

141

polynomial. Then the vector fields conjugate near the origin .

x = Ax + q(x)

and

Y = Ay +

q(y) +p(y) are ~

• Use the method of introducing additional monomial variables, likewise in Lemma 8.1 above.

8.3. Theorem. Let K and k be positive integers, ~ be a c< vector field on E, the origin being a saddle rest point, A = D~(O). If K it Qo(k) then the vector field ~ can be reduced, by means of a ~ coordinate change near the origin, to the resonant polynomial normal form n

Qo

y = Ay + E E p;l'et t-I

where p; - 0 implies at

=

(y

E

1.,.1-2

and cr _ S(k) .

• By Theorem 3.S, the vector field ~ is locally (3.3) which, in tum, is 8.2.

(8.12)

E),

~ equivalent to the vector field

~ conjugate to the vector field (8.12), by virtue of Lemma

8.4. Theorem. Let k be a positive integer, ~ be a ~ vector field on E having the origin as a nodal rest point, and A = D~(O). If k it Q1 then the vector field ~ can be reduced, by the aid of a ~ coordinate change near the origin, to the resonant polynomial normal form n

Y = Ay +

E E p;y'"et t-I

where p! - 0 implies at

QI

(y

E

E),

(8.13)

1.,.1-2

=

and cr _ S(k) .

• The validity of Theorem 8.4 follows from Theorems 3.6, 7.8, and Lemma 8.2. The next two theorems are completely analogous to Theorems 8.3 and 8.4. Therefore

142 their proofs will be omitted.

8.5. Theorem. Let k I!: 1, IE Difto(E), K I!: Qo(k) and L. Df(O) be a hyperbolic linear operator of saddle type. Then near the origin the diffeomorphism f is c! conjugate with the map n

fly)

00

= Ly + rL rL PerY'ere,

(y

E

E),

(8.14)

,=\ 1.,.1-1 where P;;I: 0 implies a,

= and

8.6. Theorem. Let IE Difto(E), k

I!:

IT

tl

21(k).

QIo and L

rator 01 nodal type. Then the diffeomorphism I is resonant polynomial normal form

Ii

c! reducible near the origin to the

(y

where P;;I: 0 implies

9,

=

and

IT

tl

E

(8.1S)

E),

S(k).

8.7. Theorem. Let k be a positive integer, ~ be a

origin, and K I!: Qo(k)

Df(O) be a hyperbolic linear ope-

c
, IT tl 21(k).

= ,

IT

tl

21(k), and

~

The proof of Theorem 8.7 is similar to that of Theorem 8.3 (see Remark 6.19 and the method of introducing additional monomial variables exposed in Lemma 8.1).

8.8. Theorem. Let k be a positive integer, ~ be a c< vector field on E, and 0 be an equilibrium all 01 whose eigenvalues have non-positive real parts. if K I!: Q\ + k, then the c! generalized resonant polynomial normal form (3.37) of ~ contains only such resonant terms Per(z)xer that IT II! S(k).

143

We leave to the reader as an exercise to formulate and prove similar theorems for local diffeomorphisms near a fixed point and for vector fields in the vicinity of a periodic orbit (without hyperbolicity assumptions).

§ 9. Linearization of Finitely Smooth Vector Fields

and Diffeomorphisms In § 9, 10, we shall prove several theorems on linearization as well as a few more general theorems concerning normal forms of fmitely smooth vector fields and diffeomorphisms in the neighbourhood of a fixed point. These theorems supplement the results presented in the previous sections, but essentially differ from them in the techiniques of proof. The last section, § 11, is devoted to a comparison of all the results presented in this chapter. In what follows, we shall freely use notation and definitions introduced in § 1 - 7.

9.1. Theorem. Let L: E -+ E be a hyperbolic linear operator and M, N, k be positive illlegers satisfYing M;>'I > k ;>." N J.l1 > k J.lm • Denote K = M + N + k, Q = M + N

+ max

{M, N}

>

K.

If f.

E -+ E is a

c" mapping which has contact of order (Q, K) with

the zero mapping at 0 e E, then the local diffeomorphism F(x)

= Lx + ft.x)

is

F c c"(E, E)

defined by

c! linearizable near O.

• Let U and V denote the contracting and expanding invariant linear subspaces of the hyperbolic operator L: E -+ E. Set LI = L I U, ~ = L I V. Then E = E $ V and L = LI $ ~. Denote Ji = pru • f, h = pry • f, where pru: U $ V -+ U and pry: U $ V -+ V are the canonical projections. By Theorem 1.4.7, we may assume with no loss of generality that Ji (0, v) • 0, h(U, 0) • O. For every numbers ro > 0 and £ > 0, one can find a > 0 and a mapping fa: E -+ E of class

c"

so that:

(1) fa(x) = j(x) if

IIxll

< a; (2) fa(x) = 0 if

IIXII it

ro;

(3) sup {IIDrfa(X)II: x E E} < £ (r = 0, I, ... , K). Henceforth, we shall assume that I is replaced by the truncation fa, where a corresponds to the numbers TO and £ to be specified later. According to Taylor's formula, write

144 j(U, v) = ./(0 ,VI~

+

D V\ no , V) U

+

•••

+

M M I (M _11) 1 D I ·1'{O , V)U •

(9.1)

J I

+

o Note that D~O, v) .•. , K - i;

i

= 0,

M.I

(1 - t) DM'j(tu v)uMdt (M - 1) I I' .

c""

E

and IID~D~O,

= O(IIVIIQ-q·,)

V)II

as

IIvll

-+ 0 (q = 0, 1,

1, ... , M - 1);

J(1(M- _ 1) I I

IIDq [ 1

M.I

t)

DM'j(tu V\UMdt]1I I

'I

= O(IIUII M )

o as

IIUII

-+ 0 (q

= 0,

1, ... , K - M);

[J (~M-_tL I I

IIDr

M.I

D~./(tu,

= O(IIXII Q·r )

v)uMdt]1I

(x

= (u,

v»

o as

IIXII

-+ 0 (r

= 0,

1, .... K - M).

Using once again Taylor's theorem and taking into account that Dkj(O, 0) (k = 0, 1, ... , K), we get

D'no v) V\'

=

J(1 I

=0

N I

- s)1)1• D'DNnO sv)~ds (N12.1\'

(.r

= 0, 1, ... , M - 1).

o Finally, apply Taylor's expansion in (9.1). As a result, we obtain M·I

v up

to order

N - 1 to the remainder term in

N·I

where

= ...! - s) • i! J(1(N-l)1 I

rp ( v)

,

o

N I

D'DNnO sv\ds 12.1\'

I

0 1, ... , M - 1) ; r=,

(.

145

( ) _

1

I/JJ U - j!

J(1(M- _ I)! I

t)

M I

-

o

= J(1 I

:t:(u v) ,

M

o

J

D ,Dlf(IU. O)~

J(1

(j = 0, 1, ... , N - 1);

I

-

S)H-I

(N - I)!

- t)M-I

(M - I ) ! ~"tDr:.tc.tu, sV)dtds • UM~;

0

(q = 0, 1, ... , M - i

+

k;

(q = 0, 1, ... , N - j

+

k; j

i:,.

=:

°,

1, ... , M _ 1) ,

°,

1, ... , N - 1) ,

(9.2)

(9.3)

(9.4) (IiUIi -+

0,

IIvll -+

0; p

+

q = 0, 1, ... , k).

Consider a mapping T: E -+ E of the form M-I

T(x)

=X + t(x) =x

+

+

h(x)

g(x) = x

+

H-I

Lht(v)ut~ + LgiU)uMyl, J-O

where ~(v)

=°and

g~(u)

=0

(the superscripts 1 and 2 refer to the projections onto

U and V, respectively). Then M-I

F

0

T(x)

= LT(x) +

LIPt(v + r(x»(tI + I ,(x»t(v +

:J. H 1 (x»

H-I

+

LI/Jiu + tl(x»(u + tl(x)t(V + r(x»J +

;t •

T(x).

J-O

Let us examine an arbitrary term of the form Applying Taylor's theorem to

IPs(v

+

r(x»

IPs(V

=IPs("

+ r(x»(u + t'(x»S"(v + r(x»H. + ~(v)~ + (r(x) - h~(v)~» at

146

the point v

+ h~(v)~,

we get

M-s-\

L c(q)Dqrps(v + h~(v)~)(u + i(x)t(v + r(x»N[r(x) - h~(v)~]q

=

qaO

M-s-\

M-\

L Dqrps(v + h~(V)VN) Lc(t;) n[h~(V)U(VNt(

=

q=O N-\

M-\

N-\

}=\

t=o

}=\

n[g~(U)UMv't} n[h~(V)U(VNt( n[g~(U)UMv't}

ueyr

(9.5) where the inner summation is taken over all multiindices t; = (€la, •.. , aM_to b\, ... , bN _\, c\' ... , CM_to do, ... , dN _\, e, f, q)

such that M-\

M-\

N-\

N-\

Lat + Lb} ~ q, Lat + Lb} + f (=\

t=o

)=\

}=\

M-\

= N

N-\

Lc( + Ld} + e =

+ q,

t=\

s;

}=o

c(t;) are certain non-negative constants.

Pick out the terms in (9.5) with b} = d} = 0 and associate them in groups according to the power of the variable u ranging from 0 to M - I inclusively. All other terms include in the remainder. Thus, we obtain

M-\ M-s-\

M-\

L L Dqrps(v + h~(V)VN) LC(II) n[h~(v)tt l-O

q=O

t-O

147 M-I

E N(4 t +ct)+4o!i+f

M-I

n[h~(v){Vv

t -I

+ iAu, v),

t-I

where the summation is taken over all

1)

= (lla, ,,-, aM_I, CI' ••• , CM_I, J, q)

such

that M-I

M-I

M-I

M-I

s

(note that the last equality implies form

~ l). The function

is(u, v) = 4I;(u, V)UM~, where 41; e C«E;

as lIuli -+ 0,

IIvll -+ 0

(0

N-I

=

~

+

p

q

~

M-I

is can be expressed in the

PM,N(U

Ell

V; E),

therefore

k). Similarly,

N-r-I

N-I

m=O p=O

J=O

L L vPI/I,.(u + g~(U)UM) Lc(~) n[gj(u){J N-I

N-I

n[gJ2(U)]d

E M(bJ+d j>+boM+f! Ju J-I

_m V

+ ;;;"',.(u, v" ,\

J=I

where the summation is over all multiindices such that N-I

J

;!:

p,

Ld J=I

J

LbJ + e =

~ r,

(b l ,

bN _1> dl> ... , dN _1> e, p)

.'"

N-I

M

IP; e C 0 one can find c5 > 0 so that if I is replaced by the truncation la, then

(9.18)

Recall that the operator !l«,/3 is contracting. Taking into consideration the special form

of the linear operator 8(v) (see (9.11) - (9.13», the equality (9.18) and the

C


!Pc ar.(Zc)·'c.

hence.

171

It remains to show that ¥>~,or.

t~,or.

E

C+ n +t - I • Observe that

n

Ii

sEc tEIlc

sEc tEIlc

From this and (9.27), we derive that ¥>~,or. can be expressed in terms of partial derivatives of order not greater than

+

Ipl - r - n

1 (recall that r = min {pi, ... , pn}).

Therefore the class of smoothness of the functions ¥>~,or. is not less than (Ipl

+ t)

- (Ipl -

9.S. Lemma. Let k be a positive integer,

DPt(O) t~,or.(z)

=r+n+t

r - n + 1) IC.

= (k,

... , k)

E

- 1.

l~j t

E

CCn+I(Z, Z) and

= 0 for p = 0, 1, ... , (k - l)n + 1. Then there exist functions = ,,~,oe(Zc) ·l' (c c I, c - 121, «E 14(IC.,. c», such that:

(2) t

L L

=

t~,or.: Z -+ Z,

t~,or.'

cCI oeE.4{rc,c) c_12I

({3 E l~,

(3)

• Let

{3

E

l~,

I{31

!is

k. If s

E

I and i

I (31

it

{3

s

!is

k).

~f3

t

_

t _/3S ~f3

,then v ~st - ~s

v t. Indeed,

172

< fill,

If ;

then JjJa!

= O.

6:

Next let us prove the equality rfl •

=

6:-lJ s

rfl. In fact,

•

Ie-I

rfl •

6: t(z) = rfl[t(z) - r a~t(z)] Ie-I

= JjJt(z)

t rfla~t(z)]

-

Ie-I

= rflt(z)

-

t

a~-fJsJjJt(z)

le-lJs-1

=

lid -

r

a:lrJIt(z)

= 6:-fJ'"

• rflt(z).

q=D

Assume that the hypotheses of Lemma 9.5 are fulfilled. According to Lemma 9.4 (with P

= Ie)

and assuming for definiteness that f3'

=0

for i

E

1\ c, we have

sEe [Ene

sEe [Ene

therefore

i.e., the condition (3) is fulfilled. Applying Lemma 9.4 for p that (1) and (2) also hold.

=

It

and

I

=

I, we see

173

9.6. Theorem. Let I E c!(Z, Z) and ifl(O) = 0 for p = 0, I, .,,' K - 1. Denote k = [(K - 1) I n]. Then the local diffeomorphism F: Z -+ Z defined by F(z) = Lz + I(z)

is c;k linearizable near the .fixed point z ~

= O.

Without loss of generality, we may assume that n

> 0 and a function

exist a number 15

i(z) = 0 if IIzll ~ 1 and

~

2. For every c

> 0 there

= t(z)

if IIzll:s 15,

i E c!(Z, Z) such that i(z)

Z and p = 0, I, .'" K - 1. Since the problem under consideration is local in nature, we may replace I by its truncation

i. Further, let >.

o :s >.(z) :s (i

1 (z

E

= 1 if IIzlI:s 15, >.(z) Z; P = 0, I, .,,' K).

Z) and

lIif>'(Z)II:S 2 (z

E

Let 0 denote the set of all multiindices w

= (wi,

=

{I, .,,' n}

I, .,,' n) and there exists a number j

= {s E 1: w EO).

= 0 if IIzll ~ I,

c!'(Z, R) be such that >.(z)

E

E

lIifi(z)lI:S c for all z

w· = k} (w Denote :E =

$

EO).

E

.,,' w") 1&

E

E

I,

W EO],

such that w':s k

1 with wJ

S =

= k.

Put c(w)

s!". Pw(Z; ZJ)

Assume we are given polynomials

[Pc.,cZ; ZJ): j

z~

V. I,

Ls-!J.W, JEI wEn

g(z, 3')

= z + >.(z)S

(1) .J

all

C

0

E I,

(2) sup{

SEC,

g(z, S) C;II

II,

Z, S;;; {~}

E

:E).

(c c: I,

C;II

II,

j E I) of class c;k+1 such that:

= L~,oc,.(Zc; SEC,

«E .4(It,

S)·zoc

VE

I), where the summation is taken over

c);

II~~ (Z· S)- II c c,OC,. c' : IIz.1I ;II 0, I ZcIK·~IIZ.1I

(3) ~,oc,. does not depend on ~

E

~,oc,.: Zc x :E -+ Poc(Z; ZJ)

9.7. Lemma. There exist mappings « E A(IC, c),

(z

S:

if

IIz,lI;II 0 (i

c(w) \

C

;II II

E

c), fJ

E z~,

IfJl:s k}


0 (i

Let j, p E l , w E 0, C C 1 and cc E "(Ie, c). If r the composite mapping formula (see subsection A.6),

E

c), f:!

E

c(w)\c

E

z~,

'It Ill,

If:! I

s

then wr

k}


k Ilm • If K ~ Q == M + N + max {M, N}, then there exists a local morphism which brings the diffeomorphism F to the polynomial normal form n

Fo(x)

t-I

where p; - 0 implies that at

=

M+N-I

L L p!xa'e"

= Lx +

C< diffeo-

(10.1)

Ia'i =2



k I-Im' Denote P

1, ... , M

+N-

1) and there are no resonant terms o/degree M

/or all j e {I, ... , n} F(x)

= Lx + j(x),

«E z~,

and all

admits a

1«1

+N

= M + N),

(i.e., 9J

;I:



0 and t

{xo = X. XI • .... Xle_I' xle = x; to. t l • .... tle _l } tt E T such that tt ill: t and p(f{xt • tt). Xt+I)

>

0 of


At;

J:5t

(2) the junction t is strictly decreasing along the trajectories of the system f out of the set A i! AI V .,. v Ale'

Bundles. Vector bundles

1.7. Def"mitions. Let us later on. A bundle is a triple ~ = p: X ... B is a continuous the total space of the

recall some notions from the theory of

bundles

needed

where X and B are topological spaces. and surjective (i.e .• p(X) = B) map. The space X is called bundle. B is called the base, and the preimage (X. P. B).

aE {x E X: p(x) = b} == Xb is called the fiber of the bundle lying over the point bE B. Let (X, P. B) and (X. P'. B) be bundles. A pair (t. 9') consisting of continuous mappings I: X ... X and 9': B ... B such that p' 0 I = 9' 0 P is said to be a morphism from the bundle (X. P. B) into the bundle (X. P'. B). Because tp is uniquely determined by t. we shall sometimes denote the bundle morphism (t. 9') simply as t. Let :BUll. denote the category of bundles. as defined above. Let ~ = (X. P. B) be a bundle and f. BI ... B be a continuous map. The bundle (XI> PI. B I ) where

p-I(b)

XI = {(b. x): b

E

B I, X

E

X, p(x) = ftb)},

PI(b, x) = b,

It:..

is called the pullback of the bundle (X, p, B) by f. BI ... B and denoted Let (X, p, B) and (X, p', B) be bundles with the same base B. A continuous map I: X ... X is called a B-morphism of the bundle (X, p, B) into (X. p', B) if p' 0 I = p. The B-morphism I: X ... X is called a B-isomorphism if the map I is invertible and I-I: X ... X

is continuous.

197

Let Suns denote the category of bundles with a fixed base B and B-morphisms as morphisms. A bundle (X, p', B) is said to be a subbundle of the bundle (X, p, B) if X is a subspace of X and p' = piX. A bundle (X, p, B) is called an n-dimensional vector bundle if each fiber Xb (b E B) is provided with the structure of a real n-dimensional vector space in such a way that the following property of local triviality holds: for each point p E B there exist a neighbourhood

V

= V(b)

and a

h: V x R" ... pol(Y)

V-isomorphism

such that the

restriction of h to x x IR", X E V, is a vector space isomorphism. Let (X, p, B) and (X, p', B) be vector bundles. A bundle morphism (L, 91) is said to be a vector bundle morphism if for each point b E B the restriction

L: pol (b) ... (P')ol(9I(b» of L: X ... X is a linear map. Let VS denote the just defined category of vector bundles. Given an arbitrary paracompact space B, let vS s denote the category of all vector bundles over B and let VS~ be the category of all vector bundles with the base B and fiber R".

=

Let (X, p, B) be a vector bundle and Xo be a subspace of X such that p(~ B and (Xo, plXo, B) is vector bundle, too. We say that (Xo, plXo, B) is a (vector) subbundle of the vector bundle (X, p, B) if the inclusion Xo c X is a vector bundle B-morphism. Given vector bundles (X, p, B) and (X,]i, B), one can define a new vector bundle L(X, X) Ii Hom(X, X). The fiber of Hom(X, X) over the point b Ii B is equal to the vector space L(Xb' Xi,) of all linear mappings 91: Xb ... Xi,. There is a one-to-one correspondence between sections of the bundle Hom(X, X) and vector bundle B-morphisms from X into X. The Whitney sum of bundles (X, p, B) and (X, p', B) is defined to be the bundle (X. X, p • p', B), where

X. X

= {(x,

(P • p') (x, x')

x'): x

E

X, X

E

= p(x) = p'(x')

X, p(x)

= p'(x)},

«x, x) EX. X).

Note that (p. p')"l(b) = Xb x Xi, (b Ii B). If (X, p, B) and (X, p', B) are vector bundles, the (X. X, p • p', B) is naturally endowed with the structure of a vector bundle. A Riemannian metric on a vector bundle (X, p, B) is a continuous map g: X • X ... IR

198

such that for each point b

E

B

the restriction g IXb

X

Xb

is an inner product on the

fiber Xb. The number IIxll = V'g(x, x) is called the norm of x E X. Usually, we shall write <x, y> instead of g(x, y). Whenever the base is paracompact, Reimannian metrics exist. Let (X, p, B) be a vector bundle and IRs E (B x IR, prl' B) be the trivial onedimensional vector bundle over B. The vector bundle L(X, IRs) is said to be dual to (X, p, B) and will be denoted (X·, p" B). The fiber X~ is equal to the space of all linear functionals I: Xb -+ IR. Fix some Riemannian metric on the vector bundle (X, p, B). To every element

y (x

EX E

put in correspondence the element

X,

p(x) = p(y».

(X, p, B)

ot, p.,

The map

y

1-+

l

y.

E/y

eX· defined by /y(x) = <x, y>

is a vector bundle

B-isomorphism from

into (X·, p., B). Thus, every Riemannian metric allows us to identify

with (X, p, B). We also note that the vector bundle dual to (X·, p., B) coincides with (X, p, B). Let Y be a vector subbundle of (X, p, B). By yl- we shall denote the set B)

yl- = {~ It is easy to verify

that

E

X·: ~(y) = 0 (y

E

Y)}.

yl- is a vector subbundle of the dual vector bundle

(X·, p., B).

Given a vector bundle (X, p, B) and a set Me B, XIM (or X[M]) will denote the

restriction of (X, p, B) to M (i.e, the vector bundle (p-I(M), plp-I(M), M).

Extensions of transformation groups 1.S. Definitions. Let (X, T, n) and (B, T, p) be transformation groups. Let p: X -+ B be a continuous map such that p(n(x, t» = p(P(x) , t) for all x E X and t E T. In this case, we shall say that p: X -+ B is a homomorphism and write p: (X, T, n) -+ (B, T, p). If, moreover, p(X) = B, then (X, T, n) is said to be an extension of (B, T, p) and the map p: (X, T, n) -+ (B, T, p) itself is also called an

extension. Let (X, p, B) be a vector bundle and let (X, T, n) and (B, T, p) be topological

199 transformation groups so that p: X -+ B is a homomorphism. The extension p: (X, T, x) -+ (B, T, p) is said to be linear if for every b E B and t E T the map XtlXb: Xb -+ Xp t (b) is linear.

Linear extensions of dynamical systems occur, for example, when linearizing smooth dynamical systems. Let B be a smooth manifold and (B, IR, p) be a smooth flow. Let (TB, 'tB' B) denote the tangent vector bundle. The flow (TB, IR, x) defined by X

t

(v)

=

t

(Tp )b(V)

(b

B, v

E

E T~)

is said to be tangent to (B, T, p). Clearly, 'tB: (TB' IR, x) -+ (B, IR, p) extension. It will be referred to as the tangent linear extension. If, in particular, B

=

IRn and the flow

(B, IR, p)

is a linear

is generated by the differential

x

equation = j(x) (x E IRn ), then the tangent linear extension (TB, IR, 'It) corresponds to the so-called equation 01 first variation:

x = j(x), t

= Dj(x)~

(x

E

IR n , ~

E

IRn ).

We would like to warn the reader that whenever B is an arbitrary smooth manifold and the fiow (B, IR, p) is determined by a vector field ~ E rr('tB) then the flow (TB, IR, x) corresponds not to the vector field n: TB -+ T(TB), as it might be conjectured, but to Jon;, where J: T(TB) -+ T(TB) is some involution defined in a standard way. A morphism of the linear extension p: (X, T, x) -+ (B, T, p) into the linear extension PI: (XI' T, XI) -+ (BI' T, PI) is defined to be a vector bundle morphism of (X, p, B) into (X.. PI' B) such that L(xt(x»

= x~(L(x»

(x

E

X, t

E

1),

Clearly, the class of all linear extensions forms a category. Therefore, it makes sense to speak about the Whitney sum of linear extensions, the pullback and so on. For (W, T, p) be a example, let p: (X, T, x) -+ (B, T, p) be a linear extension, transformation group and h: (W, T, :>.) -+ (B, T, p) be a homomorphism. Denote N = {(x, w): x E X, WE W, p(x) = hew)} and define a transformation group (N, T, p.) by p.t(x, w)

=

(xt(x) , ;>.t(w».

Then

q: (N, T, p.) -+ (W, T, :>.),

q(x, w)

=

w, is a linear

200 extension called the pullback of h: (W, T, ~) .. (B, T, p).

ot, p.,

Let p: (X, T, tr) .. (B, T, p) be a linear extension and (X, p, B).

bundle dual to

tr.(~, t)

It

can

E

by the homomorphism

p: (X, T, n) .. (B, T, p)

For ~

E

(X)b ,. (Xb)· and t

E

B)

be the vector

T. defme an element

X;t(b) by

be

easily

verified

that

(X,

T, tr.)

is

a

transformation

group

and

l: (X,

T, tr.) .. (B, T, p) is a linear extension. The latter is said to be dual to the linear extension p: (X, T, tr) .. (B, T, p).

Identify

the

dual

vector

bundle

fixed Riemannian metric on p: (X, T, tr.) .. (B, T, p) is given by

(X,

p., B)

(b

Consequently, for

with

(X, p, B). Then

E

B, t

(X, p, B)

the

E

by

using

a

dual linear extension

7).

x, Y E Xb , we have

The last equality can be expresed in an invariant (i.e., not depending on the choice of Riemannian metric) form: t

tr.(~)(tr

t

(x» =

-t

~(tr

t

(tr (x») = ~(x)

(b

E

B, x

E

Xb , ~

.,.

E Ab'

t

E

7) •

• 2. Exponential Separation and Exponential Splitting 2.1. Notation. Let B be a compact space, T = IR or T = z, (X, p, B) be a vector bundle and p: (X, T, n) .. (B, T, j..) be a linear extension. Set

201 (I e T, b e B).

In the theory of linear extensions of great use are the following Lyapunov exponents (see Lyapunov [I], Bylov, Vinograd, Grobman and Nemytskii [1]): g(n, b)

= lim

sup! In IIn!1I

t-++11111

101(n,

b)

iii

· sup -1 1n = - 1t-++11111 1m ==

inf {II: IIn!1I exp (- Ill) -+ 0 lin . tt

pCb)

II

1·1m sup -1 In

iii -

t-++11111

sup {II: IIn·~ II exp (Ill) -+ 0 P (b)

It is easy to verify that

+

(1-+

(I -+

+

III)};

11(11't,,>·1 II

III)}.

and w('If, .) are constant along the trajectories of (B, T, p) and do not depend on the w(n, b) :s g(n, b)

(b

E

B). The functions

g(n, .)

(X, p, B). Identify the dual vector bundle

choice of Riemannian metric on

(X·,

l,

B)

with (X, p, B) by using some Riemannian metric. Let p.: (X, T, n.) -+ (B, T, p) be the dual linear extension, then

=

IIn·tlX t

p (b)

(b e B, IE 7).

II

Therefore g(n., b)

= - w(n,

b),

w(n., b)

= - g(n,

b)

(b e B).

Although the exponential rates g(n, b) and w(n, b) are asymptotic in nature, they are intimately tied to uniform behaviour. This can be seen from the next two lemmas obtained by Fenichel [1-3].

2.2. Lemma. LeI lhe number II be such lhal lim IIn!1I exp (- Ill) = 0

(b e B)

t-++III

lhen lhere exisl numbers c > 0 and ; , ; < II, satisfying Ihe inequality (b

E

B, I

~

0) .

• For every point b e B, choose a number r(b) , l(b) > 0, so that

202 1In:~(b)1I


0 satisfying there exist numbers d >

°

t lin (XI)II

lint (x 2 ) II

--IIXI"

IX2 "

(xl

E

Xu

~

E

Suppose that X

X'+I' IIX1"11~1I

=

XI • ~

'#-

for

0, P(xl)

~

d exp (Ill)

= p~,

I

> 0, i = 1, ... , k - 1).

some invariant vector subbundles. If there exist

205

>

numbers d

0,

01:

>

0 such that

then the linear extension p is said to be hyperbolic. We shall also say in this situation that X Xl • Xl determines an exponential dichotomy. Because B is compact, this property does not depend on the norm on X. Without loss of generality, we may assume that d = 1 (this can be achieved by an appropriate choice of the Riemannian metric). Such a metric is said to be a Lyapunov metric (or adapted to the hyperbolic linear extension). A continuous function F: B x T -+ IR is called an (additive) cocycle of the dynamical system (B, T, p) if the following identity holds:

=

F(b, I When T

= IR,

+

'1:)

= F(b,

t)

+

F(Pt(b) , '1:)

(b

E

B; I, '1:

E

7).

a wide class of cocycles can be expressed in the form t

F(b, t)

= JG(pT(b»d'1:

(b

E

B, t

(2.1)

IR),

E

o where G: B -+ R is an arbitrary continuous function. In fact, if the cocycle F(b, I) is differentiable with respect to I at t = 0 and, moreover, the function G: B -+ R defined by the formula

G(b)

= !. (F(b, dt

t)) 1t

-

0

(b

E

B)

is continuous, then (2.1) holds. Indeed,

!. (F(b, dt

= lim

t»1 t-O

riO

F(b, '1:

+ s)

- F(b, '1:) S

= lim riO

F(pT(b) , s)

= G(pT(b».

S

2.7. Theorem. Let p: (X, T, n) .. (B, T, p) be a linear extension exponentially splitted into invariant vector subbundles XII"" Xk • Then there exist cocycles r,: B x T -+ IR, IR,: B x T .. IR (i = 1, ... , k) and a number fJ > 0 such thai a point x

206

belongs to the subbundle X,

W (b

for all large enough numbers t

= p(x»

> O. Moreover.

rt+l(b. t) - R,(b. t)

it

(b

fit

E

B. t

E

T. i

=

1..... k - 1) .

• See Bronstein [4. Theorem 6.33].

2.B. Theorem. Suppose that X

= XI

is an expo~ntial splitting of the li~ar extension p: (X. T. '11') -+ (B. T. pl. Given a number h. 0 < h < 1. o~ can choose numbers TO > 0 and EO > 0 so that every linear extension p: (X. T. A) -+ (B. T. p) satisfying the inequality III ••• ..

XI(V) ,

(F~u +

F!u • 80>(V»:

vE ~

}

(b

E

Q).

From (3.9) and (3.10), it easily follows that the operator

has an inverse for all b

E

B and, moreover, (b

U).

E

Therefore, 8t

consequently,

18t"

Recall that Gb

S

t t = (Fu. + F...

",t K

= Xe:!

(t

ill:

\ (Ft • 80)·

t uu + F.u

to>. Thus, 18t"" 0 as

for all b

E

W'(M),

vectorial set GI W'(M) u p(Q, R). Hence, GI U morphism is defined by 81 Un pt(Q)

Because 18t"" 0 as I ..

= 8t +

00,

(I

ill:

,0>,

-1

• 80> ,

I ..

+

00.

according to the definition of the

= graph (g) ,

where the vector bundle

81 Un W'(M)

= O.

we conclude that 8 is continuous at all points of

Un W'(M). Whenever bl , b" E U \ W'(M), then there exist uniquely defined numbers 'I' I" ill: 0 and points '1"" E Q such that b, = p(y" t,) (i = I, 2). Hence, if b" .. bl , then ,,," 'I' ',," 'I' Since G(Q) is a subbundle, we conclude that the map 81i"(P(Q, R+» determines a (continuous) subbundle.

Lower seml-contlnulty or the vectorial sets ~ and

X"

Let (X, p, B) be a vector bundle. A vectorial subset E c: X, p(E) = B, is said to be lower semi-continuous at the point b E B if there exist a neighbourhood U of b and a subbundle G of the vector bundle XI U such that Gb = Eb and Gz c: Ez for all Z E U. The vectorial subset E is called low" semi-continuous if it is lower semi-continuous at each point b E B.

217

If the linear extension p:

3.13. Theorem.

transversality condition then the vectorial sets

r

(X. T. n) -+ (B. T. p)

and

salisjies the

Jt' are lower semi-conrinuous .

• For a proof. see Bronstein [4. Theorem 8.48].

Coherent families of stable and unstable subbundles Let B be a compact metric space and p: (X. R. n) -+ (B. R. p) be a linear extension satisfying the transversality condition X A

=

{b

E

B:

= r + Jt'.

x: • r,,: = X

b}

According to Theorem 3.4. the set

== {b E B:

x:

n

r,,: =

{O}}

is closed and invariant. Moreover. the linear extension p is hyperbolic over A. Denote At

=

{b e A: dim

x: = i}

(i

=

O. 1•...• n).

(3.11)

Then A = "0 U AI U ••• U An. and {Ano An_I' ••.• "o} is a Morse collection. Observe that the numeration (3.11) is opposite to that used in Theorem 3.4. hence (3.12) To each point b

E

B. one can assign uniquely determined numbers i. j e {O. 1•...• n}

such that b e W(At) n ~(AJ). Since p is hyperbolic over A. there exists a Riemannian metric on (X. P. B) adapted to A. i.e.• such that for some number Ao. 0 < Ao < 1. the following inequalities hold:

(3.13)

ex E Jt'IA.

t ~ 0).

The proof of the following result is based on the technique developed by Robbin [1] and Robinson [2].

3.14. Theorem. Let p: (X. IR. n) -+ (B. IR. p) be a linear extension satisfying the At and transversality condition. ~n there exist neighbourhoods Ut of the sets

218

(continuous) vector subbundles ~ and Et i, j = 0, I, .... n the following assertions hold:

(d) ~

CD

Et

(e) IInt(x)1I

=

:s

(x

"At IIxll

~b c~,

such

that

for

Xlp(Ut , IR);

IIn-t(x)1I s "At IIxll

(0

of XI p(U" IR)

~b

C

(x ~

~I U" 0 :s t

E

E

Etl Ut , o:s t

(b

E

p(Uk

,

1),

s

s

1) for some number

"A,

0


0

is assumed to be so small that 0

It follows from (3.17) that D c: us denote r}q) satisfying

< i + £}.

.·(b)

= p(U.

[0. q]).

(3.19) Next we shall prove that D n p(W" IR+) = III. Suppose. to the contrary. that there are a point b E W, and a number t ill: O. for which pCb. t) E D. Then I - d = L(p(b. t» ill: L(b) ill: I - £. according to the properties of L and (3.18). but this contradicts the condition

W1q) =

£

< d.

Thus.

D n p(W" IR+)

= III.

consequently.

Given a point bED. let j

= j(b)

denote the smallest integet j.

I. such that b E W)q) and b til ~q) (j the point bED c U, so that

< is

I). Choose a neighbourhood Vb of

D n

III.

os j
q. hence

E

p(W,. IR+) \ ~q) (see (3.20». Then y

p-ll(y) E

W,. Since L(p°s(y»

E

E

E

Vb c U>q)

p(Wto s) for SOme

L(W,) c: L(U,) c [i - 114. i

+ 1/4]

221

and the function L is nondecreasing along the trajectories, we get J:

i - 1/4 > j

+ 114

L(Pt(y» ~ L(p's(y»

(-s s t sO), consequently, pt(y). UJ for t

E [-

q, 0], i.e.,

y • ujq)j a contradiction. Thus, (3.21) holds. Our next step is to construct the subbundle

Et.

Let b

E

B and j

= j(b)

be defmed

as above. Since j < I, the induction hypothesis (d) holds. From D C WS"(I\,) and (3.14), it follows that there exists a vector subbundle G of XI Vb satisfying the conditions (3.22) Let p: XI Vb -+ E'I Vb denote the orthogonal projection (relative to the Riemannian metric adapted to 1\). Denote

db)

(b)

Gy

= peG).

.,.u C l:.Jy

Then

(y

E

V(b».

(3.23)

Taking into consideration (3.22) and assuming the neighbourhood Vb to be small enough, we get (3.24) Let UD be a closed neighbourhood of D in L·1(1 - d) satisfying the condition UD C U, n (U [ Vb: bED]). Find a continuous partition of unity subordinate to the cover {Vb: bED}. Since UD is a compact set, we may suppose that all but a finite number of the functions fJb are identically O. Because UD C Ult (3.15) and (3.24) imply that (3.25) As it is seen from (3.25),

db)

can be represented as the graph of a vector bundle

morphism gb covering the identity map:

Define gD: E"I UD ... ~I UD ,

gD

=

L fJb8 bED

b,

222 and set

if =

graph (gD), then

if is a vector subbundle of

XI UD. It follows from (3.22)

and (3.14) that (3.26) Now let us prove the inclusion (3.27) Since the fiber ~ of the vector bundle (b Y

E E

D),




It is assumed that

s: z

-+ X is

a

Z). The inequality (5.4) should be

E

+ fs(k) II
+

J1Iv(t;)1I

2

dt;.

(5.11)

o Indeed, denote v(t;) == n:~(yo)

Now let !pet)

= < s(t) ,

= Yi;

and write

vet) >. As it is seen from (5.ll),

+

!p(t

'1:) - !pet)

;!:

0

(t

E

IR, or z: 0).

In other words, the function !p: IR -+ IR is non-decreasing. Hence, there exist a = lim !p(t) , b = lim !pet). t-++GO

t-+-GO

= b = O. Suppose, for definiteness, that {tn } -+ +GO and I/In(t) = !pet + tn). The sequence {I/I n } converges to the constant

Next we shall prove that a lim n:.(yo, tn)

= Zo.

Set

It follows from (5.11) that ,,' (t) = IIv(t)112

function I/I(t) i i a uniformly on segments. (t E ~). Therefore

uniformly

on segments.

Because

{I/In } -+ 1/1 II a,

we have

lim I/I~(t) E O.

Hence,

t

1In:.(.zo)1I == 0, whence IIZoIl = O. Since s is a bounded function, a == 1im !p(tn ) n~..

Similar arguments show that b

= O.

=

1im < s(tn ) , v(tn ) >

n~

..

= O.

Taking into account that "

is non-decreasing, we

236

get

rp(t)

condition

!II

1Iv(~)1I

O. Hence, (5.11) implies

"Yo"

= 0,

i.e.,

IIn!(yo)II

ii

0, contradicting the

O.

¢

p: (X, T, n) .. (B, T, p)

5.6. Defmitions. We might say that the linear extension is uniformly weakly regular if for each continuous section continuous section s: B .. X of (X, p, B) such that t ·t f[vosop

=s

(t

E

B .. X

IT:

there exists a

7),

(5.12)

i.e., the image s(B) is invariant under (X, T, nv)'

= IR,

Whenever T

the relation (5.12) can be written as t

f[t(s(B»

+

f[t(

J

(n- (

0

IT

0

p()(b) d~)

= s(/(b»

(b

E

B, I

E

IR).

o For T

= z,

we get (b

E

B).

Surprisingly enough, uniform weak regularity, as defined above, does not imply weak regularity. In fact, consider the simplest linear extension with the one-point base (in other words, let X = IRn and f[t(x) = exp (At)·x). This trivial extension is uniformly weakly regular iff det A ¢ O. It is weakly regular iff A has no pure imaginary eigenvalues. It can be shown that if the system (8, IR, p) does not contain rest points and periodic orbits, then these two definitions are, in fact, equivalent. In the general case, it is reasonable to strengthen the notion of uniform weak regularity in the following way. We shall say that p: (X, T, f[) .. (8, T, p) is uniformly weakly regular, if the following condition is fulfilled. Suppose that W is an arbitrary compact space, (W, T,~) is any dynamical system and h: (W, T, ~) .. (B, T, p) is a homomorphism. We demand the linear extension q: (N, T, v) .. (W, T, ;\), the pullback of p by the homomorphism h, to be uniformly weakly regular in the previous sense. Recall that

X, W E W, p(x) = h(w)} , vt(x, w) = (nt(x) , ~t(w», q(x, w) = w. Let us show that uniform weak regularity (in the adopted strong sense) implies weak regularity. Let s: T .. X be a bounded uniformly continuous map such that

N = {(x, w): x

p(s(t»

= pt(bo)

E

(t

E

7), where bo is some point of B.

Let W denote the closure of the

237

family {s-r!': T E 1} of shifts S-r!'(I) = S(I + T) (I E 1) with respect to the compactopen topology (i.e., the topology of uniform convergence on compact sets). Clearly, W is a compact space. Define the shift dynamical system (W, T,:\) by :\t(!p)(E;) = !pet + E;) (!p E W, t E T, E; E 1). Evidently, the map h: W ~ B, helP) = p(!p(O» (!p E W), is a homomorphism. Let q: (N, T, v) ~ (W, T,:\) denote the pullback of p: (X, T, x) ~ (B, T, p) by h: (W, T, :\) ~ (B, T, p). The map 5: W ~ X, 5(!p) = !p(O) , is continuous. Therefore the map 0': W ~ N, O'(rp) = (5(rp) , !p), is a continuous section of the vector bundle (N, q, W), and !T(ST) = (p\b o), ST) (T E 1). By hypothesis, there exists a continuous section

/: W ~ N which is invariant under (N, T, vcr). Since the

image of / under the projection pr\: N ~ X is bounded, we get (in the case that (5.4) holds with Xo = pr\ 0 O'(s). The notion of uniform regularity (in the strong sense) can be introduced in way. We leave this to the reader. Now let us present the definition of a Green-Samoilenko function of extension p: (X, T, x) ~ (B, T, p). Let c: X ~ X denote a continuous

T = IR)

a similar a linear function

satisfying the conditions p 0 C = p, xt. C = C • xt (t E 1) and CIXb E L(Xb' X b ). Note that we do not suppose, in general, the function b..." dim Ker (CIXb ) (b E B) to be locally constant. In other words, C: X ~ X is not necessarily a vector bundle morphism. Put C-r!' = C for

T

~

0 and CT = C - I for

map). Further, set CT = xT 0 CT c > 0 and v > 0 such that

== CT

0

1[T

(T

E

(b

T

< 0 (here I is the identity

1). Assume that there exist numbers

E

B,

T E

(5.13)

1).

Then the mapping (b

E

B,

T E

1)

is called the Green-Samoilenko function. It is easy to show that if a given linear extension has a Green-Samoilenko function, then the same is true for any pullback. In fact, given C: X ~ X, we define C: N ~ N by C(x, w) = (C(x) , w). Observe that the existence of a Green-Samoilenko function implies uniform weak regUlarity. Indeed, if T = IR and l}: B ~ X is a continuous section then the section 0: B ~ X, where

238

+ c5(b)

=

00

JG (11 T

(p-T (b») d-r

(b e B),

- 00

is also continuous and invariant under (X, IR, nl)' In the case section can be written as

c5(b)

=

L

if.

lJ •

p-n(b)

T

=

I, the required

(b e B).

Let p: (X, T, n) ~ (B, T, p) be a linear extension and rO(X) be the Banach space of all continuous sections ~: B ~ X of (X, p, B) endowed with the norm

II~II = sup {1I~(b)lI: b e B}. Let (rO(X) , T, nil) denote the transformation group defined by n~(~)

= nt

• ~ • p-t

~ e rO(X». As usual, denote

(t e T,

5.7. Theorem. Let B be a compact metric space, (X, p, B) be a finite dimensional vector bundle, and p: (X, T, n) ~ (B, T, p) be a linear extension. The following statements are pairwise equivalent: (1) p is weakly regular; (2) p is uniformly weakly regular; (3) p satisfies the transversality condition; (4) rO(X) = rOs(X) + rOu(X); (S) the dual linear extension

p": (X·, T,

n.) ~ (B, T, p)

has no non-trivial bounded motions; (6) p admits a Green-Samoilenko jimction; (7) there exists a quadratic junction ~: )( ~ IR such that ~(n!(x» - ~(x)

> 0

0, t > 0) . • We confine ourselves by considering the case T = IR. Theorem 3.S shows that the statements (3) and (S) are mutually equivalent. The implication (1) .. (S) holds by Lemma S.S. It was shown above that (2) implies (1). Clearly, (4) implies (3). According to Theorem 4.10, (3) _ (7). As it was already mentioned, (6) implies (2). So it remains to show that (3) .. (4) and (3) ~ (6). Thus, assume that the linear extension p: (X, IR, n) ~ (B, IR, p) satisfies Ute transversality condition. We shall use Theorem 3.14. Let the neighbourhoods Ut of the

(x

e

X·,

IIxll;l:

239

Et

of Xlp(U" IR) (i = 0, 1, ... , n) satisfy the The sets {p(U" IR)} cover the base B. Let {a,}

sets 1\, and the subbundles ~ and conditions (a) - (f) of this theorem.

be a partition of unity subordinate to this cover. Given 11 E rO(X) , denote By using the property (d), represent the sections ll, in the form ll, = ll,. + ll'cr(b)

E

~b

(b

E

p(U" IR); i = 0, 1, ... , n;

liS

"

L ll,. ,

=

llu

tJ'

= s, u).

=

ll, = a,ll. ll,u,

where

Set

" L ll,u .

It follows immediately from the definition that liS and llu are continuous. Let us prove

that llcr E rOcr(X) (tJ' = s, u). The proof will be carried out only for tJ' = u. It is based on the methods due to Robbin [1] and Robinson [2]. Define neighbourhoods W, of 1\, by formula (3.18). If e > is sufficiently small, then W, c U, (i = 0, 1, ... , n). Because W, is defmed by the aid of two Lyapunov functions, the trajectory of each point b E B can enter W, (and leave Wi) only once. Recall that the ex- and ",-limit sets of all points b E B under (B, IR, p) are contained in the set 1\ = Ao V 1\1 v ... V 1\". The same is true for the cascade generated by the

°

powers of the diffeomorphism

f =

Therefore {p(W, , Z): i

pl.

= 0,

1, ... , n}

is an

open cover of the compact metric space B. Let {aa denote a partition of unity subordinate to this cover. Pick numbers ~ and e, < e < 1, A < ~ < 1, satisfying

°

(5.14)

Choose a finite atlas .4

= {(U,

each chart (U, ex, IR") and each number j ( U" WJ

E

E

{O, 1, ... , n} the following relations hold:

;t f2I ) .. (

(1 - e) IIvll :s IIxll :s (1

Where the element v

for the vector bundle (X, p, B) such that for

ex, IR")}

+

e)

(5.15)

V c: UJ ),

(5.16)

IIVIl

IR" is determined from the equality ex(x)

According to the definition of {a~}, we have

= (P(x) ,

v)

(x

E

p.I(U).

240 supp (e)

!!!

{b

e~(b)

B:

E

¢

O} c p(Wj

•

U = O.

Z)

1..... n).

r such that

Because the set supp (9) is compact. there is a number r

UI(W

supp (9) C

U = O.

j )

(5.17)

1..... n).

For b E B and 'II e rO(X) define 11'11. bll = max 1I'II«(b)lI. where 0:('11. b) and the maximum is taken over all charts (U. 0:) e" with b e U. Put

= sup

11'1111 0

It is easy to verify that

'II«(b»

{ 11'11. bll: b e B}.

rO(X)

is a norm on

II • 11°

= (b.

equivalent to the initial norm

11.11 0 ,

Fix some number i E {O. 1..... n}. For 'II e rO(X) define 'Ilk

Then for all k

i!:

=

'It

ok

0

'IItu

0

pk

E

ok('IItu)

(k

'ltl/

E

z).

O. we have (5.18)

Indeed. if b e supp ('Ilk). then pk(b) e supp ('IItu). hence. pk(b) e U [pl(Wt ): -r sis r] according to (5.17). whence b for all k

i!:

E

U [plok(Wt ): - r sis r] c U [ps(Wt ): -

00

< s :s

r]

Q

E

O.

Let Vo = sup 1I'lt0IIXbli.

v = max { vo(l

+

e)(1 - ert, 1 }.

bE'S

Define r: B ~ IR by the following conditions: reb) b e p(Wt • Z) n (Wo

U

WI

U ... U

W,,);

5.8. Lemma. Let the atlas " 'II e rO(X).

r(b)

=v

=0

for b e Q; reb)

=,.,.

for

otherwise.

and the number

15

satisfy the above conditions

0

If

then (b

E

B. k

= O.

1•. 0.).

(5.19)

241

~ Let 'II e rO(X) and b e B. Find two charts (U, cr.) and (V, (3) from b e U, j(b) e V. Then

.4

so that

Recalling (S .16) and the definition of v, we get

Thus, the inequality (S.19) holds at least when r(f{b»

= v.

= Il, i.e., j(b) e p(W" Z) n WJ for some number j e to, ... , n}. '11M 1 = n;l~. Since j(b) e p(Wt' Z) n Wj C P(Ut, IR) n p(UJ , IR+),

Assume now r(f{b»

Let ~ = 'IIx' then then the property (c) yields

Et.f(b) C EJ.f(b)'

But

~(f{b» = 'IIx(f{b» = (n- x • 'IItu because 'IItu(z) e (e), we get

Etz

0

/)(f{b» e EJ.f(b)

for all z e P(Wh IR). Therefore, by (5.14), (S.16) and condition

(recall that j(b) e j(U) n V n WJ , hence V c UJ , according to (S.lS». Thus, (S.19) is true in the second case, too. Suppose now that r(f{b» = 0, i.e., j(b) fl Q. Then 'IIx(f{b» = 0 by (S.18), therefore, 'IIx+l(f)

= n-1'llk(f{b» = O.

5.9. Lemma. Denote q rcf(b»

~ Let

=v

Thus, (S.19) holds in this case, as well.

= 2n + 3r.

Given an arbitrary point b e B, the equality

is valid for no more than q values of k

b e B.

Since

{a;}

2:

O.

is a partition of unity, it follows from (5.17) that the

242 family of sets {f(WJ ): for each Wo u ...

number U

- r

t e z,

k

:5

r; j

:5

the

= 0,

collection

1, ... , n} covers the space B. Therefore

{/'''(b) , ... ,I+"(b)}

meets

the

set

WJ no 2(n + l)r

Wn . The orbit of b e B under the flow (B, IR, p) enters (and leaves)

more than once. Consequently, f(b) e Wo u ... values of k 2: O. Four cases can occur. Case 1: b

I!

Q. Then f(b)

I!

U

Wn for all but no more than

Q, i.e., rl/(b»

Case 2: f(b) e p(W" Z) for all k

2:

=0


I O(Tp, b)

(b e B),

262

=

=

=

nt lXI' n; 'Itt I~ (t ~ 0; I 0, 1, ... , k). where n~ Clearly, the notion of hyperbolicity coincides with that of O-hyperbolicity. In this

case, XI

= ](I,

~ =

r'. Since

B is compact, k-hyperbolicity does not depend on the

choice of Riemannian metrics on (X, p, B) and (TB,

'tB'

B).

6.2. Lemma. 1/ p: (X, T, n) -+ (B, T, p) is a k-hyperbolic linear extension of class ~, then

r

and K' are ~ vector subbundles.

~ Let us prove, for example, that

r' is a vector subbundle of class ~. Without loss the maps n: X -+ X and

of generality assume that T = Z. Replacing, if necessary, p: B -+ B by their iterates, we may assume that

< 1,

sup lI'ltlbll IITp·I(p(b»IIS' bEB

< 1

sup IIni!(b)1I IITp(b)lIS'

(s

= 0,

1, ... , k),

beB

where 'ltlb so that

-uS·1 = nI lAb' 'ltlp(b) = n.1 1Ap(b)' vU

p

= PI ,

b

EB.

S ber e ect i anum

C E

(0 , 1)

(6.2)

(6.3)

k, and kl be ~ smooth vector subbundles which approximate the subbundles XI. r and Xl. r', respectively, in the c! topology so that X = XI • Xl. Let P,: X -+ X,

Let (i

=

1, 2) be the corresponding projectors. Assume

(b) 11(1\ 111'1

0

0

'It I Xlbr '

II

S

'ltIXlbll :s c,

IIni!(b)1I

IIPl

0

+ c;

111'1

'ltIXlbll


0; (7) there exists a quadratic function

for all ~

E

[r(X)]·, II~II '" 0, t

(8) P is uniformly weakly

c!-

t: [r(X)f -+ IR such that t(n!.(~» - t(~)

>

0

> 0; regular (in the strong sense) .

• The assertions (I), (2), (3), (5) and (7) are pairwise equivalent by virtue of Theorem 5.7. Let us show that (3) .. (4) and (3) .. (6). The proof of these implications consists, essentially, in repeating the reasonings used in the proof of Theorem 5.7. So we restrict ourselves by indicating the new points. Theorem 3.4 being applied to the linear extension Pit states that the restriction Pit I A is hyperbolic. It then follows from Lemma 6.4 that P I A is k-hyperbolic. Suppose, for definiteness, that T = IR. Let the neighbourhoods Ut of

At

and the subbundles ~ and

Itt

of Xlp(U h IR) (i

be taken from Theorem 6.8. Further, let {B t } be a

c!-

the cover {p(U h IR)} of the manifold B. Given '"

E

I, ... , n)

partition of unity subordinate to rJc(X), define

done in the proof of Theorem 5.7. Because the subbundles ~ and

=

= 0,

",s

and

Itt

are

",u

as it was

CC

smooth,

Since ",a' E rita' (X) , we see that (4) holds. The proof of the implication (3) .. (6) is carried out exactly as in Theorem 5.7, but since the projectors

",a'

E

rlt(X) (II"

s, u).

n

t=o

smooth. Clearly, this gives (6). It is not hard to see that (6) implies (1) and (8) and (4) implies (3) (because for

259 each element ~ E [r(X)]b there is a section rr E rl«X) such that j!;(rr) = ~). Now we shall prove that (8) implies (1). For convenience, let us suppose T = I. The condition (8) means that for every section S . f ymg . I -I = rr, I.e., . satis '/(& 0 rr 0 p - '/( I 0 rr

=

A: rl«x> -+ rl«x> by A(rr)

rr • pi -

'/(1 •

rl«X) there exists a section rr E rl«x> + rr p I = s. Defime a 1·mear operator

E

rr.

0

Condition (8) guarantees that A is a

surjective operator. Moreover, this is true not only for the given linear extension p but also for any pullback of p. Assume (8) holds. According to the well-known Banach theorem, there exists a constant L

D such that for each section

rl«x> satisfying the conditions

fmd a section rr

E

IIrrlll< :s LIISIIl

= p"(bO>

(n

bo E B

11': I -+ r(X)

and

be

r(x>,

necessary, to a pullback) that the map ~: p(bo, I) -+ is well-defined and continuous.

~(b)

=

Db for

=

E

Extend

to a

Vi

function

Clearly,

there exists a section S

1, ... , m).

Let A

denote

E

pi -

=S

and

map such

that

'/(1

bounded

rr

0

~(p"(bO»

= rp(n)

~: B -+ r(X)

(n

E

I),

by letting

B \ p(bo, I), where Db denotes the origin of the fiber [r(X)]b. Let

B.

bu •.. , bm (i

b

a

0

rl«X) one can

With no loss of generality, we may assume (by passing, if

I).

E

A(cr);: rr

S E

the

collection of all finite subsets of B

ordered by set inclusion. There exists a net satisfying

lim 1!(S",)

=

rl«X) such that 1!(S)

E

=

{s",:

;PCb)

(b

IX

E

E

A}

of elements

;P(b()

partially E

rl«x>

rl«X) ,

i.e.,

s'"

B).

",EA

Let

{rr",:

IX E

A}

be

the

corresponding

net

of

elements

rr",

E

= s'" and IIII"",II:S L1is",lI. Since ;p: B -+ r(X) is a bounded section and we can apply Tikhonov's theorem on compactness of a Cartesian product of compact spaces. It follows that there exists a subnet {rr/3: ~ E AI} of {IT ... : IX E A} which 11"",

0

pi - '/(1 •

11"",

IIcr",1I :S LlIs",1I

converges pointwise to some bounded section I; of the bundle reX). Then Ib

I

~ • P ( ) - '/(l
0 and a vector subbundle W of (X, p, B), let us write C(W,

h)

= {a +

b: a

E

W,

b

J.

W,

pea)

= pCb),

IIbll:S

h lIall}.

Assume that p: (X, T, 1t) ... (B, T, p) is a linear extension, and XI' ... , X k are 1t-invariant vector subbundles such that X = XI 81 ... 81 Xk is an exponential splitting (see Defmition III.2.6). Denote XlJ = Xl .... 81 XJ' Xu = Xl (1:s i < j :s k). Clearly X = XI ..... Xl_I 81 XlJ 81 X J +I 81 ... 81 XIt (1 < i < j < k) is also an exponential splitting. 1.2. Theorem. Assume thai the linear extension p: (X, T, 1t) ... (B, T, p) is exponentially splitted into invariant vector subbundles XI' ... , Xk • Then for each number h, 0 < h < 1, one can choose to > 0 and £0 > 0 so that: (1) if p: (X, T, ~) ... (B, T, p) is an arbitrary extension leaving the zero section invariant and (to, £O>-close (in the Lipschitz sense) to p: (X, T, 1t) ... (B, T, p),

then there exist subbundles ~'~J (l:s i < j :s k) of (X, p, B) in the category !3W1.B

invariant under (X, T,~) and such that X

=~

81 ... 81

~;

(2) ~ c C(Xh h), ~J c C(X'J' h)

(1:s i :s j :s k). If the number. h > 0 is sufficiently small, then ~J is the maximal ~-invariant set contained in C(XtJ , h), i.e.,

~J

=

n ~t[XlJ' h)); tET

(3) there exists a !3W1.B-isomorphism F of (X, p, B) into itself, FIZ(B)

= id,

which

263

carries X'J onto rtJ (1:1 I :I j :I k). The morphisms F - I and F· I - I, where I denotes the identity mapping, satisfy the Lipschitz condition and, moreover, Lip (F - I) and Lip (F ·1 - I) are O(h) as h -+ O. Besides thai, F .llrt coincides with the natural projection P,: X = XI Ell ••• 48 XJt -+ X, (i = 1, ... , k). ~

For a proof, see Bronstein [4, Theorem 9.28].

1.3. Theorem. Assume the hypotheses of the preceding theorem to be fulfilled. Let f3 be a positive number, and r, and R, (I = 1, ... , k) be the cocycles constructed in Theorem III.2.7. Then for every c, 0 < c < f3 12, there exist numbers to > 0 and 150 > 0 such that for each extension p: (X, T, ~) -+ (B, T, p) leaving the zero section invariant which is (to, c5O>-close to p: (X, T, 7f) -+ (B, T, p), the following statements hold: (1) the point IIXIl

r;

x belongs 10

exp [r,(b, I) - c/]

iff

lI~t(X)1I

:I

:I

IIxll

exp [R,(b, I)

+ ct]

(b

= p(x»

for all sl4iJiciently large numbers t > 0; (2) whenever 1:1 m


0, and IIXII

exp [Rm(b, t)

+ ct]

:I

lI~t(x)1I

:I

IIxll

exp [rm(b, t) - ct]

(b

= p(x»

for all negative numbers t with a large enough modulus. ~

See Bronstein [4, Theorem 9.29].

1.4. Notation. Let B a compact space, (E, p, B) be a vector bundle, and if, p) be a vector bundle automorphism of (E, p, B). Further, let X and Y be f-invariant vector

264 subbundles such that X

Y = E. We shall assume that (g, p) is an automorphism of

$

(E, p, B) in the category ~un. close, in a certain sense, to if, pl. The zero section Z(B) is not assumed to be g-invariant. We seek conditions ensuring the existence of

g-invariant subbundles Xg and Yg close to X and Y, respectively. The proof of the next theorem is based on the graph transform method. This means that the subset Xg , for example, will be represented as the graph of some idB-morphism

tl'g: X ... Y, i.e., Xg = {(x, crg(x»: x E Xl. Let PI: E ... X and P2: E ... Y denote the projectors corresponding to the direct sum decomposition E = X $ Y. Let cr: X ... Y be some idB-morphism in the category f3un.. The mapping g: E ... E carries graph(cr) onto the set g(graph(cr» which is not, in general, the graph of any morphism from X into Y. Since g(graph(cr» = {(PI

0

g(x, cr(x», P2 • g(x, cr(x))): x e

Xl,

g(graph(er» will be the graph of some morphism iff the mapping her == PI X ... X is invertible. In such a case, g(graph(er» = graph(g,(cr», where

0

g o(id

+ er):

The set graph(er) is invariant under g iff g,l(er) = er. Thus, in order to construct the required invariant subbundle Xg , we must find an idB-morphism er: X ... Y invariant under g,. Fix some Riemannian metric on (E, P, B) so that X.L Y, then IIPIII = IIpzll = 1. Denote (b e B).

(Ll)

1.5. Theorem. Assume

(1.2)

There exists a number

fJ.

> 0 such sup Lip

that if

«g - f) IEb )


0, is satisfied. Firstly, prove that the operator Tpl • Tg • (id + 1;) is invertible for II~II s 1. Clearly, (Tpl • Tg • (id

+

~), PI • g •

(id

+

tI'»

281

is a vector bundle morphism from

(TX,

T x,

into itself.

X)

Theorem I.S, we have established that p~. g

0

In the course of proof of

+ 0' 0 to be

Jt.Y1 = jP (W E

the inequality (4.11)

TY).

Now, let Z = Z(Y) be the zero section of (TY, 'ty, y), and HE = {(u, v): U E HX, v e Z, Tq(u) = Tq(v)}. Note that HE can be identified with a vector subbundle of (TE, Tq, TB) iii (TX, Tq, TB) e (TY, Tq, TB). Since 'tE. Tq = q • 'tE' we get that (HE,

'tE'

(TE,

'tE'

tHE:

(TE,

is. ~-. vector subbundle of (TE, E) = (HE, 'tE' E) e (VE, 'tE, E) (see E)

'tE'

E) -+ (E e TB e E,

pr.,

'tE'

E).

It is easy to verify that

subsection

A.lS).

Therefore

E) is a ~-. vector bundle isomorphism. It can

also be regarded as a ~-. ids-isomorphism tHE: (TE, q • 'tE' B) -+ E e TB e E = X .. Y. TB eX .. Y in the category :Bun.. Let P denote the mapping from TE into ,Y defined by

283

As it was noted above, (id + ~): TX ~ TE is a vector bundle Consequently, it can also be considered as an idB-morphism (id + X III Y til TB til X Ell Y in the category !lun.. Moreover,

Thus,

t;: TX ~

of

instead

II~II == sup {II~I T,xXII: X e X} :s

i~B-morphism. ~):

X

til

TB

1\1

X~

TY we get the mapping ~ == P ~: TX ~ Y with 1. So, we have defined ~: X Ell TB 1\1 X ~ Y. Accordingly, 0

(4.4) becomes -

-

F... (t;) = P

where 'P...,(: X

Ell

TB

1\1

X

~

X

1\1

0

Tg

0

Y Ell TB

'P... ,(

0

X

Gl

Ell

[7PI

0

Tg

0

'P...

,d-I '

Y is given by

Next we shall verify the estimate

(4.12)

Identifying TE with X

&I

liP

Y &I TB 0

&I

X

Ell

Y via iHE , we get

7]{XI' YI' W,

X2'

whenever X and Yare sufficiently close to

yJ - j(yJII
0 fails to be smooth at the point Q, but is still homeomorphic to the circle.

p

Figure 1.3.

292 The extent of pathology is, in fact, much greater than the above examples reveal. As it was shown by Jamik and Kurzweil [1], an exponentially stable two-dimensional invariant manifold, when perturbed, can tum into a set which is not even a topological manifold. Moreover, Kaplan, Mallet-Parret and Yorke [1] have constructed an asymptotically stable torus which bifurcates into a set of non-integer Hausdorff dimension, a strange attractor. Thus, we face the problem of finding conditions for a smooth invariant manifold to persist under perturbations of the vector field. To be more exact, we seek conditions ensuring that the given isolated CC smooth invariant manifold, being perturbed, gives rise to a unique invariant manifold of the same smoothness class. This property will be refered to as

CC

persistence (precise definitions will be given later).

1.3. Notation and def'mitions. Let M be a smooth manifold, T

=R

or T

= Z,

r

~

1,

and (M, T, f) be a dynamical system of class C. Let 1 sis r and A be a compact c! smooth sub manifold of M invariant under (M, T, f). Let (TM, "CM' M) denote the tangent vector bundle and TII.M TM[A] = {v E TM: "CM(V) E A}. Because the tangent bundle (Til., "CII.' A) is embedded in TM[A], one can form the quotient bundle TM[A] / Til. which is

=

called the normal bundle of the submanifold A c M. Note that Til. is a c! smooth vector subbundle of the tangent bundle TM since the transition functions for the vector bundle Th can be obtained from that of TM by restriction to the c! submanifold heM. Hence, the normal bundle Nil. iii! TM[A]/TA can also be provided with the naturally defined

structure of a c! smooth vector bundle. The C dynamical system (M, T, f) induces a C· I smooth linear extension, the tangent linear extension "CM: (TM, T, T/) -+ (M, T, f), where T/(v) = T/(x)(v) (x E M, v E TxM, t E 7). Since the submanifold A is /-invariant, the vector subbundles Til. and TM[A] are invariant under (TM, T, T/). Therefore, (TM, T, T/) induces a dynamical system on Nh, denoted by (Nh, T, Nf), which is a linear extension of (A, T, /lh). The smoothness class of this extension is equal to min {I, r - I}. For each point x E M, let expx: TxM -+ M denote, as usual, the exponential mapping that corresponds to a certain Let I

=1

Riemannian metric on (TM,

"CM'

M).

C vector subbundle of TM[A] complementary to Th. Let rl(N) space of all C sections of the vector bundle (N, "C M' A) provided with

and N be a

denote the Banach

e

293 the

c: norm

Note that each

II' III'

the form A = {exPb(~(b»: b

E

c: smooth submanifold Anear A can be represented in

A} for a certain ~

E

rl(H).

c: submanifold invariant under the cascade generated by some diffeomorphism Difr(M). We shall say that A is c: persistent provided there exist a neighbourhood

Let A be /

E

U of A in M and a neighbourhood 'l.L(f) in Difr(M) such that for every mapping g there is a unique section 11

Ag

= "6 E rl(H) "

E

'l.L(f)

satisfying

n g"(ll) = {exPb(1I(b»: b

E

A},

"EZ

and, moreover, the mapping from 'l1(f) into rl(H) defined by g t-+ 1Ig is continuous. Note that Ag is the maximal g-invariant subset contained in U. It is assumed that A.f' = A, hence, A is supposed to be isolated. Now, let 12:2 and N be a

c: smooth vector subbundle of TM[A] complementary to TA. c!

Let rl(H) denote the Banach space of all endowed with the The

c!

norm

c! sub manifold

integer k, k

=

A

sections of the vector bundle (N,

TM'

A)

II' Ill'

is said to be

c! persistent if it is C persistent and for each

I, .,,' I, the mapping g

t-+ 1Ig

carries the set 'l.L(f) n Difrc(M) into

rc(H) and is continuous in the ~ topology.

The notion of formulations.

c!

persistence for flows can be defined similarly, but we omit precise

f 2. Normal Hyperbolicity and Persistence In this section, sufficient conditions for persistence of a submanifold are given.

smooth invariant

2.1. DermitioDS and notation. A smooth invariant submanifold A is said to be normally k-hyperbolic if the normal linear extension is k-hyperbolic (see Definition

m.6.1), i.e., there exist 1Vf-invariant vector subbundles JII and Jt4 of NA and positive numbers c and ~ such that JII. ~ = NA and

294

(2.1)

(t ~ 0, b e A; m = 0, 1, ... , k).

Let p: TM[A] .... NA be the canonical projection. Denote E: = p'I(~), Clearly, TA = f:S n

r r

El

TA

El

= p'l(i't').

By Lemma A.28 and (2.1), there exist Tf-invariant vector

r. TA,

and Jtl of 1M[A] such that ~ =

subbundles 1M[A] =

E'.

E'

E'

=

Jtl •

TA, consequently,

Jtl. Moreover, (2.1) can be rewritten as

(2.2)

~

(t

0, b e A; m

=

0, 1, ... , k).

Evidently, the just formulated condition is equivalent to k-hyperbolicity of A. Because A is compact, the conditions (2.1) and (2.2) do not depend on the choice of Riemannian metrics. Observe also that the submanifold A is normally k-hyperbolic under the flow (M, IR, 1) iff it has this property with regard to (M, I, 1). Therefore we shall confine ourselves by considering only the case of cascades. Denote (see 1lI.2.1)

rl(b)

= rl(7Jl TA,

b),

w(b)

= w(Tjl TA,

b).

The inequalities (2.2) become rl5(b)

Let

E




m rl(b)

(b e A;

m = 0, I, ... , k).

295

< m w(b) - 4e,

cS(b)

>

wU(b)

m c(b)

+

4e

(b

E

= 0,

A; m

1, ... , k)

(such numbers exist by Lemma III.2.3). Further, there exists a number satisfying

c; I exp[w(b) - e]t

:S

II

if ITAli

Fix some Riemannian metric on (TM,

:S

c£ exp[c(b)

T M'

mutually orthogonal and approximate it by a

+

e]t

(b

t

E A,

M) such that the subbundles

e

I:

XS,

c£

>

0

0) TA and

Jtl are

Let d denote the

Riemannian metric.

metric on M that corresponds to the latter Riemannian metric. Let U be a fixed small enough neighbourhood of A in M. Denote

W'"(f) WU(f) Given x

E

U, let

tends to 0 as n -+

=

{x

E

= {x

U: .f(x) -+ A

E U: .f(x) -+

A

(n -+

+ GO)},

(n -+ - GO)}.

~(f) denote the set of all such points Y

+

E

GO faster than d(/,(x), .f(z», whichever Z

U E

that d(/,(x), .f(y» A, Z

'It

x, be chosen.

As it follows from the proof of the theorem below, ~(f)

= (y

E

U: d(/,(x), .f(y» exp[-cs(x) - e]n -+ 0

(n -+

+ GO)},

Similarly, we define the set w,:,(f):

w,:,(f)

= (y E

U: dif -"(x),

f -n(y»

exp[wu(x) - e]n -+ 0

(n -+

+ GO)}.

Note that by virtue of uniform integral continuity of the dynamical system on the compact subset U c M, we have

for each positive integer

v. Replacing, if necessary, the mapping

f by some iterate,r,

296 we may assume that

SUD n7J1X:n n7]'·tIT.f(b),\lIm:s 1/3

(m

bEh

= 0,

(2.3)

1, ... , k),

(2.4) Because we are investigating the behaviour of the dynamical system only near A, we shall assume, without loss of generality, that the manifold M is compact.

2.2. Theorem. Let M be a smooth manifold, T

= IR

or T

= z,

k

~

1, and (M, T,fJ be

a dynamical system of class ~. Let A be a C smooth submanifold of M invariant under (M, T, fJ and satiqying the condition of normal k-hyperbolicity. Then: (a) The sets A, W(f) and W'(f) are (b)

W(f) =

U W:(f), bEA

(c) The manifold A is

d'-

submanifolds;

W'(f) =

U ~(f), bEA

~ persistent, and if (M, T, g) tends to (M, T, go> in the ~

topology, then W(g) -+ W(go,) and W'(g) -+ W'(go,) in the same sense; (d) Near A, (M, T, fJ is topologically conjugate to (~ •

It', T, N/).

• At first sight, the most natural way of proving this theorem should be as follows. First, by using a tubular neighbourhood, the given dynamical system should be carried from a neighbourhood of A into a neighbourhood of the zero section of the normal bundle N- NA. Then one should apply Theorems IV.4.2, IV.2.2 and IV.3.S, and the proof would be complete. Unfortunately, the dynamical system induced on N is not, in general, an extension of the system defined on A, so these theorems are not directly applicable. Therefore, we are forced to proceed in a roundabout way, namely, instead of dealing with the neighbourhood of A, we must consider the neighbourhood of the diagonal tJ.(A) • {(x, x): x E A} in A x M. The proof will be carried out for cascades. Given r > 0, define U(r) = {(x, y): X" A, Y EM, d(x, y) < r}, TAM(r) = {v E TAM: nvn < r}. Clearly, U(r) is a neighbourhood of tJ.(A) in A x M. There exists a number Co > 0 such that the mapping

297

Exp: TAM(cO> -+ U(cO> defined by Exp(v) = (X, expxv) (x E A, v E TxM) is a C diffeomorphism. In general, the mapping Exp is as smooth as the submanifold A. Denote fa = f x /I A x M. Choose a neighbourhood V of lI(A) in A x M so that if x .f)(V) c U (recall the equality ft.A) = A). There exists a number C1' 0 < C1 < Co, such that the formula 70 70: TAM(c 1) -+ TAM(cO>.

= EXp-l

• if x.f)

0

Exp is meaningful and defines a mapping

Let g: M -+ M be a diffeomorphism close enough to f (in the

C topology) so that the

mapping go: TAM(cl) -+ TAM(co> , go = EXp-l • if x g) • Exp, is well-defined. Note that = w, where W is determined from the relation exp.f(lC)w = g(expxv) (here x = '1:M(V) and, consequently, J(x) = '1:M(V». Thus, '1:M 0 g = f· '1: M • go(v)

Let c be a small enough positive number. Choose a C smooth vector bundle isomorphism t: TAM -+ TAM so that l17J{x)(v) - t(V)1I < (c/lO)IIVIl (x E A, v E TxM). Assume IIgo(Ox)1I < c and IIDvgo(Ox)(v) - 7J{V)1I :5 (c/lO)IIVIl (x E A, v E TxM) (this can be achived by choosing g to be sufficiently C near to /). Define a mapping ~: TAM(cl) -+ TAM by ~(v) = go(v) - t(v). We have IIDv~(Ox)1I < cIS (x E A), hence there exists a number cr,

o < cr
0 such that

299

= {v E

X3

X2

TAM: IIF -n(V)1I :s de IIvll exp[- wl.l.(b)

= {v E

+

2c]n, b

= 'tMcr(V» ,

TAM: d;1 IIvll exp[w(b) - 2c]n :s IIF'(V)II :s de IIvll exp[n(b)

n

+

it

O}

2c]n,

According to the choice of c > 0, we have nS(b)

+

2c




0 is

belongs to

satisfying the

condition

0 is small enough, we get g 0 se = Nf 0 se' consequently,

p •

g

0

se

= P • NJ.

only one section

T/

se

= c-I

=/lA. . se 11 -

E

Thus, for each section v

E

r'"(N) there exists one and

r'"(N) satisfying the equality

NJ

0

11 • (j1 Arl

= v.

(3.2)

In other words, the operator id - /,: rm(N) -+ r'"(N) is invertible. Given

tI' E

rm(N), one can define the corresponding section j'n(v) of the vector

bundle p"(N) (see Definition A.14) by the formula x ..... J':(v) (x

E

A). Let rb(P"(N»

denote the Banach space of all bounded sections of p"(N) equipped with the norm 1I~lIb

= Sup{II~(X)II:

x

E

A}.

For each element

(E

rb(p"(N»

there

exists a net

{vex: « E A} of elements Vex E rm(N) such that J';(vCX> -+ (x) for all x E A.

Let

{1Jex: « E A} be the net of elements lIex E rm(N) satisfying the relation (id - /,)(1ICX> = Vex' By Tikhonov's theorem, there exists a subnet {1J/3: {3 E B} of the net {lIex: lie E A}

such that

{In.

11/3: {3 E

B} converges pointwise to a certain element (E rb(P"(N»

302

and, moreover, lim J";«(1"~)

= ~(x)

(x

= (id - fi)(-rr~).

(1""

A), where

E

Define

Then

= lim

[;11'1

1)~(x) -

0

~

Thus,

id - 1,11'1)

maps rb(P"(N)

r

onto itself.

0

Nf

0

1)~

0

(flArl(X)]

In other words, the linear extension

(P"(N) , p"(Nf» is weakly regular. By Theorem m.S.7, the dual linear extension has no non-trivial bounded motions. To finish the proof, it suffices (by Theorem m.3.6) to show that P"(Nf) does not have non-trivial bounded motions, too. With this end in view, let us prove the following proposition.

3.2. Lemma. Assume A to be I:

>

c! persistent.

Thenfor each a > 0 one canfind a number

0 and a neighbourhood ti of the diffeomorphism fin Difft(M) such that whenever

1 :s m :s k, ~

E

Nm,m-I(TM[A] I TA), II~II
a o•

Let ti = ti(ao> be the neighbourhood mentioned in the definition of c! persistence. Pick out a diffeomorphism g E ti having the above property. Let XI E A be the source of the jet ~,and

X2

= i(xl)'

It suffices to consider the case when

g'(xl) '" XI

for

i '" O. Choose a small enough neighbourhood WI of XI in M and denote W2 = g'(WI ). There exist surjective c'< diffeomorphisms ",: W, -+ {y E IRP x IRq: lIyll < 1} satisfying the conditions ",(x,) = 0 and flt(Wt n A) = {y E IRP x {O}: lIyll < 1}. Here p is the dimension of A and p + q = dim M. Moreover, we may assume that T"t ITx,M are isometric

303

= 1,

operators (i Let i\

E

2).

(0, 1), ~

= {y E IR P

parameter i\ is sufficiently small, then

i\},

a ~ function 1/1: IR -+ IR such that 1/1(0) = 1, I/I(t) = 0 for for all t

E

IR. By

It I

it

II/I(t) I =s 1

1, and

t.;: IRP -+ IRq denote the homogeneous polynomial of degree m that

~ E H~;m-I

corresponds to

~ = i(Wi'). If the i = 1, ... , I. Choose

Wi' = "il(B~), Wi' n l(Wi') = ~ for all
0, cr. > 0 and (:J > 0, the following inequalities hold: IIN/INxll :s

C

exp(- cr.t)

(x

E

A, t

I:

(4.1)

0),

(4.2) Condition (4.1) means that A is exponentially stable in the positive direction with respect to (M, T, /). Condition (4.2) signifies that the exponential rate of contraction in the normal direction is greater than that along A. By Lemma A.28, it follows from

]f c 7M[A]

(4.2) that there exists an uniquely determined vector subbundle

if-invariant

for all

1

E

which is

T and complementary to TA. According to Theorem 2.2, there

exists a neighbourhood V of A in M which is invariantly fibered by w:. == W(x) (x E 1\) so that YEw:. iff difCy),/(x» :s c£ exp[(- cr.

+

e)l]

(I

I:

0)

305

for each

£

> O. It will be no loss of generality in assuming that I(V)

4.2. Dermition.

Y

Define a mapping

W'"'ex), then set . / = cf IA) • I

E

l(y)

= x.

as follows:

I: V -t A

Since 1(W'"'ex»

C

wcfex» (x

E

A, 1

C

V (I

2:

if

x

A

2:

E

0). and

0), we have

I (I it 0). The mapping I: V -t A is called the asymptotic phase. Note that II A = id. Reversing the direction of time, we get the notion of asymptotic phase for a submanifold exponentially stable in the negative direction.

4.3. Smoothness of the asymptotic phase. Let U be a neighbourhood of the zero section ZeAl of the normal bundle (N, p, A) and h: U -t M be a (partial) tubular neighbourhood

C. Let

of class

h(ll)

I: V -t A be the asymptotic phase. With no loss of generality, assume

Let k::5

= V.

T.

We shall say that

tubular neighbourhood h: U such that

I.

h

= pi U.

-t

V of class

I

C".r

is of class

C".r

whenever there exists a

(see Bourbaki [1, subsection 15.2.4])

Recall that h is of class

C".r

iff for each point lEU there

exist a neighbourhood WI of x == pel) in A, a neighbourhood W2 of w == hell in M, a vector bundle chart (WI,!p, Rq ) for N, a chart (WI' !/I, RP) on A, and a chart (W2' A, IRp + q ) on M such that the mapping H: !/I(WI )

=

)( Rq -t IRp

+q

defmed by H(a, b)

possesses partial derivatives D~D~H for all pairs (u, v) of non-negative integers such that U::5 k, u + V ::5 T. A • h • !p-I (!/I-I (a) , b)

Denote

= cf IA)

l = h-I

• I •

./ •

h: U -t U (I

it

0). Then p •

l =p

h = cf IA) • P (I > 0) and Wex) = I-lex)

Thus, if the asymptotic phase

1

is of class

C".r

2:

./ •

h

and h denotes the corresponding tubular

neighbourhood, then for the semidynamical system {l: I {/: I

= 1 ./ • h = hlp-lex) n ll) = h(Nx n ll). • h-I

2:

O}

O} via h, the sets Wex) coincide with Nx n U ex

E

which is conjugate with A). In orther words, the

mapping h straightens the subsets W'"'ex). Let

II

c TM[A] be some vector subbundle of class

sufficiently close to subbundle

II.

If.

The normal bundle N

E

C

complementary to TA and

TM[A] I TA is

Therefore we shall not distinguish between N and

Assume that the asymptotic phase idA-morphism !/I: U

-t

TA of class

I

is of class

c!'.r

such that

c!'.r.

C isomorphic to the

N.

Let us show that there exists an

306

= {expx(v,

W(x)

lII(v»:

v

E

Nx "V}

(x

E

(4.3)

A),

where expx: TxM .... M denotes the exponential mapping corresponding to the Riemannian structure on M. By Exp we shall denote the mapping defined by Exp(v) = ('t'M(V), exp v) for all v in a small enough neighbourhood Vo of the zero section of the tangent bundle

c!' diffeomorphism from Vo onto a neighbourhood of the diathat there exists a C vector bundle isomorphism TM[A] N \I TA.

TM. It is known that Exp is a gonal t. c: M x M. Recall

= pr2,

Define lII(v)

R$

EXp-1('t'M(V), h(v» (v

0

C!"'.

that III: U .... TA is of class

E

U). It follows from the above considerations

Since (v, III(V»

= EXp-I('t'M(V), h(v» , we have W(x) = h(Nx " U) = {expx(v,

(TM(V), h(v» = ('t'M(V, III(V» , exp(v, III(V»), therefore III(V»: v E Nx "V}. Conversely, if there exists an idx-morphism III: V .... TA of class

c!.r

such that (4.3) holds, then the mapping h: V .... V defined by h(v)

(v

E

Nx "U) is a morphism of class

I

=P

0

h-I •

c!.r

and

W(x)

= expx(v,

= h(Nx "U),

This means that the asymptotic phase I is of class

lII(v»

consequently,

c!.r.

Now let us look at the meaning of smoothness of the asymptotic phase from one more viewpoint which turns out to be most helpfull. But at first, we need to show that if (~, P2,' X) are

(EI' PI' X) and

can define a equal to

c!'

Let"t be a vector bundle atlas of class

(VI' !PI' IR")

~1(P1(y»(y»

(y

~I(X): E lx .... IR"

vector bundles then for every positive integer r one

vector bundle (C(EI' E2 ), tt, X) whose fiber [C(E I , ~]x at x

C(Elx , Elx ).

(i = 1, 2),

c!'

E

E"10

pjl(VI»,

and

(V2" !P2' IRm) E~, X

!P2(Z)

I: tt-I(VI " VJ .... (UI " VJ

by

(z

Ix(~)

0

0

:;I:

121.

Let us show that! is a

C(IR", IRm» and (VI" V2 , III, C(IR", IRm» and define ~: W .... L(IR" , IR") J1(x)

= ~2(X)

0

formula A(x)1I

and

Finally, define

= J1(x)

11

0

[~(x)rl

c!'

where

p;,I(UJ) ,

Further,

E

"1' (V2, !P2' IRm)

vector bundle atlas. Let both (UI L(lRm ,

IRm)

= VI "V2 ~(x) = ~I (x)

by

A: W .... L(C(IR", IR m), C(IR" , IRm»

(x E W). Then

define

1'It«()(~». By ! denote the fa-

"

E

~,

V2 , t,

"VI" Vz

belong to !. Set W

J1: W ....

[~2(X)]-I. 0

E

[~I(X)]-I.

mily of all triples (VI" V 2' I, C(IR" , IRm» where (VI' !PI' IR") and VI" V2

for (Et, Pt, X)

are linear isomorphisms. Define a mapping

= ~2(X) ~ x C(IR" , IRm) by I(~) = (tt(~),

Ix: C(Elx , ~) .... C(IR" , IRm)

X is

U I "V2• Then !PI(Y) = (PI (y) ,

(P2,(z) , ~2,(P2,(z»(z»

=

~2(X): ~ .... IRm

E

c!'

E

0

[~I (x)]"!, by the

307

Using some results from global analysis on manifolds (in particular, the ",-theorem; see, for example, Leng [I, p. 171-178]) it is not hard to prove that the transition function A is of class

e.

Recall that N ~

Thus, (C(EI' E,),

N

'11',

X)

is a

c: vector bundle.

and TA are vector bundles of class

c:

uniquely determined structure of a

C.

Provide Nand TA with the

vector bundle compatible with the structure of

C. Denote the so obtained vector bundles by Nand TA, respectively. Define the vector bundle (C(N, tA), '11', A), as described above. class

Now we shall show that the asymptotic phase

t: V -. A is of class

c!.r

e

iff there

exists a section 0': A -. C (N, tA) such that (4.4) and, moreover, for each integer p, 0 ~ p ~ k, the mapping cr: A -. COP(N, TA) is cP smooth. x

E

Indeed, let 111: N -. TA be an idA-morphism of class c!.r satisfying (4.3). Given A, define a mapping o'(x): Nx n U -. TxA by o'(x)(v) = 1II(v). According to Bourbaki

[1, subsection 15.3.7], the mapping

0': A -. COP(N, TA)

is a cP smooth section

(0 ~ p ~ k). Conversely, assume that the section 0': A -. C(N, TA) has the indicated properties. Then the mapping 111: N -. TA defined by 1II(V) = O'('rM(v»(v) is a morphism of class c!.r (see Bourbaki [I, subsection 15.3.7]). In the remainder of this section, we shall consider only cascades. Without loss of generality, we shall assume that (4.1) and (4.2) hold with c = 1 (to this end replace, if necessary, the diffeomorphism integer).

f.

M -. M

by

.r,

n being an appropriate positive

4.4. Theorem. Let M be a smooth manifold, r ~ I, and (M, T, /) be a dynamical system

308

of class C. Let A be an f-invariant C submanifold of M. Assume that there exist numbers « > 0 and ~ > 0 such that

for all x

E

and t

A

it

O. Then the asymptotic phase of (M, T, /) defined in the

vicinity of A is of class (fl'''. In other words, ~(f) are C manifolds depending continuously on x • Let I:

E

A.

Besides that, Tx~(f)

> 0 be sufficiently small and k:s

asymptotic phase is of class (fl.Jc. For x IIvll :S

El,

= N! • E!

E

A,

r.

(x

E

A) •

We shall prove inductively that the

let Nx(E) denote the set {v

E

Nx :

and ~(Nx(E), TxA) be the Banach space of all

satisfying !p(Ox)

= Ox

supplied with the norm

c!- mappings 11': Nx(l:) -+ TxA of uniform c!- convergence. Define a

Banach vector bundle ~(lV(I:), til.) with fiber [~(lV(E), tA)]x = ~(lVx(I:), TxA) (x E A) in a manner described above (henceforth, we shall omit the tildes over N(E) and TA). Given ~

E

~(Nx(E), TxA),

II~II:S 1, define an element

by where

prl and pr" correspond to the decomposition TM[A] = N. Til.. The element ~. is welldefined whenever I: > 0 is small enough. Since Nf is a contraction, in virtue of (4.1), by making the relation

E

> 0 smaller we get that

Consider first the case k

= O.

~.l is also contracting.

Note that I satisfies

Instead of the vector bundle ~(N(E), Til.) , we shall

309 use the bundle Coip(N(e), TA) whose fiber over the point X E A consists of all mappings '1': NAe) ~ TxA, !p(Ox) = Ox, satisfying the Lipschitz condition with Lip(!p) :s 1. Defme the norm of 'I' by the formula 11'1'11 = sup {II!p(V)1I I II vII : v E Nx(e),

c!

IIvll - O}. The corresponding topology is stronger than the topology of uniform convergence. It follows from (4.2) that if mapping

I:

Coip(N(e), TA) ~ CoiP(N(e), TA)

Lip(/) :s exp(- (3/2)

> 0

e

satisfies

is sufficiently small, then the the

Lipschitz

< 1. Hence there exists a continuous section

II IT II :s I, which is / -invariant, i.e., / (IT(x)) = lTif -I(x» satisfies the condition (4.4). For x

Next let us consider the case k = 1. I-jets, j~(i;),

denote the set of all condition t;;(v) = Ox x L(Nx, TxA),

E

TxA. Identify N with

N and

j~(t;;) = (v, t;;(v), Di;(v»

then

and

A

v

E

E

with

A ~ ~P(N(e), TA),

A). Consequently, Nx(e) ,

mappings (: Nx(e) ~ TxA

C

of

E

(x

IT:

condition

let

IT

H!,o(e)

satisfying the

i(Nx(e), Txh) with Nx(e) x TxA

t;;

for every

diffeomorphism f carries the mapping t;; to the mapping I(i;)

E

E

C~(Nx(e), TxA).

The

Co(Nrl (x)(e), Trl(x/)

and induces a mapping ie/) on the set of I-jets, which can be written as ie/lev, i;(v) , Di;(v»

= (~(v),

",(v), (v»,

where ~ and '" are defined above and the element i;(v)

E

L(Nrl(x)' Trl(x)A)

is given

by the formula i;(v) = pr2

0

Dexp~\x)(z)

0

Df -I(y)

0

Dexpx(v, t;;(v»

[pr l

0

Dexp - II

0

Df -I(y)

0

DexpAv, i;(v»r l ,

o

r

(x)

where y = expx(v, t;;(v» , z = f -I(y),

TxM

=

N!

III

TxA and Dexpx(Ox)

(z)

x E A.

v E Nx(e) and

= id (x

E

A). Whenever

N

Recall the equalities is invariant, we have

therefore

t;;(Ox) = Dif -I I A)(X) (here just like before, we identify N'"

0

Dt;;(Ox)

0

Nfl Nr

I

(x)

(x

E

A)

N). Thus, whenever N is invariant, we get

310

i(/)(o, 0, >.)

= Df'\

>.

0

0

Nf

(>.

E

x

L(Nx, TxA),

E

(4.5)

A).

It follows from (4.2) and (4.5) that

Let Po: TM -+ If and PI: TM -+ that IIPo - PIli < c, then

If

denote the idA-projectors with kernel TA. Assume

where A: N~ -+ TxA is a linear operator with sup {lIi(/)IN~,o(O)II: x close enough to

If.

E




= O(c)

E

A}

< 1

0 is sufficiently small. By the fiber contraction theorem (see Theorem E

A) of the section

IT:

in fact, to the space C~(Nx(c), TxA) and, moreover, the section continuous.

> 1, the proof proceeds by induction. Let

k-jets j~(~) of

Therefore

By continuity,

A.25), we get that the values IT(x) (x

For k

as c -+ O.

1 even if the subbundle NI is notf-invariant but only

sup {lIi(/)IN~,o(c)lI: x whenever c

IIAII

c!

A -+ ~(N(c), TA) belong, IT:

N~,Jt'I(c)

A -+ ~(N(c), TA)

is

denote the set of all

mappings ~: Nx(c) -+ TxA having contact of order (k - 1) with the

zero mapping. Let N~,Jt'I(O) be' the subset of N~,Jt'I(c) consisting of all jets satisfying the condition v

= Ox'

The linear space N~,Jt'I(O) is isomorphic to the space PJt(Nx

of all k-homogeneous polynomials. Whenever

therefore

If

IIJJt(/) IN!,Jt·I(O) II

to be close enough to

If

S

exp(- (3)

and c

(x

If

E

, TxA)

is 1j-invariant, we clearly have

A) by virtue of (4.1) and (4.2). Assuming

> 0 to be sufficiently small, we get by continuity (4.6)

311

Suppose that o'(x)

E

~.l(Nx(e), TxA)

(x

E

A). The inequality (4.6) enables us to apply

once again Theorem A.25 and to prove,. in this way, that o'(x) (x

E

E

~(Nx(e), TxA)

A).

4.5. Notation. According to m.21, let

=

n(NJ, x)

lim sup

n-++CD

~ In

IINjINxll,

w(1J1TA, x) = - lim sup -nlln IIV·nIT nAil, n-++CD

n(1J1TA, x)

=

f

lim sup -nlln IITjlTxAll

n-++CD

(x)

(x

E

E

A)

A).

By Lemma II1.2.2 and Lemma III.2.3, the inequalities o'(Nf, x)

< 0,

o.(Nf, x) - w(7J1 Til., x)


- c, where c is a small enough positive number. ~

Apply Theorem 4.6.

Proofs of some theorems stated in Chapter 1 At the end of this section, let us present proofs of Theorems 4.3, 4.7, 4.8, 4.11 and 4.12 stated (without proof) in Chapter I. 4.9. Proof of Theorem 1.4.3. Let t: E .... E denote the to-shift along the trajectories of the vector field ( for a sufficiently large number to > O. Set L = Dt(O), f = t - L, then .1(0) = 0, Dj(O) = O. Given a positive number IJ, there exist a number c5 = c5(IJ) and a function f/J. E c!'(E, E) such that f/J.(x) = j(x) for IIxll :IIi c5,f/J.(x) = 0 for IIXII ~ IJ and sup {max [llf/J.(x)II, IIDf/J.(x)II]: x E E} < c. We shall regard E as a trivial vector bundle over a singleton, then (L, id) is a vector bundle automorphism and (L + f,..., id) is a weakly non-linear automorphism of E. Apply Theorem IV.4.2. We have /30. IILI~II Take ~

$

~

< I,

«0

= IIL'II~

$

~II

:IIi

I, if to is sufficiently large.

and ~ in the capacity of L-invariant subbundles Xc, and Yo, respectively.

c!' subbundle WCu invariant under L + f/J. and tangent to ~ $ ~ at the origin O. Consequently, there exists a C submanifold W'u locally invariant under ( and such that To W'u = ~ $ ~. A similar reasoning proves the existence of the sub manifold W'S' with the required properties. Because the manifolds W'u and W'S' cross transversely at the point 0, we conclude that W'. W'u n W'S' is also a locally ~-invariant C submanifold, and ToW' = ~. By Theorem IV.4.2, there exists a unique

In order to prove the rest of Theorem 1.4.3, we must apply Theorems V.2.2 and V.4.7

taking the center manifold W' in the capacity of A. Unfortunately, we cannot apply this theorem directly because the submanifold W' is not compact. Therefore, we are forced first of all to compactify W'. This will be done as follows. According to the proof of

313

WC u

Theorem IV.4.2, the invariant manifold

C

it' • ff

smooth function 1/1: (x,

y, z)

carries the vector field

-+ P;-. The change of variables

(x - I/I(y, z),

H

~

can be represented as the graph of a certain

C

to a

y, z) (x

E

P;-, y ~

vector field

= ff.

WC

canonical form. Let B

IR P )

z

E

it'

WC·,

WC·

We shall also assume that the matrix Ac E L(RP ,

it',

having

unstable manifold. A similar reasoning can be applied to out loss of generality, we shall assume henceforth that

E

ff), (9

ff as its center-

as well. Therefore, with-

= P;- •

= A Iff

ff, WC U

=

it'

(9

ff,

is reduced to the real

be any block of the matrix Ac. Since the eigenvalues

of Ac are pure imaginary then either p

=2

and B

= [0 - Col

Col

1

0'

Col'"

0 or B is an

,

elementary nilpotent block. In the first case, the corresponding linear vector field on Rl is a center.

(z E IRl)

Compactify IRl by adding

to 1RIP3

=: IRl U 1R1P2

RlPl

in the standard way.

z = Bz

and extend the vector field

We get a vector field on 1RP3 all of

whose trajectories (except for the rest point 0 E 1R1 are periodic. The Lyapunov exponents of the linearized vector field are equal to zero for all points (in both directions). Consider now the case where B(u\I ... , up) = (~, ... , up, 0). The vector field is given by space

u = ~, ... , Up_I = up, l

RIP P

to

IRP ,

words, we embed

IRP

and in

up

= O. We compactify IR P

by attaching the projective

then extend projectively the indicated vector field. RP + I

In other

and extend the given vector field by adding the equation

v = 0; after that we pass to homogeneous coordinates. So we get a vector field on =: RP

In

IRtP p +1

u IRIPP , which can be written in each local chart as follows:

this

chart,

the vector

field

11

has

infinitely

many

equilibria

that

fill in

314

the axis Os. At each of these points, the eigenvalues are equal to O. (2) Let 2:s q :s p - 1,

... ,

=

wp

=v/

s

up / u q ,

u q '"

0,

=

WI

uq .

ul

wq _1

/ uq ,

= u q _1 /

Let us show that the vector field

=

=

Wp _1

=

uq + 1 / uq ,

2 Wq+2 -Wq+I' ••• ,

has no singular points in this chart. In fact, if

lJ

w

= 1. Otherwise, the existence of a singular point implies s = wp " = ... = Wq+2 = O, hence W· q + 1 = - Wq2 + 1 '" 0 ,a contrad"lction. 0 then

(3) Let ... ,

Wq+1

Then =

Wq+1

uq ,

q _1

0,

up '"

W p _2

=

Wp-I'

=

WI

Wp _I

UI / up, ••• ,

=

I,

s = O.

Wp-I

=

s = v / up. Then WI = W2, has no rest points in this part of

Up_I / up,

Clearly,

lJ

IRIPP+i.

Thus, the projective vector field

lJ

on IRIPP + 1 that corresponds to the Jordan block B

has only singular points with zero eigenvalues. All other trajectories tend toward rest points as I -+ + .... Let

A

be equal to the Cartesian product of all projective spaces that correspond (in

the above sense) to the Jordan blocks of Ac. Consider the naturally dermed vector field on A (i.e., the product of the corresponding projective vector fields).

Clearly, the

I -+

+ ...) of the tangent linear extension are equal to 0

IIxll ~ 11-,

we see from the above construction that the

Lyapunov exponential rates (as for all points of A.

=0

Since fll.(x)

for

diffeomorphism I/J == (L can choose II-

>

+ f~lJ?

117A(z)1I :s 1

Set M = ~ x

induces a C diffeomorphism A: 11.-+11.. Given e

If'"

x A.

+

e,

117A-I (z)lI:S 1

+

e

(z

(4.9)

A).

E

Define a vector bundle morphism r of (M, pr3' A) by

r(x, y, z) = (L.,x, LuY, A(z»

Along with

> 0, we

0 so small that

(x E~, Y

r, consider the following C diffeomorphism

'II

E

If'",

z e A).

from M to M:

315 _ { (L

+ f,.J(x,

y, z)

if (x, y, z)

E;

E

'II(x, y, z) -

rex, y,

if x

z)

The vector bundle TM[A] decomposes as

TM[A]

E

E", y

E

= 1:: elf' •

~, Z E A\E.

TA

where 1::

= A x E",

If' = A x ~ and, moreover, the subbundles 1::, If', and TA are invariant under IT. If c > 0 is sufficiently small, then (4.9) shows that TM[A] = 1:: $ If' e TA is an exponential splitting (see Definition m.2.6). Because'll tends to r in the C topology as Il -+ 0, we deduce from Theorem m.2.8 that for sufficiently small '" > 0, the vector

r,

bundle TM[A] can be represented as the Whitney sum of 7'{t-invariant vector subbundles

It', and

r

TA, where

is close to 1:: and

It' is close to If'. If

c

>

0 is small enough,

then by (4.9) we get

for all z

E

A and

m

= 0,

1, ... , r. In other words, the manifold A is normally

r-hyperbolic under the cascade (M, 'II). According to Theorems 2.2 and 4.7, there exist smooth manifolds W(A) and WU(A). Moreover, W(A) =

UW:,

WU(A) =

zEA

are

=~

Wo = ro- = E"

!Ii

and

=~. It is not hard to show that the manifold W(A) coincides with

WU(A) coincides with

and WU

U~; w: and ~ zEA

C smooth submanifolds depending continuously on z E A;

~

C

Wi:

d(¥'(z) , 'IIn(y»

are s

WC u in the vicinity of the point

c£ d(z, y) exp(- «zn) (n

one can show that inf

the

{«z: Z E

E

E c: A.

~-invariant. Besides, if

C smooth and

positive numbers. Repeating

0

=

arguments

A}

!!!

«

>

ZE A

and

YEW:,

and Cz are some in the proof of Lemma III.2.2,

1, 2, ... ), where used

«z

0 and that there exists a number c E~,

then

> 0

>

0

WC then d(rpt(z) , !pt(y» s c d(z, y) exp(- a:t). A similar statement holds for ~ (z E WC). If c > 0 is sufficiently small, then Theorem 4.8 implies that and ~ are C smooth in z E WC. satisfying

the condition: if

z

E

WC, Y

WC·, and The manifolds W;;; ~

t

and !p(z, [0, t]) c:

w:

316

4.10. Proof of Theorem 1.4.7. Let I; be a A

= Dt;(O):

C

smooth vector field on E, 1;(0) = 0,

E -+ E, and rp be the phase flow of 1;. Let ~. E', and If denote the

A-invariant linear subspaces of E

that correspond to the eigenValUes of A

Re ~ < 0, Re;\ > 0, and Re ~ = 0, respectively. Denote As = A I~,Au The principal part of the vector field t; can be expressed as I;(x,

C

E", y

E E', Z E If)

smooth functions of order O(IIXIl 2

+

lIyll

It follows from the above proof of Theorem 1.4.3 that after a suitable

C

where p, q and p are IIzll -+ 0.

= A IE', Ac = A IIf.

y, z) = (A,sX + p(x, y, z), AuY + q(x, y, z), AcZ + p(x, y, z» (x E

+

satisfying

change of variables, we get W's p(O,

y, z)

= q(x,

= E" 19 0, z)

If,

=0

W'u = E' (x E

+ 19

E", y

lIyll2

If,

+

IIZII2) as IIxll

W' = E", consequently,

E E', z Elf).

C smooth, If -+ If that

According to the same theorem, the asymptotic phases for W's and W'u are

i.e., there exist such C mappings ~I: E" ~\(O, z)

=

~2(y' u) =

~2(0, z)

=z

(z Elf),

19

If -+ If

and

~2: E'

1&

w: = {(x, 0, u): ~\(x, u) = z},

~

=

{(O, y, u):

z}. Make one more change of variables, namely, h: (x, y, z) ~ (x, y, ~2(y' ~\(x, z»)

(x E

E",

Y E E', z Elf).

Then h(W:)

=

h({(x, 0, u): ~\(x, u)

= z}) =

{h(x, 0, u): ~I(X, u)

=

= {(x, 0, ~2(0, ~\(x, u))): ~\(x, u) = z} = {(x, 0, z): x

Recall that rp(W:) =

W:(Z) ,

rp(~) = ~(Z) (z

E

z} E

r}.

If). Hence

tot t t) rpt{,x, 0 ,z) = ('P\(x, z), , IP3(Z» , 'Pt (0, y, z) = (0, 'P2(Y' z), 'P3(Z) .

Thus, passing to new coordinates, we obtain p(O, y, z) = p(x, 0, z) == p(z)

y E E', z

E

If). Set R(x, y, z)

=

p(x,

y, z) - p(z), p = p, Q = q.

(x

E

E",

317

4.11. Proof of Theorem 1.4.8. Apply Theorem 2.2 to the manifold A constructed in subsection 4.9. 4.12. Proof of Theorem 1.4.11. According to Theorem 1.3.9, the principal part of the local representative of the vector field ~ can be written as (x, y,

z, e) = (Ast + PVc, y, z, e), Au>' + AcZ Vc

E

+

E:, y

z, e),

y, z, e), 1 + cr(x, y, z, e»

p(x, E

q(x, y,

It',

Z

E

~,

e

E

IR),

where p, q, p and cr are c! smooth functions, 2w-periodic in e and such that p(O, 0, 0, e) ii 0, q(O, 0, 0, e) II 0, p(O, 0, 0, e) II 0, cr(O, 0, 0, e) ii 0, D,p(O, 0, 0, e) • 0, D 2q(0, 0, 0, e) E 0, D 3P(0, 0, 0, e) II O. As before (see the

weS' = ~

proof of Theorem 1.4.7), we may assume that therefore, p(O, y, z, e)

weS'

phases for = p(x, 0, z, e) proof, put P

ii

0, q(x, 0, z, e)

we

u and ii 91o(Z, e),

= p,

Q

= q,

ii

III

~

III

IR,

we = It' u

III

~

III

IR,

O. The existence of c! smooth asymptotic

allows to obtain the following equalities: p(O, y, z, e) cr(O, y, z, e) = cr(x, 0, z, e) II !/Jo(z, e). To finish the

R

= r - 910'

S

= cr - !/Jo.

4.13. Proof of Theorem 1.4.12. Compactify and then apply Theorems 1.4.11 and V.2.2.

we after the manner used in subsection 4.9

Bibliographical Notes and Remarks to Chapter V Theorem 2.2 can be found in the book by Hirsch, Pugh and Shub [1] and in the papers by Fenichel [1-3], but our proof is more detailed and more explicitly based on global analysis methods. Osipenko [1-3] has studied the notion of weak persistence (when the given invariant manifold is not isolated). Theorem 3.1 is proved by Mane [2] for k = 1. The case k > 1 is examined by Bronstein [6]. The notion of asymptotic phase for invariant manifolds is investigated by Hirsch, Pugh and Shub [1], Fenichel [1-3], Robinson [1], Kadyrov [1,2], Aulbach, Flockerzi and Knobloch [1]. Our exposition follows the paper of Bronstein and Kopanskii [4].

318

CHAPTER VI NORMAL FORMS IN THE VICINITY OF AN INVARIANT MANIFOLD

§ 1. Polynomial Normal Forms (the Nodal Case)

In this section, conditions for smooth conjugacy of two dynamical systems in the neighbourhood of their common smooth compact asymptotically stable invariant manifold are given. An illustrative example is presented. 1.1. Standing assumptions. Let M be a smooth manifold, k I!: 1, T

= IR or T = I, and

(M, T, /) be a dynamical system of class ~. Let A c: M be a compact sub manifold of

t

class ~ invariant under (M, T, /). Denote = /IA (' E 1). Henceforth, we shall assume that A is exponentially stable (for definiteness, in the positive direction). Let N

= TM[A] I

N/: N -+ N (t

E

TA. The linear morphism

T/:

TM[A] -+ TM[A] induces a morphism

1). As usual, denote

O(Nf, x) w(Nf, x)

= - lim sup!t In H+CD

0(7]1 TA, x) w(7]1 TA, x)

= lim sup!, In t-++CD

IIN/INxll, IIN/"tIN

t

f (x)

II,

= lim sup !, In IIT/ I TxA", t-++CD

= - lim sup !, In H+ CD

117]' -t ITt f

ex E

All (X)

A).

In what follows, we shall assume that O(Nf, x)

< 0,

O(Nf, x) - w(7]1 TA, x)

we obtain g = g\ • exp(O'z

+

P(O'\. 0':0). Denote 0'3 = O'z

+ P(O'\.

0':0. Then gil

0

g = eXpO'3 and

(2.8) The promised estimates (2.7) follow from (2.4) and (2.8).

C
K satisfying

sup{- w(T/, b) + r·n(T/, b) + [(Q - K) - r]·c(7J1X'". b)} < 0

bEll.

:S

CXo

+k

338 (r = 0, 1, ... , k).

Applying Lemma m.2.2, choose a integer v so that IITr" IT" Ell .1I1'j IEbl(' 1I..,.p IX'::bII (Q.K).r f

'J

(b)

(b e A;

:!S

(2.9)

114

r = 0, I, ... , k).

Denote

(z e E \ y),

1 i m sup lilt, w w

e

0

-+ z

gn(W)1I I IIlt,(w)1I

(z

E

Y);

E\Y

Bn(z) = IITgn (z)1I

(z e E).

Since the subbundle X is close to r and the mapping II Xb is close to D(j1 Xb)(Ob) (b e B) in the vicinity of the zero section, we conclude that IlflXbli is close to

1I1]l~II. Further, the mappings g and gl are assumed to be arbitrarily c!' close to f (at least, in a sufficiently small neighbourhood of the zero section). Therefore, we obtain (2.10)

For simplicity of notation, assume v Let I'"

== I"'(-r E )

=

1.

be the vector bundle of m-jets of sections IT: T -+ TE. Let r°P"

denote the Banach space of all continuous bounded sections of I'" (m = 0, I, ... , k). Let

r,m-,: I'" -+ I"'-l

a function

Pm

and ltm : rOpn -+ rOpn-l denote the canonical projections. Defme

on the fibers of ltm as follows: if ltmCl = ltmc? then

339

(note that

Pm

can take infinite values).

Let b:

JIl

-+ TE denote the projection On the

=b

target of the jet. Each element c E rOJll can be identified with a section Co E -+ TE of the bundle (TE,

TE,

For

m

= (z,

expi,l

0

element of rOpn with

and h

E

0

eXPa

gl·1

0

0

eXPa

r'"(71i.) is an arbitrary

=

E rOJll by the rule: to(c)

= g(z) ,

g(z» , a

0

small enough, then tm(c)

IICllo

-I = J""( z eXPb

satisfying J":(h)

Co

0

C

b

= gil

c*,

=

0

g(z) ,

where

z E E.

define an operator from rOpn into itself as follows: if

= I, ... , k,

C.(Z)

gil

0

hog), a

= c·,

= g(z) ,

c is an

where

b

= gi I

0

ze E

g(z) ,

C" smooth section of the vector bundle

(TE,

1["/

• Consider first the case

In

=

O. Assuming ...

0 , 1, ... , k).

(2.11)

= I, We obtain by virtue of (2.10) that

Po(toe l , toC'-) = sup {lIt..rI(Z) - t..r2(z) II II II'~Q K) II'" II'" x·: -I

lIexPb

. II II-(Q-K)·

. z

x

-I

0

gl

- ( = x,

:s sup {Dodl(z) IIbcl

. sup {[lIl[l(g(z»1I /

0

eXPa

bel

Q

y) e E,

II~II

g(z) -

h

k

< 0

(K - r) C(Tf-IIX", b)}

bEJ\

for r

= 0,

1, ... , k. Put Q

= Q(K).

Apply Lemma 2.4 to the diffeomorphisms r l and

gil, h being replaced by WS(f). We get that f

-I

and gil are

d'-

conjugate to one another

in the vicinity of WS(f) and, consequently, in the neighbourhood of h. This would finish the proof of Theorem 2.3, but unfortunately Lemma 2.4 is not directly applicable because X is not compact. To get over this difficulty, we compactify the space X by attaching the projective space PXb to each fiber Xb (b E B) and then extend the mappings f and glover X· x Y in a natural way by taking into account the fact that j{z) for 111.11 ~ roo

= gl(z) = Lz

2.6. Remark. It is seen from the proof of Theorem 2.3 that the number Q can be determined as follows. First choose the smallest integer K satisfying K > k and ~~ {- w(Tf- I , b)

+ r C(Tf-I,

b)

+

(K - r) C(Tf-IIX", b)}

< 0

(2.12)

(r = 0, 1, ... , k),

then pick out the smallest number Q such that Q sup {- w(T/, b)

+

r C(Tj, b)

+

> K + k and

[(Q - K) - r)]

bEJ\

(r

= 0,

ccmr, b)}


ot l

+ ... +

(in other words, to obtain the sequence {ot(})} , write

otk_l}

1 Cli

loti = ot,

+ ...

(l:s j :s loti)

times, then write

2

«,.

times, etc.) Let E I , ••• , Em F be Banach spaces. By It" we shall denote the direct sum of the family of spaces Ea.(,}) for 1::5 j ::5 IClI. Let La.(EI' ••• , En; F) denote the Banach space of all continuous symmetric IIX I-linear mappings f; from It" into F endowed with the norm

Let Pt be the canonical projection from E

==

EI

$

••• $

En onto E t • Given

(l

e Z~, let

Pot. denote the mapping x ~ (Pa.(,})(x» from E into It". The mapping f E -+ F is said to be a multihomogeneous polynomial of multi-degree (l, if there exists an element (l e La.(EI' .•• , En; F) such that f = u 0 Pot.. The image of the space LOt.(E" ••• , En; F) under the linear map u ~ u 0 POI. is denoted by Pa.(E)o ••. , En; F). The value of the

344 polynomial I

E

POI.(E\t ... , En; F)

at the element (x\t ... , xn)

•

01.

E

EI

81 ... 81

En

will

•

OI.n

01.1

be wntten as ft.xl' ... , xn) - I ' x - I . XI ..... xn • Whenever the dlrect sum decomposition E = EI 81 ••• 81 En is fixed, we shall write POI.(E; F) instead of POI.(EI , ... , En; F).

= XI 81 ... 81 Xm ,

Let X, Y and Z be Banach spaces, X

fl

E

Z:.

Y = YI 81

... 81

Yn ,

Ot E

Z':',

A.2. Lemma. There exist canonical Isomorphisms

7

A.4. (i

there are polynomials I~.a

A.3. Lemma. Given IE POI.(X; Z), such thai

E

p~.a(X,

X; Z),

+ a = Ot,

=

f(x

+ y) =

L I~a

Lemma. If IE

P/3(Y; Z),

Ott

1, ... , n),

then

the /unction

E

• x~ya

z,:"

Ut

(X, y

= (Ot~,

E

X).

... , Ot7) and

h: X ... Z defined lTy

h(x)

gt

= j{gl(x),

E

POI.t(X; Z)

... , gn (x» ,

n

belongs to the space P~(X; Z), where 7J

=

LfltOt{

(j

= 1,

... , m).

t-I

A.S. Notation. Let E and F be Banach spaces and k be a positive integer. By Lk(E; F) we shall denote the Banach space of all continuous k-linear symmetric mappings from Ff into F. Given (E LJc(E; F), the mapping f. E ... F defined by f(x) = (x, ... , x) is called a homogeneous polynomial of (full) degree k on E with values in F. The set of all such polynomials is denoted by Pk(E; F). For k = 0 define Po(E; F) = F. Denote pk(E; F)

=

k 81

P,;(E; F), ac(E; F)

=

k 81

P.(E; F) .

• -0

Let

E

= EI

81 ... II

En.

Clearly,

PJc(E; F) is the direct sum of the spaces

345 Pa.(E\ • •..• En; I') for all Il

E

Z~.

IIlI

= k.

f E C(U. /) and l:s k :S r. denotes the k-th iterated derivative of f at the point x. Recall. that

Let U be an open subset of a Banach space E. x Dk.~)

As usual.

Dk.~) is an element of Lk.(E; 1'). In particular. D~)

= E\

Let E

• •.. • En and Il

E

U.

E

L(E; 1').

E

z~. The iterated panial derivative

is denoted by Da.~). Clearly. D"1Cx)

E

La.(E; 1').

A.c). Composite mapping formula. Let E. F and G be Banach spaces; U c E and V c F be open subsets; g

E

f

C(U. V);

Dk.(f. g)(x)

E

C(V. G);

l:s k

L LI1'k.W Dqj(g(X»

=

:S

r;

X E

U. Then

. D'\g(x) ..• D'qg(x).

\:Sq:Sk. ,

where the inner summation is over all such collections i of positive integers it • ...• iq that i\ + ... + iq = k. and the coefficients 11'k.(I) are certain positive integers. In particular. for k = I we have the chain rule DC!. g)(x)

Assume. additionally. that E fJ

E

Z:'.

l:s fJ

:S

Ij! I

I!:

E\ •.•.• Em. F

. Dg(x).

=

F\ •...• Fn. 8

=

(g\O •••• 8n).

.Ie l.

E

r. Then

Here T = (T\ ••..• Tn) moreover.

=

= Dj(g(x»

E

j = {j.k. .:

n Z+;

1 (1:5 k :5 n. 1:s

S :S

1:s k:5 n.

1:5

S

} :5 Tk..

m Z+

d an.

Tk.). The summation extends on all such pairs

n

(T.l) that

1:s ITI:s IfJl

and

LC!~

+ ...

+iZ:k.)

= fJ.

Thecoefficients

e(T.l)

take positive values. A. 7. The differentiation formula for a polynomial. Let

Il E

z:'.

E = E\ •...• Em

346

If tI e

and f e Pa.(EI, ... , Em; F). •.. , m), then

Z':'

and tI:s

CIt

(i.e.,

tlt:s a.t for all i = 1, (A.1)

where 1 is the polylinear mapping that corresponds to the polynomial f and e(a., (3) are

> a. p for some number p e {I •...• m}. then iff ==

definite positive integers. If 13 p

o.

A.S. The differentiation formula for a polynomial with variable coefficients. Let E and F be Banach spaces, a. e

z,:" E

of class C. Define A: E -+ F 13 e

z,:"

= EI

ED •••

by A(X)

=

CI)

Em' and ;\: E -+ Pa.(E; F) be a mapping

[;\(x)](x) == ;\(x) . xa.. For each multiindex

I:s 1131 :s r, the following formula is valid:

r

Df3 A(X)·hf3 =

(A.2)

d(CIt. 13. JL)(if-Ji;\(x) . hf3 -Ji ) . xa.-Ji . hJi.

",:Sf3

",:Sa.

• Define a function f E

$

E -+ F by f(u, v)

= ;\(u)(v)

==

;\(u)· va.. Then A(X)

= j{x,

x).

Clearly,

vaA(X)

• h(3

r

(A.3)

C(JL, v) Ii;D'U(x, x) . hVhJi ,

Ji+v=/3 where

JL,

v e z':' and c(JL, v) are positive integers. Since f(u, v) = ;\(u) . va., we have

D"J(u, v) . h~ therefore for

JL:S

= (Dv;\(u)

. h~)(v) == (Dv;\(u) . h~)va..

a. Ii;(D"J(u. v) . h~)(u, v) . h~ = Ii;[(Dv;\(u) . h~) . vj . h~ (A.4)

= ~(Dv;\(u)·h~)(v)·h~ = e(a., 1J.)(Dv;\(u)·h~)·va.-flh~

by (A.l). If there exists a number

k e {l, ... , m}

such that

JLk

>

Cltk'

then

Ii;D'U{u. v) == O. Substituting (A.4) with u = v = x, hi = ~ = h in (A.3), we obtain the equality

347

ifA(x)

. hf3 =

which is much the same as (A.2).

Jets of Banach space mappings A.9. Dermition. Let E and F be Banach spaces, U be an open subset of E, a E U, and k be a positive integer. Let I and g belong to C(U, F), and j{a) = g(a). One says that

I and g have contact olorder k at the point a if x -+ a.

iff

as

C(U, F) and O:s k :s r. The mappings I and g have contact DSj{a) = DSg(a) lor s = 0, I, ... , k.

A.10. Lemma. Let j, g 01 order k at a

I!f{x) - g(x)n = o(nx - an k )

E

A.11. Lemma. Suppose the hypotheses 01 the preceding lemma are fulfilled. Denote

j(x) = j{x that

.f

+ a) (x

E

U - a).

There exists exactly one polynomial Pk

E

r(E; F) such

and Pk have contact olorder k at the origin. In lact, k

Pk(h)

L -{ D1f..a)

=

. h(.

I.

(=0

A.12. Definiion. Let a

E

E,

b

E

F,

and k be a positive integer.

Assume that

= g(a) = b.

The functions I and g are said to be equivalent, if they have a contact of order k at the point a. Clearly, this is indeed an equivalence

j, g

E

CC(E, F) and j{a)

=

,relation. The equivalence class containing I is denoted by l" j!W and called the k-jet of the function I at the point a. We also say that a is the source and b is the

target of the jet. We write a

= s(l)

and b

=

b(/c). The family of all k-jets

l"

with

s(/') = a and b(l') = b is denoted by ~(E, F)b' Let U and V be some open subsets of E and F, respectively. Denote

348

r(U,

U UJ!(E, F)b'

V) =

aeu bey

According to Lemma A.II, there is a one-to-one correspondence r(U, V)

(A.S)

U X V x rI(E, F).

0 and JI. e (0, 1) that sup 1111("(x)lIS' n7/'l1E

1I:s c JI." (n = 1,2, ... )

-n

h

xeA

(x)

for s = 0, 1, ... , k. A section u: A -+ EIA is said to be in X and a

eft

eft

smooth if there exist a neighbourhood U of A

smooth section u.: U ~ EI U such that u.(x)

A.36. Theorem. If the morphism

=

u(x)

for x e A.

f. E -+ E is k-contracting over the set A, then there

363

exists a /-invariant section 11'f: A

-+ E 01 class

cf .

• It is well-known (see Husemoller [1]) that for the vector bundle (E, p, X) there

exists a vector bundle (Eo, no, X) such that

E

$

Eo

is trivial. Define a k-contracting

vector bundle h-morphism 10: Eo -+ Eo of class cf. Then 1 $ fa is k-contracting over Hence it follows that without loss of generality we may assume the vector bundle (E, n, X) to be trivial. Repeating the arguments used in the proof of Theorem A.33, one can show the existence

A.

of a section

"1

e rO(r(E) IA)

which

is

the globally

mapping ~(f). Let 11'0 e rl«E) and ITn = j

(j~n}n=I.2.... converges to as ITn(x)

=

(x, u(n) (x» ,

"1:

0

ITo

0

attracting

fixed

point of the

h-n (n = 1, 2, ... ). Then, clearly,

A -+ E uniformly on E.

u(n) maps A into IRd , where d

The section ITn can be written

= dim

Ex, x e X. Denote

and set

Lu~n)(x) -

u~n)(x, y) = II

L u~~k(y) (x - y)~ I /3! II

IIX _ yll/-I