Bifurcation of maps and applications, Volume 36 (North-Holland Mathematics Studies)

BIFURCATION OF MAPS AND APPLICATIONS This Page Intentionally Left Blank NORTH-HOLLAND MATHEMATICS STUDIES 36 Bifu...

Author: Author Unknown

15 downloads 367 Views 8MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

BIFURCATION OF MAPS AND APPLICATIONS

This Page Intentionally Left Blank

NORTH-HOLLAND MATHEMATICS STUDIES

36

Bifurcation of Maps and Applications G.IOOSS lnstitut de Mathematiques et Sciences Physiques Universite de Nice, France

1979

NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM

NEW YORK. OXFORD

0North-Holland Publishing Company, I979 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 85304 9

Pu blrshers : NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM* N E W YORK *OXFORD Sole distributors for the U.S.A. and Canada: ELSEVIER NORTH-HOLLAND, INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

Library of Congress Cataloging in Publication Data

Iooss, GQrard. Bifurcation of maps and applications.

(North-Holland mathematics studies ; 36) 1. Nonlinear operators. 2. Mappings (Mathematics) 3. Bifurcation theory. I. T i t l e .

w329.8.148 ISBN 0-444-85304-9

515 ' .72

PKlNTED IN THE NETHERLANDS

79-9345

CONTENTS

Introduction

v i i

CHAPTER I

S t a b i l i t y o r i n s t a b i l i t y o f a f i x e d p o i n t o f a map i n a Banach s p a c e

CHAPTER I1

Bifurcation o f fixed points i n

CHAPTER I11

B

9

1.

Fixed points

2.

Points o f period 2

12

3.

The P o i n c a r i ! map - o r b i t a l s t a b i l i t y

17

Hopf b i f u r c a t i o n i n

9

R

2

27

1.

Standard H o p f - b i f u r c a t i o n

27

2.

Non-standard H o p f - b i f u r c a t i o n

44

3.

R o t a t i o n number o f t h e d i f f e o m o r p h i s r n r e s t r i c t e d t o t h e i n v a r i a n t b i f u r c a t e d c l o s e d c u r v e and weak r e s o n a n c e 2 Hopf-bifurcation f o r f i e l d s i n R

47

4. 5.

B i f u r c a t i o n i n t o a 2-dimensional i n v a r i a n t t o r u s f o r a non-autonomous d i f f e r e n t i a l e q u a t i o n

6. B i f u r c a t i o n i n t o a 2 - d i m e n s i o n a l i n v a r i a n t t o r u s f o r an autonomous d i f f e r e n t i a l e q u a t i o n

a.

71 7a

a5

99

7. E x e r c i s e

CHAPTER I V

1

Domain o f a t t r a c t i v i t y and u n i q u e n e s s o f t h e invariant circle

100

Subharmonic b i f u r c a t i o n s o f f i x e d p o i n t s i n R 2 - s t r o n g resonance

105

1.

The g e n e r a l s t u d y

105

2.

Subharmonic b i f u r c a t i o n s f o r a non-autonomous d i f f e r e n t i a l equation

123

3.

Subharmonic b i f u r c a t i o n f o r a n autonomous d i f f e r e n t i a l equation

126

4.

R e l a t i o n w i t h t h e p a p e r o f A r n o l d a n d comments

127

V

vi

Con tents

CIIAPTER V

CHAPTER

VI

131

Invariant manifolds and applications 1.

The hyperbolic case

132

2.

The central case

145

3.

Application to bifurcation problems

157

4. Applications to differential equations

169

4.1.

The non-autonomous casB

169

4.2.

The autonomous case

180

Bifurcation of an invariant circle into an i n v a r s 2-torus for a one parameter family o f maps

201

1.

Introduction.

202

2.

Main theorem and comments

205

3.

Center manifold theorem

208

Definitions

4. Proof of the maln theorem. Step 1: Reduction to the dimension 2 5. ti.

7.

Proof of the main theorem. Step 2: Persistence o f invariant circles for P r o o f o f the main theorem.

21 1 # 0

217

Step 3: Bifurcation

222

An example

226 229

BIBLIOGRAPHY

*********

1NTRM)UCTION

These n o t e s c o v e r and e x t e n d t h e c o u r s e g i v e n by t h e a u t h o r a t t h e U n i v e r s i t y o f Minnesota d u r i n g t h e f a l l 1977 a d i g e s t o f a j o i n t work o f A.

.

The r e s u l t of CHAPTER V I i s

M-IENCINER and t h e a u t h o r d u r i n g 1977-1978 and

improved w h i l e t h e y s t a y e d a t t h e U n i v e r s i t y o f C a l i f o r n i a a t B e r k e l e y i n J u l y 1978 ; a l l t h e d e t a i l s o f t h i s work are d e v e l o p e d i n

r4] and

.

[4 h i s ]

The s p i r i t o f t h e s e n o t e s i s , as f a r as p o s s i b l e , t o g i v e e x p l i c i t f o r m u l a s f o r t h e b i f u r c a t e d o b j e c t s a n d t h e simplest p o s s i b l e way t o use them. It i s w i s h a b l e t o e n c o u r a g e Mechanicians, P h y s i c i s t s , C h e m i s t s , B i o l o g i s t s a n d o t h e r ( u s e f u l ) p e o p l e t o compute ( w i t h a computer) t h e p r i n c i p e l p a r t , f o r i n s t a n c e , o f t h e b i f u r c a t e d o b j e c t which t h e y s e e k and t o compare t h e i r corrputed r e s u l t s

e i t h e r w i t h t h e r e s u l t s o f b r u t a l n u m e r i c a l c a m p u t a t i o n s or w i t h real e x p e r i m e n t s . O t h e r w i s e t h e y may n o t b e s u r e t h a t t h e phenomenon t h e y are l o o k i n g f o r is really a bifurcation

.

The p r e s e n t a t i o n i s mainly a n a l y t i c a n d , f o r example, no

of t h e t y p e of t h o s e of J.A.

P.H.

RABINDSJITZ f o r f i x e d p o i n t s or o f

qlobal result

J.C.

ALEXANDER

YORKE f o r c l o s e d o r b i t s are g i v e n here. The r e a d e r i s a s k e d n o t t o open

t h i s l e c t u r e n o t e s if he is only i n t e r e s t e d i n g l o b a l r e s u l t s . T h e r e are many books which d e a l w i t h b i f u r c a t i o n p r o b l e m s , w i t h s t u d i e s

of t h e s t a b i l i t y of b i f u r c a t e d s e t s and w i t h some p h y s i c a l a p p l i c a t i o n s too see f o r i n s t a n c e

[2C] , [291 , [ I 8 1 , [ 2 5 ] , [ I 2 1 , [ 3 2 ] , [ 3 4 ] , [35]

.

The s e t o f p a p e r s on t h i s t o p i c i s n o t c o u n t a b l e , s o we c a n n o t refer t o a l l of them and we ask t h e r e a d e r t o l o o k a t t h e b i b l i o g r a p h y of t h e c i t e d

books t o o b t a i n s p e c i f i c r e f e r e n c e s

.

vi i

,

-

viii

Introduction

I

Except i n CHAPTER

, each

chapter c o n t a i n s e i t h e r new r e s u l t s o r new

f o r m u l a t i o n o f some known r e s u l t (see t h e comments a t t h e end o f t h e chapters), Nevertheless, i m p o r t a n t p a r t s o f CHAPTER I11 and V

J.E. WRSDEN

found i n the book o f

O.E.

of

and M.

The a i m here i s mainly pedagogic, advanced student.

MCCFIACKEN [ 2 5 ]

a p a r t o f $ 111.6

LANFOFD [ 2 2 ]

o f these notes can be and i n t h e paper

.

coming f r o m S. STERNBERG [31]

t h e a u t h o r i s i n t e r e s t e d i n t h e average

That i s why, elementary p r o o f s o f t h e r e s u l t s a r e s t a t e d

as f a r as p o s s i b l e

.

Some open problems a r e g i v e n f o r the i n t e r e s t e d reader.

F i n a l l y i t i s perhaps u s e f u l

t o note t h a t b i f u r c a t i o n f o r maps has no p r a c t i -

c a l advantage f o r steady b i f u r c a t i o n s , even though t h e a u t h o r has presented formulas f o r t h i s case.

([29]

that

or

[32]

I

The r e a d e r i s asked t o l o o k a t t h e r i g h t p l a c e f o r

[121

.

f o r instance)

NOTATIONS.

E

Let

11

. /IE

or

be a Banach space on

11

,

I(

R or 6:

We s h a l l denote t h e norm by

Z(E)

if t h e r e i s no p o s s i b l e confusion.

Banach space o f bounded l i n e a r o p e r a t o r s i n

&(El

. E

i s t h e n the

and t h e standard norm i n

is

We s h a l l denote

d(E1 ;

El

t h e Banach space

E2)

t h e Banach space o f bounded l i n e a r o p e r a t o r s f r o m

t o t h e Banach space

E2

,

w i t h t h e standard norm. We

s h a l l assume a l o n g these notes t h a t t h e r e a d e r is f a m i l i a r w i t h t h e n o t i o n o f d u a l space (x,y*)

for

E*

of

x E E

, t h e p r o d u c t of d u a l i t y w i l l , y* E E* and i t i s l i n e a r i n

E

be noted

x

(x,

y”)

, semi-linear

or in

y

*

.

ix

Introduction

It i s t h e u s u a l scalar p r o d u c t i f

E

is a H i l b e r t space. W e s h a l l also

assume t h a t t h e r e a d e r knows t h e d e f i n t i o n o f t h e a d j o i n t o p e r a t o r L : El

a l i n e a r operator

is dense)

.

of

when i t c a n b e d e f i n e d ( f o r i n s t a n c e i f

+E2

E

i s l i n e a r unbounded i n

L*

* L

one c a n d e f i n e

L

p r o v i d e d t h a t t h e domain of

L

The most i n v o l v e d t h i n g we s h a l l use e x t e n s i v e l y i s t h e n o t i o n of s e p a r a -

t i o n of t h e s p e c t r u m f o r a l i n e a r o p e r a t o r satisfies a02

= o1 U o2 w i t h a n o p e n

P2

such t h a t

The r e s t r i c t i o n s

oi(i = 1,2)

.

PI

Li

W e as s u m e too

,

+ P2 of

in

, such

E

such t h a t :

O2

= Id

O2 3 o2

.

O2

n o1 = d ,

T h i s l e a d s t o t h e d e f i n i t i o n ff two p r o j e c t o r s

( = 1)

and

Pi

commutes w i t h

t o t h e i n v a r i a n t s u b s p a c e s P.E

L

t h a t its spectrum

number o f "circles" (we c a l l c i r c l e a n y c l o s e d

being a union of a f i n i t e

curve diffeomorphic t o a circle) P,,

L

L ( i = 1,2)

.

have t h e spectrum

1

t h a t t h e r e a d e r is familiar w i t h t h e e l e m e n t a r y r e s u l t s

of t h e p e r t u r b a t i o n t h e o r y o f i s o l a t e d eigenvalues of a one parameter f a m i l y of bounded o p e r a t o r s .

T. KATO

[I91

or to

1127

Now, for a n o n - l i n e a r

.

F o r a l l t h e s e n o t i o n s , we r e f e r t o t h e book of

differentiable operator

F : El

-.+

E2

b et w een two

Banach s p a c e s we s h a l l write t h e F r e c h e t d e r i v a t i v e

F'(X One of t h e main t o o l

1

I

DF(X~)I D ~ F ( x E ~ )~ E , ;

for c o m p u t a t i o n s w i l l

.

be t h e implicit f u n c t i o n t h e o r e m

which t h e r e a d e r w i l l f i n d i n a s u i t a b l e a n a l y s i s book or i n

[I21

.

W e s h a l l assume f o r t h e a p p l i c a t i o n s t h a t t h e r e a d e r knows t h e e l e m e n t a r y r e s u l t s on d i f f e r e n t i a l e q u a t i o n s s u c h as t h e d i f f e r e n t i a b l e dependerice i n t h e i n i t i a l d a t a o r i n a p aramet er, and t h e F l o q uet t h e o r y f o r e q u a t i o n s w i t h

Introduction

X

For these r e s u l t s we r e f e r t o t h e book o f

periodic coefficients.

.

HALE [7]

We s h a l l note

,

space

class

p

Cp

E N

,

CPfcr(A ; E) cy

A

where

i s a s e t of

Rn

and

E [ O , 11 t h e Eanach space o f f u n c t i o n s DPf

such t h a t

is

HMlder continuous o f exponent

has a c l a s s i c a l Eanach s t r u c t u r e

(for

cy = 0

we n o t e

Cp(A

J.K.

E

a Banach

f : A ~y

.

4

E

of

This space

; EP

ACKNWLEDGEENTS.

I

am indebted t o

A.

CHENCINER,

R. k GEHEE, D. JOSEPH, and

f o r the h e l p t h a t they gave me f o r w r i t i n g these notes.

H. WEINBERGER

T h e i r h e l p was o f

v a r i o u s types : a s k i n g o r answering good questions as w e l l a h e l p i n g me t o w r i t e t h i s i n t h e standard Queen's E n g l i s h . t h e q u a l i t y o f t h i s l e c t u r e notes

.

This work was supported by the g r a n t s G-0122,

and, f o r t h e l a s t chapter, by a

a t t h e U.C.

Berkeley

.

This has s i g n i f i c a n t l y improved

MCS

73-08535 A 04 and

DA AG 29-77-

N S F g r a n t w h i l e t h e a u t h o r was s t a y i n g

I

-

STABILITY OR INSTABILITY OF A FIXED POINT OF A MAP I N A BANACH SPACE.

E

I n the following

denotes a Banach space on I? o r

Definition 1. The f i x e d point i f f f o r every neighborhood

such t h a t

U

of a map

0

of

,

0

2

F"V c U Vn

F

:E

-+

E

.

C

i s Lyapunov s t a b l e

t h e r e e x i s t s another neighborhood

.

0

An exercise l e f t t o t h e reader c o n s i s t s of showing t h a t s t a b l e i f f every neighborhood of

F

0 for

Definition 2.

U

The f i x e d point

The fixed point

of

0

3V

F :E

E

-+

, i s asymptotically s t a b l e

such that V x E V , Fn( X ) + 0 , n

3V , Y >

0 i s exponentially s t a b l e i f f

such t h a t

Definition 3.

0 contains an i n v a r i a n t neighborhood

.

i f f it i s Lyapunov s t a b l e and

kE(0,l)

of

0 i s Lyapunov

0 i s Lyapunov s t a b l e and

The f i x e d point

0

F :E -+ E

of

VxE V

-+a

0 and

\lFn(x)IlE5 ykn

,

i s c a l l e d Lyapunov unstable,

Lyapunov s t a b l e . This means t h a t 3 c and 3 n

>

0 with

Example 1. E = R for

>

such t h a t

0

IIFn(x)II >

6

v6

>

0

, 3 x such t h a t

I/xI( 5 6

.

, F(x) = Ax-x 3

/ A { < 1 ,0 i s exponentially s t a b l e

Ihl > 1 , 0 i s Lyapunov unstable

1 = 1 , 0 i s asymptotically s t a b l e A, = -1 ,0 i s Lyapunov unstable. Comment.

I n the paper of J. Scheurle [30] an example i s given i n RL

0 i s Lyapunov unstable, b u t a l l points of a neighborhood of

that 3 p

, $ ( x ) = 0 (hence Fn(x)

-+

0

1

,

n

-+

a)

.

, where

0 a r e such

There i s no contradiction

Bifurcation of Maps and Applications

2

between t h e Lyapunov u n s t a b i l i t y and t h i s f a c t . Theorem 1. L e t

L

-1

be d i f f e r e n t i a b l e a t

E

F ' ( 0 ) = LE 4 E )

and l e t of

F :E

0 and s a t i s f y

be i t s Fre'chet d e r i v a t i v e a t

0

l i e s i n a compact subset of t h e open u n i t d i s c , then

.

F(0) = 0

,

If t h e spectrum 0

i s exponentially

stable. I n t h e case when E

Remark.

n a t u r a l extension on

C

, we

i s a vector space on R

and we extend L

consider i t s

t o t h i s extended space i n t h e

standard way i n order t o d e f i n e i t s spectrum. Proof of Theorem 1.

Let us choose a norm i n (/LII = k

and

0

N

, and we choose

E

1

such t h a t

6

a/l+q

>1

.

> 0 i s chosen s u f f i c i e n t l y small

Stability or instability of a f i x e d point

5

n v = x u . F'(y.)xE

(4)

j=1 J

J

( 5 ) I/v\l 2 a'//xII , a '

aj

being independent of

-

> Y j

To prove t h i s , we consider

We note t h a t

P2Lx

chosen s o small t h a t

(4)

Hence

IIvII 2

We assume t h a t

If xE gsn closed, If

F(x)E K

k F (x)

6

0 because 2

6(l+q)

i s proved. IILXII

#

< pq

xE (q

k

gives

#

0

.

When 6

is

f i x e d ) , i t follows t h a t

is

Furthermore

a IF Uj(FI(y.)-L)xlI2[= J i s s o small t h a t

a' =

-

81 IIxII

and by

( 5 ) , \ ~ F ( xL) a~ 'l\xll ~

gsn i

for

A t a c e r t a i n s t e p F"(x)E K but

by

(3)

- 6 >1 . l+s

, the Riemann sums a r e i n is i n

P2x

.

k = 0,1,. . , n - l

by a'

(4)

>1

*

Now

.

F(x) =

Since K

1 F'(tx)x dt, 0

is

.

, then iiFn(xfii 2 afniixli

I/Fn(x)I/ cannot remain

O(b+s)

El

:

.

An e x e r c i s e l e f t t o the reader c o n s i s t s of showing t h a t t h i s i s a norm

.

Stability or instability o f a f i x e d point

, t h e two norms a r e equivalent i n El

l i m 3lLy\l = b n-tm

Since

111~1~1111

= SUP

n30

n+l llLl xlll (b+c)n

We now choose a new norm on E2 because

7

04 o2 = spectrum of

= ( b + € ) sup n>l

.

Moreover,

n llLlXlIl

- < (b +4111~1111 . (b+e)n

-

-1

i n a s i m i l a r way b u t using

L2

.

L2

which e x i s t s

We d e f i n e

-n

lllx2111 =

SUP n 30

llL2 x211

- ,

We obtain a n equivalent norm i n

E2

since

lim n+m

=

a-1

and

hence

, t h e n we have

because of t h e equivalence of t h e norms i n of

P1

E( 11.11)

and

c *

P2

in

E

E( 111

. 111 )

.

a s it can e a s i l y be seen.

.

and

E2

and of t h e boundedness

Hence, the canonical l i n e a r embedding i s a Banach space, E( Ill * 111 1 Hence t h e Banach-isomorphism theorem shows t h a t

i s continuous.

t h e i n v e r s e map i s a l s o continous. equivalent on E

El

Then t h e two norms

I \ * \ \ and 111 *II/

are

B i f u r c a t i o n of Maps and Applications

8

Exercise 1. L e t us consider the equation i n Example 2 and c a l l the s o l u t i o n of the Cauchy problem

, when X(0)

(1) X ( t ) = X(Xo,t) the map Xo-X(t) enough. IRn

i s defined for

t E [-T,T]

that i f

Xo

F

X ( s ) = X(Xo,s)

i s a l s o a fixed point of

i s an i s o l a t e d f i x e d point of

F

in

F

.

Deduce

, then AXo+ N(Xo) = 0 , i . e .

Let us consider t h e equation i n Example 2 , and define

Assume t h a t t h e r e i s an i n v a r i a n t c i r c l e for

1

//Xol/ i s small

i s a s t a t i o n a r y point of t h e d i f f e r e n t i a l equation.

Exercise 2 .

T

provided t h a t

. Show t h a t

Xo

,

Xo

i s a fixed point of the map Xo+-+X(Xo,to)

Xo

Assume t h a t

=

=R Ib

i s the one dimensional t o r u s .

i s a l s o an i n v a r i a n t c i r c l e f o r F Deduce t h a t i f

y

1 F : Y : {vc-=Xo(cp);~ET

3

Show t h a t

where

'?I

Y, :{ q n X ( X 0 ( q ) , s ) ;cpE

.

i s an i s o l a t e d i n v a r i a n t c i r c l e for F

, then

Y

a t r a j e c t o r y of the d i f f e r e n t i a l equation. Hint:

Write

x(x0(cp), s )

where

S -+ 0

when

vector f i e l d a t

s

Xo(cp)

-+

= x0[ f(cp,s ]

.

0

.

Then

dxO

,

-(q)

dv

and remark t h a t

f(cp,s) = cp+S(s,cp)

has the same d i r e c t i o n a s t h e

is

9

BIFURCATION OF F I X E D POINTS IKlR

11.

Let us consider a map F

P

neighborhood. of verifies

, where

in R

i s a r e a l parameter i n a

p

-

d = 0 and DF ( 0 ) = F (0) = k(p) P F h P dX Moreover we assume t h a t - ( O ) # 0 (Hopf c o n d i t i o n ) .

, such t h a t F (0)

0

.

Ix(0)l = 1

dw

We s h a l l see t h a t the s i t u a t i o n t h a t we consider here i s t h e one which occurs of chapter I , escapes frcm the

when a simple eigenvalue of the map F'(0) u n i t d i s c by passing through the point Let us remark t h a t i f

F

-1 a s a parameter

1 or

i s a r e a l map i n the r e a l Banach space

varies.

p

,

E

the

simplest s i t u a t i o n s f o r the escaping of some eigenvalues fran the u n i t d i s c a r e the one considered here and t h e case when two simple complex conjugate This l a s t case w i l l be considered i n

eigenvalues escape from t h i s d i s c .

chapter I11 (Hopf b i f u r c a t i o n f o r maps). 1. Fixed points.

_______

meorem 1. Let

k

C ,k

_>

A(0) = 1

,

(p,x)~-+F~(x) :R2+ R be of c l a s s

, D E (0)

and s a t i s f y Fp(0) = 0

= k(p)

with

, near

2

k'(0) > 0

Then t h e r e e x i s t s a unique b i f u r c a t e d branch of fixed points

for s

i n some i n t e r v a l

, x'(0) #

p(0) = x(0) = 0

that Ck-l

near

0

p'(s)

0

-

p

: F

P(S)

(p(s),x(s))

, such

[x(s)] = x(s)

0 and t h e functions*

i s unstable f o r

keeps a constant sign f o r

bifurcated fixed point unstable i f

p

p

.

and

x

are

.

The fixed point If

, for F

(-C,E)

0

0 and s t a b l e f o r

sE(0,c)

p = p(s)

or f o r

sE(-6,0)

is stable i f

p

>

0

.

p

2 -

.

Eliminating t h e fixed point

0 by

, we obtain

We can now solve ( 2 ) f o r i s of the form

h(p,O) = 0

p

f(p,x) = 0 = 0

.

by the i m p l i c i t function theorem.

, with

f(0,O) = 0

We then obtain

,

af

-(O,O)

p = p(x)

=

ap

of c l a s s

I n f a c t (2)

>

Ck-l

provided

0

Hence we can choose the parametrization i n theorem 1 a s

The s t a b i l i t y or i n s t a b i l i t y of t h e fixed point

0 of

Fp

results

from theorems 1 and 2 of chapter I. To study the s t a b i l i t y of the b i f u r c a t e d fixed p o i n t s , l e t us write

them x = s

i n JR

.

,

,

k'(0)

p = p(s)

and introduce the new coordinate:

The new map for y

i s then:

y-Y

with

B i f u r c a t i o n of f i x e d p o i n t s i n IR

71

We use the i d e n t i t i e s

t o obtain

(5) Hence

where

s+ 0

Because of

k'(0)

x = s , p = p(s) Now

.

~ (l-) I 0

> 0 , we then know that i s s t a b l e , while i f

has the sign of

s p'(s)

change of sign f o r

p

, or

sE(0,~)

Remark 1. It can happen t h a t S

p(s) = l o t 2 s i n

of

2 ~ ' ( s ) = s s i n l/s

for

~ ~ ( =x (1+p)x )

-

It i s easy t o see t h a t

s

for

sE(-E,O)

>

near

and is

p( s)

s

s p'(s)

for

p(s)

1 7 ds , then

0 near

s p'(s)

0 t h e fixed point

0

the f i x e d point i s unstable.

0

,

if

p'(s)

does not

,

s ~ ' ( s ) have opposite signs: i f

and

C1 0

X

2 1 xJos s i n -dx S 2

if

F :R +I3 i s

.

p(s)

has not always t h e sign

Consider the map

. C2

near

0

.

I n t h i s example

the s t a b i l i t y of the bifurcated fixed points is determined by t h e sign of

s'sin

l/s

.

12


Points of period 2 .

2.

_-

Let

Theorem 2 .

and s a t i s f y

F (x) :XI2

-+

P

F (0) = 0 k

, DFP(0)

=

be of c l a s s

R

h ( p ) with

Ck

,

,

k 1 2

,

h ( 0 ) = -1

near

0

.

h'(0) < 0

Then there e x i s t s a unique one sided b i f u r c a t e d branch of fixed points of order 2

( p ( s ) , x j ( s ) , j = 1,2) f o r

,

FP

such t h a t

x (-s) = x2(s) , ~&l ( 0 = )1 , x.(O) = 0 1 J Ck-l functions p and x are near 0 j

for

< 0 , unstable f o r p >

p

or for

sE(0,e) p p

>

0

and

, unstable x

,

sE(-e,O) if

p

0

,

n T(p)+ 16)

n+

exponentially,

-6 m

5 KIIYoll 2

-

Because of (15) and ( 9 ) , t h e r e e x i s t s a c o n s t a n t

-

T(p)

K1 > 0 such t h a t

1 5 KSllF$Yo) ]I2 5 Klk 2pl/Yol\2

ITP hence t h e sequence n-1

c

( \ - T(kL))

p= 0

converges exponentially (geometric s e r i e s ) and

IT(^)-

n T(p)

from which (18) follows. Now, we have by c o n s t r u c t i o n

hence, thanks t o (15), (18) and t o t h e r e g u l a r i t y i n

t

So, by (18) and (19) we o b t a i n t h e r e q u i r e d r e s u l t (13).

of

X(t)

,

Bifurcation o f f i x e d points i n iR

25

Comments on Chapter 11. About theorem 2 , the problem of f i n d i n g a change of v a r i a b l e s such t h a t t h e new map

F

P

satisfies

Fp(xk- x g,(x) i s open.

I

g CL (4 = g,(-x)

Of course, what we d i d i s d i f f e r e n t .

Let us n o t e t h a t i n t h e case

when t h i s study comes from a d i f f e r e n t i a l equation with ( s e e t h e paper of G. IOCGS, D.D. 2T

T -periodic coefficients

JCGEPH [15]) t h e equation which g i v e s t h e

p e r i o d i c b i f u r c a t e d s o l u t i o n has a u t o m a t i c a l l y t h e evenness property. About examples of b i f u r c a t i o n s of maps i n lR

One of t h e l a s t one i s

[6] where

, t h e r e a r e a l o t o f papers.

t h e reader w i l l be a b l e t o have o t h e r r e f e r e n c e s .

The r e s u l t on t h e l i m i t phase i n $11.3, on t h i s form comes mainly from t h e i n d i c a t i o n s given by V . I . IUDOVICH i n [ 161. There e x i s t o t h e r proofs i n i n f i n i t e dimensional spaces ( s e e [ 111) .


111. HOPF BIFURCATION I N R 2 .

1. Standard Hopf b i f u r c a t i o n .

L e t us consider a map

that

0

F

i n B2

)L

i s a fixed point of

F

)I

, where

and t h a t

p

i s a r e a l parameter, such

D F (0) = A

X P

following assumption. H.l.

ho#+l

-

has two conjugated eigenvalues

.A

and

Xo

s a t i s f i e s the

w

xo

, with

, we assume t h e so-called Hopf-condition: l e t h ( p ) ,

eigenvalues of and a r e

=1 ,

.

To be sure t h a t some eigenvalues escape from t h e u n i t d i s c when 0

lXol

A

P

for

whenever

p

near

0

, such

h(0)

be the

x(p)

= k

0

crosses

.

They e x i s t

from the c l a s s i c a l perturbation theory 1191.

F

is

C2

W e can w r i t e , if F

is

c2 near

C1

that

p

Then, we assume

o

:

and define

W e can choose a b a s i s

*

i n R2

*For t h i s , we choose the b a s i s

such t h a t

{Xr,Xi] 27

where

F

P

j

written as:

A (X + i Xi) = x(p) (Xi + iXi)

i

~

r

.

B i f u r c a t i o n of Maps a n d A p p l i c a t i o n s

28

where

We i d e n t i f y B2 and

z

by s e t t i n g

C

a s independent v a r i a b l e s .

r

LemL.

The map

F

F

i s of c l a s s

assumptions H.l, H.2 hold, and t h a t k-4

exists a

:C

C

-t

i s now

F

Ck

An

0

#

, k 2 5 near

1 for

0

,

that the

n=1,2,3,4

.

Then t h e r e

p-dependent change of coordinates, such t h a t i n t h e new

C

coordinates

P

has t h e form

Fpb)

(5)

with

R'

5

can make

where

+ i y and considering z and

Canonical form.

Let us assume t h a t

Proof.

P

z =x

of c l a s s p(p) = 0

Ck-5

,

and

z = r e2 i q

.

Let us w r i t e

AG(p,z)

can w r i t e

is homogeneous of degree

4. i n

(2,;)

, G

= 2,3,4

.

We

,

2

Hopf b i f u r c a t i o n i n iR

.

W e make t h e change of coordinate i n

C

:

i s homogeneous of degree

C

in

where

5

are

P9

y&(p,z)

P

(7)

by s o l v i n g

(8)

z =

for

p

near

(2,;)

, G2

2

; we w r i t e

satisfies:

t h e new map F'

z'

Ck-&

0

and t h e

29

-

2'

:

.

y i ( p , z ' ) + O(/Z'IG+l)

This l e a d s t o F P' ( z ' ) = FiL (z')

(9)

Let us t a k e

,

&=2

A;(I.L,z')

- k(p)yt(p,z')

+Y&[P,A

the new q u a d r a t i c terms are

= A2(W')

- h(P)Y2(C1>Z1)

+

Y2[P,A(W)Z'1

,

that i s t o say f o r t h e new c o e f f i c i e n t s :

Kpq 1

(10) Now, because near

0

:k

#

-

-

Ipq

1 for

, such t h a t

5'

Pq

-

(A-APX9Y

Pq

n = 1 and (p) = 0

:

3

, we can choose Yps€

Ck-2

for p

30


Ypq(P)

(11)

=

(Id

5

!dxq( p)

h( d -AP(

( t h e denominator i s never

A f t e r t h i s change of v a r i a b l e s (7) with

terms of order > 2

, we

spq(p)

previous change of v a r i a b l e s .

rd

=

# 1

4,=4

3

, are

Ckm3

Then, because

even a f t e r t h e

A:

#

A'(p,z)

3

1 for

Fpq(p)

,

order, we make another change of v a r i a b l e s

remarking t h a t

4 (modified by t h e previous changes).

. .

5,,

(14) a r e now Ck-4 f o r Pq We can again choose t h e

5

We can choose

Y~~ , Y~~ ,. Y~~ , y13

yPs

yO4 a s (11) if

Hence t h e lemma i s proved.

i s defined by ( 5 ) as a f u n c t i o n of t h e c o e f f i c i e n t s

Exercise 1. If a ( p )

where

.

p+q = 3

After t h i s change of 3-

a s (11) f o r

A;

=

as (11) f o r yj0 , Y= , ya3 . For yPs t h e denominator i n (11) vanishes f o r p = 0 and we cannot suppress t h i s

of t h e form (7) w i t h

+q

, p +q

$=3

4 , we can choose

and

term.

p

has modified t h e

We obtain f o r t h e c o e f f i c i e n t s of

t h e same equations (10) with

yP1

)

can make another change of v a r i a b l e s (7) w i t h

L e t us remark t h a t t h e new

n = 2

, which

L=2

0

Show t h a t

= 5pq(0)

.

Let us now write our map F 2incp

z = r e

,

~

~

P

(

i n polar coordinates:

=2 R ) e

2irrl

.

Hopf bifurcation in W

2

31

where

a:

=

-Re ( a ( o ) X o )


a: f 0

.

Then the map

Fo

i n p o l a r coordinates, t a k e s

t h e form

2 4 R = r ( l -ar ) + O ( r )

* = c p + ~ ~ + O2() r

and if

a: >

0

, t h e o r i g i n i s asymptotically s t a b l e , rhereas

if

a:

0

3

and we change coordinates t o o b t a i n t h e p r i n c i p a l p a r t of t h e i n v a r i a n t closed curve of t h e map (14) :

32

B i f u r c a t i o n of Maps and A p p l i c a t i o n s

X1 and

Ipl

are

, where

[ -1,1]x Tlx [ 0,6) means t h a t

Ck-5

X1 and

@,

are

with respect t o

T I = Xl,hZ,

x = u(tp,p)

form

for

cpE T1

and

p

i n t h e domain

i s t h e one dimensional t o r u s , which

1-periodic i n

We now want t o show t h a t t h e map

'I2)

(x,q,p

tp

.

(17) has a n i n v a r i a n t manifold of t h e small enough.

I n t h e case when

we can make t h e s i m i l a r change of coordinates

I n t h i s case we have

where

0

, and assume

, uniformly i n s m a l l

p : let

Hopf b i f u r c a t i o n in W L

35

which means

and t h e same for

@l

*

Let us w r i t e

tu

and show t h a t For

_>

cp

,

cp'

i s a b i l i p s c h i t z -homeomorphism of

if we consider t h e map

/2

@il($+1)

=

*;I(;)

.

Then

1Pu

iPu

in W

T1 onto T1

( i n s t e a d of

T

1

i s s t r i c t l y increasing i n R

This leads t o the e x i s t e n c e of

4-l :R + R

u

,

),

, and such

+I , and

(21)

Hence we can reconsider

@,

&I€

T1

NOW

.

4-l on U

T1 , which a l s o s a t i s f i e s (21) , f o r


36

defines a function X =

for

p

6(+) ,

where

u :T A

1

-t

R

.

16(*)1 5 1 - 2pRe hl

5

+ PP3/2

small enough

5 I@1-@21 if

(l-;-1pite hl + 2 p d 2 ) ( 1 - 2 p p

small enough.

Hence

6€U

.

'I2) -' h1 > 0

38


Let us show t h a t

I;(*)

u,

GE

I -< 1-2C

if uE UL

:

- 1 for c small enough

0

, hz #

, k 2 6 , we

1

, except t h e

know t h a t

Moreover, we can a l s o s a y t h a t p-

,

(X1(*;,p)

hence continuous i n

k-6,l

C

3

The choice of t h e constants i n t h e f u n c t i o n space i s

L = 1

I$! a'

u,...

, P l > l

@l(*,s,p,))

.

is

Co

taking values i n

Let us remark t h a t

Ckm5

,


42

3rd-steg.

a >

Stability result for

, h50 # 1

0

.

We have a l r e a d y proved more than t h e s t a b i l i t y of t h e i n v a r i a n t c i r c l e of

F

.

P

x = u(cp)

I n f a c t , we have shown t h a t i f

i t e r a t i n g t h e map, we f i n d that t h e manifold

*

uniformly t o t h e graph of that

lxol

5 1 , we

I

i n f lxn- u (cp) CppE T1

F

P

i n R2

.

1

Gn(cp); cpE T

Hence f o r a s t a r t i n g p o i n t x = xo

xo

u

as

tends

,

in

(p,

such

Uo

.

.

, kz # 1

We have t h e system (17') with the map

, then,

Uo

, and t h e s t a b i l i t y i s proved.

0

-+

n+

Case a < o

bth-step.

{xn =

can consider the constant

3+

Hence

.

u

is i n

,

@(o)= 0

Let us consider

F

-1 !J

< 0 , which r e p r e s e n t s

p

which i s of t h e form:

3 9

a >0 ,

We can do e x a c t l y what we d i d i n t h e case for

< 0 ($e Al> 0)

.

p

>

CL

t h a t t h i s i n v a r i a n t c i r c l e i n r e p e l l i n g for

O(p)

vo .

,

Po

w

P

I n any t u b u l a r neighborhood of

0

-+

n+

m

y

P

of

.

Po?\

(x1 9 ~ 1 )i n

dist(F-"(xl,cpl),YP)

F

.

But, t h i s means

I n f a c t l e t us consider

of t h e i n v a r i a n t c i r c l e

Consider a t u b u l a r neighborhood

Take a p o i n t

, b u t here

Hence we f i n d a n i n v a r i a n t a t t r a c t i v e c i r c l e f o r

F-l , with t h e r e g u l a r i t y p r o p e r t i e s of t h e previous case.

t h e domain of a t t r a c t i o n

0

yP

yP

.

I t s width i s

, s t r i c t l y included i n

We have

. , we can t h e n f i n d

(X~,Q,)

such t h a t

2

Hopf b i f u r c a t i o n i n Z?

does not 'belong t o

Fn(x ,cpo) G

O

y

u n s t a b i l i t y of Case -

5th-stee.

7,

for

n

43

l a r g e enough.

This i s e x a c t l y t h e

.

P

~5 0

= 1

.

I n t h i s case we s h a l l f i n d a change of v a r i a b l e s t o p u t t h e map i n t h e form (18) f o r

a >

0

, o r t h e corresponding form

if

a

P = 0

0 is attractive for

0

, 0 i s repelling f o r

IJ. = 0

, and t h e r e e x i s t s a l e f t

i n which t h e r e i s a n i n v a r i a n t r e p e l l i n g c i r c l e f o r

b i f u r c a t i n g from t h e f i x e d point loses its s t a b i l i t y f o r

p = 0

p

>

0

.

0 which i s s t a b l e t h e r e .

Note t h a t

Moreover, i n a s u i t a b l e system of

coordinates corresponding t o t h e normal form ( 3 ) , t h e i n v a r i a n t c i r c l e can be

I

expressed a s :

'


46

I

1

where

ro i s t h e s o l u t i o n of t h e equation:

I p j l = ~6~

,

j

=1,2 , b j > 0

,

Remark 1. If t h e r e does not e x i s t

L

small enough.

a2q+l # 0

problem of f i n d i n g an i n v a r i a n t c i r c l e

yP

t h e n even i f

FE Cm

, the

and t h e 'problem of i t s s t a b i l i t y

i s open. Example : the map

i

R = (l+p)r

+ p r3

B = ( p + B

has no i n v a r i a n t n o n - t r i v i a l c i r c l e f o r for

li. =

0

#

0

, b u t it

.

Proof of Theorem 2 . Let us s t a r t with t h e map i n t h e form (3) and put

We e a s i l y o b t a i n :

has i n f i n i t e l y many

H o p f b i f u r c a t i o n in W

2

47

Now, (4) g i v e s us

.

where

p

has t h e s i g n of

'a2q+l

This l e a d s t o t h e map

where

p

has t h e s i g n of

- C X ~ ~ , + ~ X1

and

Ipl

are

Ck-4q-3

m e same technique a s i n theorem 1, then gives t h e theorem 2. A p r e c i s e computation of t h e i n v a r i a n t closed curve i s p o s s i b l e .

Remark 2 .

For t h i s , see t h e paragraph 7 of t h i s chapter.

3 . Rotation number of t h e diffeomorphism r e s t r i c t e d t o t h e i n v a r i a n t b i f u r c a t e d closed curve and weak resonance.

L e t us consider t h e map "circle"

yP

of paragraph 1 which has a n i n v a r i a n t

F

li.

b i f u r c a t i n g from t h e f i x e d p o i n t

(a i s defined by t h e equations of t h e i t e r a t e s

Fi(n)

,n

-t

pi

" r o t a t i o n nuthber" of t h e map on

0

, in

t h e case

0

#

0

(12) and (15) of paragraph 1). The behavior

is related t o the P We g i v e i n t h i s paragraph some r e s u l t s

of any point

y

I-1

.

xE y

on t h e r o t a t i o n number which l e a d t o the n e c e s s i t y t o cofnpute p r e c i s e l y t h e i n p o l a r coordinates.

closed curve

So, we a l s o give a systematic way t o

o b t a i n t h e d e s i r e d p r e c i s i o n on t h e i n v a r i a n t curve, even i f Let us w r i t e t h e curve t h e map

f

P:

cp++

,d

on

yP

i n p o l a r coordinates

yIJ. can be considered i n

IR

n 0

r = r((p,p) :

= 1

.

,n25 Then

.


48

where

.

g P ( ( p + l ) = gli.((p)

Thanks t o t h e equation

(17) of paragraph 1, we

have :

where

gl

i s lipschitz-continuous i n

a homeomorphism of R homeomorphism of

T1

Q

,

and

lpl

small. Hence

passing t o t h e q u o t i e n t R/a = T1 which preserves t h e o r i e n t a t i o n .

f

P

is

, and l e a d s t o a

We now g i v e two

important r e s u l t s u s e f u l h e r e a f t e r , t h e proofs of which a r e c l a s s i c a l , and can b e found i n [ 8 ] i n more g e n e r a l and more p r e c i s e s t a t e m e n t s . Theorem (H. Poincarg) Let

B

? be a homeomorphism of T1 , whose l i f t i s a homeomorphism of

o f t h e form

f n - Id number of Moreover

?

f = I d + g with a Z - p e r i o d i c

converges uniformly t o a constant

.

p(?) = p/q

p(?)q Q

i f f t h e map

i f f the map

The r o t a t i o n number

p(?)

V H

Then when

p(?)

n

0)

, c a l l e d the r o t a t i o n

'pefq((p) mod 1 has a f i x e d p o i n t

q

for

f

),

mod 1 has no p e r i o d i c p o i n t .

fq((p)

'p

,

P(?)a Q

.

i s i n v a r i a n t under a change of the v a r i a b l e

and i s a continuous function of

r-I

.

(Id z i d e n t i t y ) .

( p e r i o d i c point of o r d e r and

g

?

i n the

Co

topology.

Theorem (A. Denjoy) Let

b e a diffeornorphism of

Then t h e r e e x i s t s a homeomorphism 6

T1 of c l a s s of

C2

T1 such t h a t

, and

let

Hopf bifurcation i n R

?

- -1R p o h-

, where R p :

= h

cpH

f

So, by a change of v a r i a b l e , t h e map

i t e r a t e s of any p o i n t

a r e dense on

tp

2

49

rp+ P(7) mod 1

is just a rotation

T1

. , and

Rp

the

.

These theorems g i v e an i n t e r p r e t a t i o n of t h e r o t a t i o n number of terms of a r o t a t i o n asymptotically equivalent t o

f .

? in

The main t o o l of t h i s paragraph i s t h e following Lemma 3 .

7

1

Assume t h a t t h e homeomorphism f

where

0

>

0

t a k e s t h e form

, and g i s uniformly bounded when

r o t a t i o n number

of

p(p)

P ( d = e(P)

Proof.

P

+

f

iL

/pI

i s small.

Then t h e

satisfies

o(lPla)

-

This follows d i r e c t l y from t h e formula

We can now prove Theorem 3 . Let t h e map

.

p = 1,2,. . , n - 1

F

P

be of c l a s s

,n25

.

Ck

,k > - n + l , and assume t h a t :1 #

Then t h e r o t a t i o n number

p(p)

of

f

P

1

is a

continuous f u n c t i o n o f p i n t h e neighborhood of 0 where y& e x i s t s , and n-2 i s a polynomial i n p of degree p(p) = Ol(p) + O( p [ y, , where 0 1(P)

[91

I

( i n t e g e r p a r t of -_

2 n-3 1.

,


50

Proof. Let us f i r s t assume t h a t

kT3

= 1

,

p

2

1 , k 12p+4

.

Then

proceeding i n t h e same m y a s i n paragraphs 1 and 2 t o o b t a i n normal forms, we make a change of v a r i a b l e s i n FP

C

, which leads t o t h e new form of the

:

Fp(z) = A ( P ) [ Z +

(3)

P

m=l

a2m+l(pL) z

m+l-m

z +b

2P+2

(pL)~2p+21 +O(lzl

2P+3)

I n polar form, t h i s leads t o

a2p+l

(cp,p)

and

both a r e of t h e form

!32P+l(rp,p)

A cos[2n(2p+3)cp] + B sin[2n(2p+3)tp] Let us note

2 then a2(o)rO(p)

ro(w) t h e unique

+ p R e X1

>

0

.

s o l u t i o n of the equation:

+ higher order terms = 0

(recall that

Cx

2

(0)

#

0)

Let us do now the change of v a r i a b l e s :

where

e

a2(o) = -lcr2(o)l

f o r a s u b c r i t i c a l one).

(Q

=+1 f o r a s u p e r c r i c i a l b i f u r c a t i o n ,

E

= -1

.

Hopf b i f u r c a t i o n in B'

51

The map takes the new form:

o( I

+

To be s u r e t h a t

x(cp)

i s bounded we may do the same a s we d i d f o r

l.2 = 1

.

So we change variables

where

xo(cp) has the form ( 5 ) and s a t i s f i e s t h e d i f f e r e n t i a l equation

where

6,

i s defined b y

The map i s now:

form ( 5 ) .

The technique of the proof of theorem 1 a p p l i e s here because and

= u(cp)

3/2

i s then bounded by 1. So, by using t h e lemma 3 , we g e t

- Y> 1 ,

52


where we only keep the terms of degree

5p

in

el(&)

.

This i s t h e r e q u i r e d

result. Now, if we assume t h a t

:k

# 1 ,m

. . , 2 p +3

= 1,.

, t h e r e a r e no

a2 p + l ’ P2p+l i n ( 4 ) , and (13) holds i n an e a s i e r way, because i n t h i s case

Let us now assume t h a t

= 1

, p

2 1,

k

2 2p+5

.

We do t h e same a s

previously t o o b t a i n

where a 2 ( o ) =

(17)

-Q:

# 0 , and

012p+;!(~.~)

B2p+2

C + A cos[2~(2p+4)lp]+ B sin[21~(2p+4)(pI

We can define

ro(p)

( c p , ~ ) both a r e of the form

.

by (6) and do t h e change of v a r i a b l e s :

The map t a k e s the form:

Hopf b i f u r c a t i o n i n iRL

53

A s i n t h e previous case, w e pose

where

xo(cp)

has t h e form

(17), and

Q,i s defined by

(11). The map

becomes:

where

has t h e form, (17). A s previously t h i s l e a d s t o a bounded

with

el(p)

polynomial of degree

Now, i f we assume t h a t %p+2

B2p+2

i n (lg), and

p

in

Am # 1 , m xo(cp)

s(cp)

p

.

and t o

This i s t h e required r e s u l t .

.

= 1,. .,2p+4

,

i s now independent of

t h e r e a r e no cp

, so

54


el

with

of degree

p + l

in

p

( t h i s r e s u l t i s a l i t t l e b e t t e r than i n

I. theorem 3 , because here we assume i n f a c t

Theorem

4

2

k

2p + 5 = n

.

(weak resonance).

Let us make t h e assumptions of theorem 3 h2p+3 = 1

( i ) if

)

0

O,(p)

f

,p 2

1

,

Then,

i s of degree

el(p)

.

p

The c o n d i t i o n

, which means t h e n u l l i t y of p c o e f f i c i e n t s , i s

BO(= m/2p+3)

necessary and i n g e n e r a l s u f f i c i e n t t o g e t the r o t a t i o n number i n independent of

of p e r i o d i c points of p e r i o d

2p+3

.

I n t h i s case

and t h e r e e x i s t t w o f a m i l i e s

p

for

p(p) = Oo

f

.

P

On t h e curve

yP

one

family of points i s a t t r a c t i v e , t h e o t h e r one r e p e l l i n g .

!(ii)If

I

iI

I

!

I _

~ 2 O p +=~ 1 , p 2 1 , 0

6 (p)

1

p(p) = Oo

i n general

.

i s of degree

P(p) =

B0

The c o n d i t i o n

, and we have the two f a m i l i e s of p e r i o d i c p o i n t s

a s i n (i) , and one i s a t t r a c t i v e on The n e c e s s i t y of t h e condition

theorem 3 .

.

If furthermore a n a d d i t i o n a l i n e q u a l i t y (33) i s r e a l i z e d ,

-

Proof.

p

i s necessary but i n g e n e r a l not s u f f i c i e n t t o g e t

0 ( = m/2p+4) 0

(P)

Assume t h a t

Y

P

el(+)

Xo2P+3 = 1 , p

2

, t h e o t h e r one r e p e l l i n g . 5

0

1 and

0

i s obvious, because of Ol(p)

f

Bo

, t h e n thanks

t o ( 1 2 ) , we have

Assume that

B2p+l(d

i s not identically

0

, which i s t h e g e n e r a l case, s o

Hopf b i f u r c a t i o n i n lR2

with

55

cpio'

# 0 . Let us introduce t h e two s o l u t i o n s

IAl + IBI

and

'p2 (0)

NOW t h e equation

because of t h e L i p s c h i t z c o n t i n u i t y i n

'p

thanks t o the c o n t r a c t i o n p r i n c i p l e i n R continuous i n

p

.

, uniformly i n , and t h e cpi(p)

To s t u d y t h e s t a b i l i t y on

f;('pi(p))

is

The r e s u l t follows. properties!

of

,

f

P

i = 1,2

, and are

So, we have two f a m i l i e s of p e r i o d i c p o i n t s of p e r i o d

2p+3 b i f u r c a t i n g from the o r i g i n f o r t h e map

so

p

y

12

F

P

.

, we can j u s t remark t h a t

> 1 f o r one family,

< 1 f o r t h e o t h e r , because of (28).

The r o t a t i o n number i s t h e n

O o = m/2p+3

from i t s b a s i c


56

ky4

Assume now, t h a t yp

= 1

,

>1

p

and

0,(p)

B0

E

,

t h e map on

i s now:

where

e2p,(cp)

(32)

=

c +A

cos[2n(2p+b)q~l+ B sin[2a(2p+b)(p]


(33)

(A2+

B2)1'2

,

1 5 cp < 2pi4

0

has two s o l u t i o n s

w i l l be s t u d i e d i n theorem 5 .

a s It can be e a s i l y seen i f we i n t e r p r e t t h e equation i n t e r s e c t i o n of t h e l i n e ';p+2

(O))

('Pi

Ax+By

+C

= 0

0

2P+2

(0)

'91

'

(0)

9

Now we have

('9) = 0 as t h e

, with t h e u n i t c i r c l e and

a s t h e dot product of t h e normal t o t h e l i n e with t h e tangent t o

the c i r c l e a t the intersection points. Hence, the proof of the existence (and uniqueness!) of two f a m i l i e s of p e r i o d i c points of period

2p+4

, b i f u r c a t i n g from t h e orgin f o r F

the same a s i n t h e previous c a s e . same because o f ( 3 4 ) , and of course Remark 1. The cases when

, is

The r e s u l t on t h e s t a b i l i t y i s a l s o t h e p( p) = B o = m/2p+4

hn = 1 f o r 0

w

n

54

chapter, they a r e t h e "strong resonance" case.

i s independent of

w i l l be s t u d i e d i n next

p

2

Hopf b i f u r c a t i o n i n IR

If t h e b i f u r c a t i o n occurs f o r

Remark 2.

p

,

0

4 a r e f u l f i l l e d , t h e r e i s one a t t r a c t i v e family

of p e r i o d i c p o i n t s .

It remains t o g i v e a very p r e c i s e r e s u l t on t h e r o t a t i o n number P ( w ) of when t h e condition of theorem n theorem 3, when h = 1 , n 0 Theorem 5 Let t h e map (i)

If

P

i2p+3 = 1 0

identically

(34)

F

el(p)

.

be of c l a s s

Ck

11 , assume

Oo ; so t h e r e i s = eo+pqe

9

,

k

l a r g e enough.

t h a t t h e polynomial

q € [ l , p ] and

0

#

4

0

where y = inf(3p-q,3~-29+3) and = 1

not i d e n t i c a i l y

1

,p2

1

eo ,

el(p)

i s not

such t h a t

.

+

Then, t h e r o t a t i o n number of the diffeomorphism f

(ii)I f

P

4 a r e not s a t i s f i e d . This r e s u l t completes t h e

25

p

f

E

P

satisfies:

=+,1 according t o t h e s i d e of t h e b i f u r c a t i o n .

, assume t h a t e i t h e r t h e polynomial el(,) or

e1( p )

I

80

and

being defined by (32). Then, t h e r o t a t i o n number of

f

P

satisfies

ICI

>

( A 2 + B 21/2 )

is

, A , B , C


58

#

Remark 1. We saw i n theorem 3 t h a t if :k has an asymptotic expansion i n powers of Remark 2.

1 for a l l

n

, then

P(p)

.

p

h2p+3=l, the theorem 5 g i v e s a p r e c i s e idea on

I n t h e case when

0

t h e asymptotic .expansion of

i n powers of

p(p)

I pI1I2

.

For i n s t a n c e , we

have f o r

if

= Go

9

1

(see theorem 4)

= 0

,

and f o r =

o

2 o +3p 2

+pel+p 0

=

[

B + p e 2 + p3 e + p

3

0

if

= O0

.

p

4+ p e.4 + p7B + (ep)11/2~6+

5

e4 + ( € P ) 9 / 2 85 + . . . i f

2

e +Jo

e

=

eo+p o2+p

=

OO

0

+pe + p

2

1

2

if

Proof of Theorem 5 .

3

3

e +...

el

= 0

6 Xo =

+...

and

Let u s f i r s t assume

i s t o e x p l i c i t a t e the map

N+1) (p

,d on

e2# 0

i s always an expansion

(cI

> (A2 +

2 1/2 B )

2 2 1/2 (A + B )

h2p+3 = 1 0

1

+o

O ~ = O and

ICl

1

.

h r f i r s t aim

, with no unknown f u n c t i o n yp (see Lemma 4 f o r t h e r e s u l t ) . f p : 9-

r(q)

o

Hopf b i f u r c a t i o n i n W

Using t h e usual change of v a r i a b l e s i n

i1

~ ~ ( =z i ()p ) [ z

(35)

+

2P k$o ‘2p+2k+4

where t h e

a

j

3p + l

+

c

m=l

zm+l-m a 2m+1

2p+k+kzk

+

C

,

2

we can w r i t e

’

2p+l +

59

k=0 b2p+2k+2z

p-l k-bp++k I:0d4p+2k+5z k=

k-2p+k+2

+

F

P

as:

+

’

p-2 z4~+k+7;k k=0e4p+2k+7

,b , c ,dj , e a r e r e g u l a r f u n c t i o n s of j j j

p

+

.

I n p o l a r coordinates, t h i s l e a d s t o

where a c a r e f u l examination shows t h a t a l l t h e f u n c t i o n s of @2m+l

a r e p e r i o d i c of period

have a

0 mean value.

’2 p+2k+2

and

except

a6p+3

a2m+l , @2m+l

hence of mean value

a2m+l

a r e combined with t h e terms i n

‘2p+2k+4

and

and such t h a t

:a

and

a

B.2m+l

a

2m+1

or with

’ @6p+3 t o get the a2m+l and @2m+l * are of t h e form A c o s 2 n . ( 2 p + 3 ) ~ + B s i n 2 ~ ( 2 p + 3 ) r p ,

a6p+3

@6p+3 which a r e products of 3 similar expressions, 0

p

2m’ 2m+l’ 2m’

This i s due t o t h e form of (35) where t h e terms with

themselves (only for t h e terms Hence, a l l the

1/2p+3

rp

.

60


Let us note

then, because of

#

a,(O)

2 ro(P) = EP

(38)

t h e unique p o s i t i v e s o l u t i o n of t h e equation:

ro(p)

( b a s i c assumption i n a l l t h i s paragraph 3 ) ,

0

m a

Re

hl

+ 0(g2)

,

with

E =-sgn(a2(o))

.

Let us do now t h e change of v a r i a b l e s (analogous t o ( 7 ) ) .

The map i s now expressed a s :

where

k2

3,k 6 , h l ,

h 4 , h6

B s i d 2 r [ ( 2 ~ + 3 ) ~, ] and a l l t h e

neighborhood of zero.

a r e a l l of the form

h a r e r e g u l a r f u n c t i o n s of p j' j The form (40), (41) of t h e map F i s j u s t an k

11.

improvement of ( 8 ) . Let us do t h e change of v a r i a b l e s (42)

x = xo(cp) +

( 5 ) A cos[2r[(2p+3)cp] +

E

,

in a


where

xo(cp)

!d = cp+zl(d

(45)

where

has t h e form

-

k2

-

,

:Plh,(cp,~)x

.

,

-

-

, k3 , k6 , hl , h 4 ,


where

x,(cp)

p

61

(5) and solve (lo), i . e

+ ( 6 ~ p+l/2) hl(cp,F)

+

2

+ $ph5(p)-2x

h6

+$pc2(cp3P)

+(EL4

(cdp+1’2h3(p):

3p-1’2h6(cp)


22 ,

+

(Xo(cp)

+z)2+ O( I pi3’)

(5).

t h e n we do t h e change of v a r i a b l e s :

has t h e form (5) and solve

+

,

62


2 ,

where

If

5 ,$ ,6, ,

P =1

hd


(5).

, the map (44) , (45) gives

i n s t e a d of (42). A l l t h e functions

\(cp)

a r e of t h e form ( 5 ) , and t h e y a r e

s o l u t i o n s of d i f f e r e n t i a l equations a s (43) and (47). t h e map f o r

p

_>

2

is

The new expression of

Hopf b i f u r c a t i o n i n R

where

- - _ k2

, k3 , hl , h4 , h6

2

63

(5).

a r e of t h e form

We do now a change of v a r i a b l e s of new type:

(54)

1/22

where

2 0

(55)

i s p e r i o d i c of period

x P

1/2p+3

,

b u t i n g e n e r a l of mean value

, and i t v e r i f i e s 2ReA

1

x ( c p ) + 6 x’(cp) P 1 P

-

eF,+(’p,o) = 0

.

It i s not d i f f i c u l t t o s e e t h a t t h e r e always e x i s t s a unique s o l u t i o n of (55) (even i f

Pl

= 0 )

.

I n t h e case

p

_> 2 ,

(57)

where

G2 ,

5 ,-hl , -

I n t h e case

h4

, h6

p = l

,

a r e of t h e form (5) t h e map i s

I

x,(cp)

the map i s now expressed:

B i f u r c a t i o n o f Maps and A p p l i c a t i o n s

64

.-. k 2 , fi1 ,

where

L4 , h6

a r e of t h e form ( 5 ) .

We now do a g a i n a similar change of v a r i a b l e s :

xptlis p e r i o d i c o f period

where

So, i f

p

xp+l = x2

22

i s of t h e form ( 5 ) , b u t i f P+l has a non-zero mean value.

?

El

, and v e r i f i e s

, x

I n t h e case

where

1/2p+3

p = 1

I n t h e case

p

, i n general

, t h e map becomes

= (1-2pRehl)2

i s of the form

p =1

+ O(lpI 3/2)

( 5 ) and x2((p) has i n g e n e r a l a non-zero mean value.

22 ,

t h e map becomes

Hopf b i f u r c a t i o n i n R

’;3 , Cl, I i 4 , h6

where

I n t h e case

p

_>

a r e of t h e form 2

2

65

(5).

, we d i d t h e two changes (54) and (60) t o

t h e formulation (52), (53) t o t h e formulation (63), ( 6 4 ) . do

go from

I n f a c t , we can

(p-1) times t h i s double operation, s o t h a t we have

i n s t e a d of (54) and (60).

I n (65) t h e f u n c t i o n s

a r e o f t h e form ( 5 ) because of p e r i o d i c of period lk,,+((p,

p)

.

1/2p+3

5(3(cp,

PI)

x p+1

’ Xp+3 * . . , xp+;lk+l’ * *

, and x , xpe,. P

b u t i n g e n e r a l of mean value

..,xpek

#

0

, because

where

2

of

They a l l a r e s o l u t i o n s of equations of type ( 5 5 ) and (61).

The map ( 5 2 ) , (53) becomes a f t e r t h i s change of v a r i a b l e s

(66)

are

= (1-2p R e

fil, i4, h6

Al):

+ ?kl(

p):

a r e of t h e form

+ ( cP)~’~$((P,

(5)

P)

+

$;8(Cp)

(p

+

22

) :

o( I P15/2)

9


66

p = 1 t o g e t (58), (59) and (62) from

Now, we can do t h e same as when

(491, (so):

and x a r e p e r i o d i c of period 1/2p+3 , and i n g e n e r a l x 3P-2 3P-1 of non-zero mean value, and s o l u t i o n s of equations o f t h e type (61). The

where

new expression of the map i s :

where

fi

1

has t h e form ( 5 ) .

if we do i n (62) t h e change

This form of the map i s a l s o v a l i d f o r

.

= (cp)’y

Now

p =1

,

(69), (70) g i v e a bounded

a s i n t h e u s u a l proof, s o we have proved:

y(cp)

L e t t h e map 0

=

1 , p

F

P

11

be of c l a s s

.

,

Ck

Then t h e map

f

Ir,

k

large enough, and assume

on t h e i n v a r i a n t curve

yli.

, takes

t h e form

(71)

fP(cp)

=(p+Z1(d

+ (6P)p+1/2hl(cp,p)

+Zph2(cp,p) +

+

+ o(pN+l) where

hl

, h2 , h3

are periodic i n

cp

of period

1/2p+3

,

polynomials i n


2

67

Remark 1. We changed t h e n o t a t i o n s t o w r i t e (71), s o hl , h e , h3

are new

functions here.

I n (70) w e had a term

Remark 2 .

more r e g u l a r i t y on

F

P

.

O ( l ~ 3p-1/2+y) l

It i s c l e a r t h a t , w i t h

, we can push t h e expansion a s f a r a s we wish.

An

i n t e r e s t i n g problem would be t o know, i n what c a s e s we can see t h a t t h e term can be incorporated i n t h e o t h e r terms, i . e . i n what c a s e s t h i s term

O(pN+’)

has period

1/2p+3

?

The idea i s now t o change of v a r i a b l e

N+1

a map with no cp up t o t h e o r d e r

p

.

rp

in

, such a s t o o b t a i n

T1

Hence, by lemma 3 t h i s w i l l

give the r e s u l t of theorem 5 i n c a s e ( 1 ) . A s sume t h a t

t o not b e i n t h e case of theorem

4.

The new v a r i a b l e i s defined by

where

h

w i l l be of t h e form ( T ) , and r e g u l a r i n

diffeomorphism of R becomes

GI--+

with

p

which passes t o t h e q u o t i e n t on

.

So, we have a

T1

.

The map (71)


68

then

(74)

+

a d d i t i o n a l terms of same kind a s those i n

We choose now

of t h e form

h(cp,p)

(5), as hl

as a s o l u t i o n of t h e equation

, t h i s i s always p o s s i b l e because eq(o) #

and because t h e terms w i t h d e r i v a t i v e s of order S o t h e map

’9, 2,

2,

and h

4

g2 ,

c3 ,$ a r e polynomials

9 .

have

1-11

f

i n factor.

i n 1-1 and p e r i o d i c o f period

has a 0 mean value.

Now we can change t h e v a r i a b l e of

22

0

f w t a k e s t h e new form

where the new f u n c t i o n s 1/2p+3 i n

( ~ p ) ~ ” ’ - ~ and

(p s o t h a t

Let us consider t h e f i r s t s t e p

2

(Gyp) becomes independent

Hopf b i f u r c a t i o n in a7

where

4,

i s t h e s o l u t i o n of mean value

Doing t h i s type of change of

G+

@ again

2

69

0 of t h e equation

times we o b t a i n i n f a c t a map

p-1

such t h a t A

&q+e

(80) where

- ( q , p )+p3'+ldq P +( Ev ) 3p+lJ2-q h3

61(p)

c 2 ( q , p )+(

i s now a polynomial of degree 3 p - q

h4(y, p ) +0(uN+'

3p-2q+5/2

and

hS(j,O) s t i l l

has a 0 mean

value. In t h e case when q 2 3, we may do a similar change o f v a r i a b l e on t h e c o e f f i c i e n t of

( E ~ L ) ~ ~ - by ~ ~a +term ~ ' ~of

order p

. Then,

-

'Q

t o replace

by a s t e p by s t e p

change of v a r i a b l e i n c r e a s i n g t h e power i n ( ~ p by ) 1/2 a t each s t e p , we g e t t h e

r e s u l t of ( i )of theorem 5. Let us now consider t h e case evenness of such t h a t

2p+4 n+m

= 1

, p

21 .

Because of t h e

it i s c l e a r t h a t i n F ( z ) we w i l l have powers CL

i s odd: n - m - 1 = 4,(2p+4)

.

So, i n t h e p o l a r form t h e map i s expressed as

)

znim


70

where a2p+2 and

B2,,

are of t h e form

We saw t h e changes of v a r i a b l e s (18) and (20).

I n f a c t we can f i n d t h e

following change :

where

ro(d

defined a s p e r i o d i c s o l u t i o n s of period

l/2p+4

(21).

After t h a t t h e map t a k e s t h e form

where

6

2Pt.2

xk a r e s u c c e s s i v e l y

i s defined by ( 6 ) , and t h e functions

i s a polynomial i n

i s defined by ( 2 3 ) , and

p e r i o d i c c o e f f i c i e n t s of p e r i o d

of equations of t h e type

p

.

1/2p+4

with

Let us f i r s t assume t h a t 31(~) =

oo

+

15 5

Pqeq(w)

then we can do t h e same change on 32p+2 , 8

P

9

a s i n t h e case

cp

i n functions independent of

cp

.

SO,

+o

(0)

A2p+3

61( p )

3

00

= 1 t o transform

i n t h i s c a s e t h e theorem

i s proved.

Let us now assume t h a t

0

,

, we know t h a t


(85)

c +A

=

e2p+2(lp)

2

71

cos[2fi(2~+4)cpl+ B sin[2fi(2p+4)(pl

.

To avoid t o be i n t h e case (33) where t h e r e i s weak resonance, we assume t h a t

then

0

keeps

(cp)

2P*

8

constant s i g n when

cp

So we do t h e change

varies.

of v a r i a b l e s :

where

h

has period

l/2p+4

, i s of mean value

0

,

and i s t h e s o l u t i o n

of t h e equation

where

K

#

0

i s d e f i n e d by

We v e r i f y t h a t

1+h'(cp)

>

0

, so

we have a diffeomorphism.

The new

expression of the map i s :

(90)

-@ = c -p

+

0

+ p P + l K + ppe8(G,p)

and u s u a l change of v a r i a b l e on

+ O ( p N+1)

(P l e a d s t o a

8

3

independent o f

(p

,

and

t h e theorem 5 i s proved.

4.

Hopf b i f u r c a t i o n f o r f i e l d s i n R

2

.

There e x i s t many ways t o t r e a t t h e problem solved h e r e a f t e r .

We j u s t

want t o use t h e t o o l of t h i s c h a p t e r t o f i n d t h e closed t r a j e c t o r y and t h e period of the b i f u r c a t e d s o l u t i o n .

Bifurcation o f Maps and Applications

72

Let us consider the following d i f f e r e n t i a l equation i n R2

(1)

dx =

L

dt

P

L

where the

are

Npyq

,

X + Np(X)

where we assume

and

N

:

sufficiently regular i n

2

q - l i n e a r symmetric i n 1R

p

and

X

.

We w r i t e

.

We denote t h e s o l u t i o n of the Cauchy problem f o r (1)with

X(0) = Xo

,

by (3)

X(t) = X(X,,P,t)

*

It i s well known t h a t (3) i s defined f o r

small enough [ 7

1.

t E [-T,T]

provided t h a t

Moreover we can f i n d (3) i n the following way:

I/Xol/

is

(1)i s

equivalent t o

(4)

X(t) = e

t L

'x0

+

J

t (t-s)L e

0

'

N&X(s)lds

,

and we can solve (4) by the fixed point theorem i n a s u i t a b l e function space. This leads t o a m c t i o n X s e r i e s of

( 51

X

near

X(Xo,

(O,O,to)

)I,t ) =

which i s regular i n

(Xoyp,t)

i s given by:

Loto e xo +B(Xo,Xo)

+

+AIXo + (t-to)A;Xo

.

The Taylor

2

73

Hopf b i f u r c a t i o n i n lR

where

is

BP””

r - l i n e a r symmetric i n

B2 ,

and

i

This givks t h e p o s s i b i l i t y t o e x p l i c i t e l y use t h e map

t o look f o r t h e b i f u r c a t i o n from t h e f i x e d p o i n t system (1). If t h e i n v a r i a n t closed curve by t h e r e s u l t of e x e r c i s e 2 of chapter

F

0 )I

FLL

:

of a c y c l e for t h e

i s i s o l a t e d , then following

I, t h i s w i l l b e a c l o s e d t r a j e c t o r y f o r

(1)*

For t h i s study, we have t o assume t h a t +koo -

.

with

s(0) = 0

For

p

near

,

0

,

~ ( 0 =coo )

L

To f i n d t h e eigenvalues of

(9)

A

P

has two conjugated eigenvalues

has t h e eigenvalues

P

>

Lo

.

0

D F (0) = A * P P

= eLoto + p A 1 + O ( p )2

.

,

we remark t h a t

74

Bifurcation of Maps a n d Applications

The eigenvalues of eLoto

kn0 # 1 for n

=

are

iwoto

A

= e

0

1

and

.

1,2,3,4 (this is true for almost all

to to realize this). Writing A(,)

(10)

= Xo(l +P

hl)

2

+ O(P

) = e

Y(I4tO

,

we obtain

and the Hopf-condition H . 2 becomes

Let us define the eigenvectors of A

where A*

P

(1’1 1

is the adjoint of A

xo =

z

c(,)

+

z’ S ( d

will be still written F (z) 1L

.

12

Now we write

9

.

IJ. : < ( p )

We have

,

c(p)

We can assume that to

, and we choose


2

75

and by i d e n t i f i c a t i o n :

where as i n .paragraph 1, Ak(p,z)

z i z j (i + j = k)

of

We need t o compute e

Lit

*

-iw0t

co=e

*

,

Y

, 5.

Sij = s i j ( 0 )

.(p)

1J

.

, I , , , Zo2 , t2,

due t o the f a c t t h a t

Lot * Lot (e ) = e

,

being t h e c o e f f i c i e n t s i n

For t h i s we can use

**

*

LOCO =-i0 0 50

Let us note

then

Now, Lo? Lor NO’*(e cO,e c O ) d T = 0

+-

-

C

3uu0

2iws

(e

0

-e

-iws 0

2kos

a (e lu! 0

15,

3

0

- e

icus

and t h a t

76


gives us

+2ab (1- A o )

(2x0-1)

2

-

(I)

(I)

0

0

We can now compute t h e p r i n c i p a l p a r t of t h e i n v a r i a n t c i r c l e :

where a. i s given by formulas (15) and (l2) of paragraph 1. We have

a = - Re(cY(o)io) , ReAl

which i s independent of

to

=

clto

, (assume a # 0) ,

.

The expression (22) has i t s p r i n c i p a l p a r t e x p l i c i t l y known, and independent of

to

.

Moreover, because of t h e uniqueness of t h i s i n v a r i a n t closed curve f o r F

c1

, t h i s i s i n f a c t a c l o s e d t r a j e c t o r y f o r t h e system (l), s o a p e r i o d i c

bifurcating solution. It i s now e s s e n t i a l t o be a b l e t o compute t h e p e r i o d of t h i s p e r i o d i c solution.

Hopf b i f u r c a t i o n i n ii?

L

77

To do t h i s , we have t o consider t n e angular p a r t of

where

go =

1

= 211 -1u t

e

-a t 211 0 0 '

, Real

i n v a r i a n t closed curve i s expressed a s in

T

1

.

51to

r = r(cp, p)

:

12

So, when t h e

i n ( 2 4 ) , we have a map

.

$ = g(tp,p,to)

: 'p*

=

F

The n a t u r a l idea t o o b t a i n t h e period would be t o consider

t

t o look f o r assumed

moto =

e 231

i r r u0

such t h a t 0

#

n =

1 for

.

+ O(p)

.

g(O,p,to) = 1

and

The t r o u b l e i s here t h a t we

1,2,3,4 , and t h a t

So we look f o r

cp = 0

toE(0,25r/ho)

g(O,b,to)

= 1 leads t o

9P

and consider t h e map

which a l s o have t h e same i n v a r i a n t c i r c l e and corresponds t o t h e map (7) with

5t0

,

9P

The angular p a r t of

For cp = 0 , t h e equation with r e s p e c t t o

.

to

i s noted

!b5)= 1

g5)

,

and we have

gives t h e period

T = 5t0

, by s o l v i n g

We o b t a i n

Exercise l e f t t o t h e reader. (i) Compare t h e r e s u l t s obtained here i n (22), (23), (26) with those

obtained by t h e method of Lyapunov-Schmidt [17]

, [29l ,

[121

(ii)Show, using t h e study of paragraph 3 , t h a t t h e expansion of contains powers of Hint: obtain

assume

p

(no

t o €(0,211/m0)

T =nto

.

I

for instance). with

n

l a r g e , and s o l v e

.

$

6.)

only

= 1

to


78

Remark on the s t a b i l i t y . We know, by t h e g e n e r a l theory t h a t t h e b i f u r c a t e d closed curve i s

>

a t t r a c t i v e i f it b i f u r c a t e s f o r

0

.

I n f a c t , we have more: f o r any

i n i t i a l data c l o s e enough t o t h i s closed curve, t h e r e e x i s t s a “ l i m i t phase”

6

such that

llx(t) exponentially, where and

Xo(t,p)

- X0(t+6,P)ll X(t)

+

0

t + w

i s t h e s o l u t i o n of t h e e v o l u t i o n problem (l),

the bifurcated periodic solution.

For t h e proof, s e e chapter

11.3. 2 -dimensional i n v a r i a n t t o r u s f o r a non-autonomous d i f f e r e n t i a l equation.

5 . Bifurcation i n t o a

We consider t h e d i f f e r e n t i a l equation i n R

where we assume t h a t

L

and

N

2

.

are sufficiently regular i n

depend p e r i o d i c a l l y , with a p e r i o d

T

, on t h e v a r i a b l e

t

.

p,

t ,X

and

We assume

also that N P( t , X ) = N 0 , z (t;X,X) +No,3(t;X,X,X) + I J . N ~ , ~ ( ~ ; X , X+ )O(I/X1/*(1

(2) where

2

and

It1

5

T

Z(Xo,p, 0) = Xo

f o r i n s t a n c e , where

.

X(X0,p;)

The fundamental matrix

+

liXll)2)

.

are q - l i n e a r symmetric i n IR P>9 We have a map i n R2: X o H X(XO,p,t) , which i s d e f i n e d f o r N

PI

Xo

near

i s t h e s o l u t i o n o f (1)w i t h Sp(t)

satisfies :

0

,

Hopf b i f u r c a t i o n i n R 2

Then we have

+I

,

X( 0, p , t ) = S ( t ) P

0

(4)

X(X(Xo, k T ) , P , t )

79

and because of the

=

X(xo, P, t + T )

T

- periodicity

.

To study the asymptotic behavior of the t r a j e c t o r i e s near

, we can study

0

where

,

S (T,s) = S (T).S-l(s) P P P

NP(tZ) =

N(*)(t$T) c1

+ NP (3)(t$,X,X)

Now t o e x p l i c i t a t e t h e map i n E2 SP(T,s)

.

, we need t o know more on the a d j o i n t of

, because of a l l the s c a l a r products t o be done f o r projecting ( 5 )

on a b a s i s . Lemma 5 .

+ o(llxl14)

We prove the following lemma:

*

[ S P ( t ) ] = ['s,(t)]-l

where

( t ) i s the fundamental matrix of the

P

l i n e a r system:

(61

E-: - -

-... -.-

N.B.

*

LP(t)X

__ .-

-ic

, where L ( t )

The solutions of (6) a r e

P

i s t h e a d j o i n t of

X ( t ) = gU(t)XO

.

LP(t)

.

80


Proof of Lemma 5 .

Let us consider t h e s c a l a r product

sp(~)[8p(t)l-1y) for

x

and

w

sp(T)[sp(t)l-lY) + (sp(T)X

I

,

i n pi2

Y

f(T)

=(Sp(7)X

,

f ' ( 7 ) = (L ( T ) S ~ ( T ) X,

then

- L ~ ( ~ ) ~ , ( 7 ) [ 8 ~ ( t ) ] - l=Y0)

.

P

Hence

f(o) = f ( t )

and

Let us do now the necessary assumptions t o g e t a Hopf b i f u r c a t i o n f o r t h e map

F

w

(5) when

The eigenvalues

crosses

p

h(p)

,

x(p)

.

0

of AP a r e t h e Floquet m u l t i p l i e r s of

t h e l i n e a r i z e d system from (1). We note A A

P

= S

P

(T)

* .=. ,Sp(-T) P

for

p

near

, such t h a t

To o b t a i n t h e map

,

0

and

(c(p),c

F

P

in

*( p ) ) c ,

t h e eigenvectors of

< ( p ) , t) = W(+cp)

where W

i s Z2 p e r i o d i c .

+

Po

t $

,

The lemma i s then proved, s i n c e

Po$ Q

.

.

2

85

Hopf bifurcation i n iR

Comments. The r o t a t i o n number

.

p(o) = Q o

that

either

(i)

=

Q0

p(p)

of

?

P

i s a continuous f u n c t i o n of

p

such

Then t h e r e a r e two p o s s i b i l i t i e s :

&Q 9

and t h e conditions of theorem

( s e e paragraph 3 ) a r e

4

r e a l i z e d , and t h e r e a r e two one-sided b i f u r c a t e d p e r i o d i c s o l u t i o n s , t h e period of which i s

qT

.

Only one of t h e s e p e r i o d i c s o l u t i o n s can be

a t t r a c t i v e , and only when the b i f u r c a t i o n occurs f o r (iij

o r , i n t h e o t h e r case

p(p)

p

>0

.

i s n o t c o n s t a n t i n a neighborhood of

0

.

I n t h i s case, when p ( p ) E Q t h e r e a r e some p e r i o d i c s o l u t i o n s corresponding, a s f o r ( i ) , t o closed t r a j e c t o r i e s on t h e t o r u s considered a s b i f u r c a t i n g t r a j e c t o r i e s . po

.

Now, when

P(po)

4Q

3P

, b u t they a r e not

The period depends of course on

, t h e r e g u l a r i t y of t h e diffeomorphism allows

us t o use t h e Denjoy theorem ( s e e paragraph 3 ) and we can then use

3c1 .

Lemma 6 t o s a y t h a t we have a q u a s i p e r i o d i c s o l u t i o n on t h e t o r u s

The i n t e r e s t i n g f a c t , i n t h e case ( i i ) ,i s t o know f o r what values of we have

p(p)E

of M. Herman [

of

p

Q or $ Q ! An i n d i c a t i o n f o r t h i s i s given by t h e result

9

such t h a t

] which l e a d s t o t h a t i n a neighborhood of P(p)k

, the s e t

Q has g e n e r i c a l l y a p o s i t i v e Lebesque measure.

more c l a s s i c a l r e s u l t i s t h a t t h e s e t of general

p= 0

a p o s i t i v e Lebesque measure too.

p

such t h a t

P(p)E Q

A

has i n

So we cannot ignore t h e s e both

p o s s i b l i l i t i e s i n order t o i n t e r p r e t experimental r e s u l t s .

6.

B i f u r c a t i o n i n t o a two dimensional i n v a r i a n t t o r u s f o r an autonomous d i f f e r e n t i a l equation. L e t us consider t h e d i f f e r e n t i a l equation i n R3

2 dt

= Gp(X)

,

p

86


s u f f i c i e n t l y r e g u l a r , and assume t h a t

G

tb+ Xo(t,p)

is a

T(p) - p e r i o d i c

s o l u t i o n of (1). Let us put,

then (1) becomes

where

L ( t ) i s a l i n e a r operator and P

t

T(p) -periodic i n

B ( t ,* ) P

a non-linear one, which a r e

, and such t h a t

Let us c o n s t r u c t the Poincare' map as i n d i c a t e d i n c h a p t e r 11.3. S ( t ) and we know t h a t 1 i s a n eigenvalue of

matrix i s noted

P

The fundamental S [T(p)] P

.

We assume now: S 0[ T ( O ) ] has t w o eigenvalues

[".' Then f o r

xo#

0

p

5 such t h a t

*

S&T(p)]

(p) t h e eigenvector of

S&T(p)]

(c(p),
2 .

A.

= e2in/3

, and

The property (13) l e a d s t o

can be calculated d i r e c t l y from

F

P

:

Fp(z) = h ( p ) z + A2(p,z) + o ( l z I 2 )

, if

k 1 2

,

.

Let us write


110

The equation (11)gives us

so

a,(o)

of ( 1 2 ) .

where

.

xoSo2

=

See i n exercise 1 t h e way t o f i n d quickly a l l c o e f f i c i e n t s

gf Ck

,

a =a,(o)

I f we w r i t e

2incp

z = r e

and

, g ( ~ , r , c p + 1 / 3 )= g(P,r,rp) =

Theorem 1. Assume t h a t

F

i s of c l a s s

assumptions H.l, H.2 hold, and t h a t when we w r i t e

F

P

in

, (16) takes here the form:

2

O(r()pI +r) )

Ck

,

k

23

.

near

.

h3 = 1 , Lo# 1 0

, that

0

Assume t h a t

the

s,#

by choosing a good b a s i s for the l i n e a r p a r t .

C

0

,

Then

t h e r e exikts a s i n g l e one-parameter family of fixed points of order 3 b i f u r c a t i n g from

0

:

[(cL(E),

1 functions

p , rp

2 x = z ( € )+ O ( € )

x(z(E),B(E),c))

and

, 14

x =

are

,5

Ck-2

I--IE+O(€

hl

; z(c) = c

2

e2iflCP(S)]

,

i n a neighborhood of 0 , with 1 arg(-)mod 0 1/3 + O ( s ) ) , rp(s) = -

-v 5,

0

.

, becomes unstable f o r

p

6

near

The f i x e d

>0 ,

b i f u r c a t e s on both s i d e s

Subharmonic bifurcations

Proof: (22 1

The equation (14) d e f i n e s a change of v a r i a b l e s of c l a s s x1 = y,(z)

such t h a t i f

P

c g (p,z,Z) q=l

Ck

:

,

i s a f i x e d p o i n t o f order 3, then

x1

= X0z +

P

-1

Hence, t h e f u n c t i o n

Fp

yp

n-1

Z

q=l

yp =

Xz-'

P

F

gq(p,z,z) = y ( h z)

,

c 1 0

i s such t h a t

.

= xoz

P,(z)

n-1

z+

F [ V ( z ) ] = x2

(23)

111

But, t h e equation (23) i s equivalent t o (11) (we choose one equation i n s t e a d of a combination, and it is independent of t h e

(n-1) o t h e r combinations

.

-

1 a l r e a d y considered t o f i n d t h e g (p,z,z) = - 0 ( p , n z , n z ) ) I n our c a s e 9 n q we have found t h a t (11)g i v e s ( 2 0 ) . Hence t h e new map, a f t e r t h i s "change of v a r i a b l e s " i s w r i t t e n as

To look f o r f i x e d p o i n t s of order 3 , i t was seen t h a t we have t o look for solution

z of ( 2 3 ) , and then

,

z , koz

0

z

w i l l be t h e 3 fixed points

of t h e family. Assuming t h a t Eliminating t h e cp

=v0 +vl

(25)

9

3cp0 =

(l+P1)

5,,#

, we o b t a i n an equation of t h e form ( 2 1 ) .

0

0 s o l u t i o n , and w r i t i n g

-e

1

-h I

ad-

-6i r r cpl

52 o +

p = el-I

)(mod 1)

gl(wl,cpl)

,

= 0

52 0

(1+ p),

X1 we o b t a i n

3

,

r = e

,


112

k-2 glE C

with

Because

and

gl(O,pl,cpl)

function theorem.

where

gl(cypl,(pl+

hE C

k-2

1/31

, we can solve (25) f o r

= 0

, h(O,O,O)

= 0

(17).

by t h e i m p l i c i t

(%,(pl)

,

, cpl(e)

: %(c)

=6irr

~ah ( O , O , O ) = 1 ,-(O,O,O) ah

%

always gives a unique s o l u t i o n Ck-2

a s i n d i c a t e d by

Writing ( 2 3 ) i n t h e form

The d e r i v a t i v e i s i n v e r t i b l e :

class

,

= gl(s,pl,(pl)

pl,cpl

.

*l

i n R2

which solve ( 2 5 ) .

,

hence

If

F

3

two f u n c t i o n s of

i s a n a l y t i c , we f i n d

by t h e a n a l y t i c version of t h e i m p l i c i t f u n c t i o n theorem, a n a l y t i c functions gq

and a n a n a l y t i c map

functions

pl,V1

a. .

F

P

, hence a n a n a l y t i c gl

i n ( 2 5 ) ,. and a n a l y t i c

a

The s t a b i l i t y of t h e f i x e d point

0 comes from t h e study o f chapter I.

Let us now study t h e s t a b i l i t y of t h e b i f u r c a t e d f i x e d p o i n t s of order 3.

For

t h i s we cannot use t h e "change of variables" (22) because i t i s adapted t o t h e search f o r t h e f i x e d p o i n t s of order 3 , and i t i s not n e c e s s a r i l y a good change of v a r i a b l e s f o r t h e map i n whole a neighborhood of

0

.

I n s t e a d we

use here t h e form (3) which we found by t h e method of chapter 111. We f i n d

which leads t o

Subharmonic b i f u r c a t i o n s

113

Now, l e t us consider

where

X(E)

i s one of t h e 3 f i x e d p o i n t s

#

0

.

The new map i n

2'

can be

written a s :

Hence t h e d e r i v a t i v e a t t h e o r i g i n can be represented by a matrix

Id +3&

where

II

The eigenvalues of

I

Ul

*u2

=

A

, o1 and o2 a r e such t h a t :

-3ko21

2

of the r e a l p a r t of the eigenvalues i s the same a s

plRe X1+

2 Re a1

gives a l l the r e s u l t s of theorem 3, because t h e eigenvalues of

L = 1,2

O&

Study of the case

.

h: = 1

, then

o1*u2 > 0 f o r t h e other branch, and t h e sign

0 f o r one branch and

1 + 42uk ~ +

0

n

25

DF'(0) c1

.

This

are

( s e e chapter 111.3 t o o ) .

The equation (12), i s now of the form = 0

CLhlZ+f(CL,Z,Z)

with

f(p,z,Z)

If we put

(45)

=

2

2-n-1 al(p)z z + a 2 ( p ) z +a3('i)z3i2 +O(lz/"+ )p121z1+1z17)

p = e pl(c)

%kl

9

,

+ a1+a2E

Z = E

e

n-4 -2nincp e

, we obtain the equation: + O ( E2 ) = O

.

.


122

i s not r e a l it i s impossible t o o b t a i n a b i f u r c a t e d family

Hence, if al/kl

of f i x e d points of order

a1 Tm-=O

If

,

.

then f o r

.

n

, it s u f f i c e s t o assume

n=5

#

a2

0

t o i n s u r e t h e e x i s t e n c e of

one branch of f i x e d points of order 5 o f t h e form )I=

'p

For

n

- e

3

F:

hl

= (Po + O(E) mod

2 6 , it

r

--

2 a1

i s necessary t o assume more on t h e c o e f f i c i e n t s of

method c o n s i s t s of p u t t i n g reasoning.

Theorem

4.

p

(B)

1

=

-$+ &

1

We can then say: Assume t h a t

F

i s of c l a s s

Ck

assumptions H . l , H.2 hold, and t h a t

6

2

.

f

The

and i t e r a t i n g t h e preceding

p2(c)

,k 24

near

0

, that the

kn0 = 1 , n > 5 ( A o # 1,-1)

.

Then,

i n general, t h e r e does n o t exist any b i f u r c a t e d branch of f i x e d p o i n t s of order

Remark.

n

near

0

.

According t o chapter 111, t h e r e i s a n i n v a r i a n t c i r c l e

b i f u r c a t i n g from

0

p o i n t s f o r values of v a r i e s with

p

.

The r e s t r i c t i o n map )I

f

P

on

a s c l o s e a s we want from

yp 0

yP

for

F

) I '

may have p e r i o d i c

, b u t t h e i r period

and they are not considered as subharmonic b i f u r c a t i o n s .

According t o t h e study of chapter 111.3 (see theorem

4 of

t h i s chapter),

we know t h a t if a f i n i t e number of a l g e b r a i c conditions a r e r e a l i z e d we have a "weak resonance", i . e . two f a m i l i e s of f i x e d points for i n v a r i a n t c i r c l e , b i f u r c a t i n g from

0

.

Fn

P

,

on t h e

The method used here g i v e s a n

o t h e r way t o compute e a s i l y such weakly resonant s o l u t i o n s .


123

Exercise 1. How t o quickly f i n d the c o e f f i c i e n t s i n equation (12): L e t us put

x

= y (z)

P

=- z +

c

g (p,z,Z)

q=l

where

We s e e by ( 2 2 ) - ( 2 3 ) t h a t i t i s possible t o f i n d

g E Ck 9

such t h a t the

equation FJYP(Z)I

becomes

Yp(AoZ)

=

3

, with f ( p , k o z , i o z ) = h o f ( p , z , z )

xoz = A(p) z + Xof(p,z,z)

.

L e t us write

(i) Assume give (ii)

Assume

0

= 1

.

Calculate

up t o the order

f

and give (iii)

A.3

4=

h

0

f

1

.

g

q ’

(zI3

Calculate

q = 1,2 up t o the order

g q = 1,2,3 up t o the order q ’

1zI3

,

4 . IzI

up t o the order

I n both cases, the assumptions of theorem 1 o r 3 being r e a l i z e d , c a l c u l a t e

s

of the fixed p o i n t s .

Subharmonic b i f u r c a t i o n f o r a non-autonomous d i f f e r e n t i a l equation. Let us consider t h e d i f f e r e n t i a l equation i n R

(1)

, and

.

t h e two f i r s t terms of t h e s e r i e s i n powers of 2,

iz13

= L (t)X at iL

+ Np(t,X)

,


124

where

N ( t , . ) a t l e a s t quadratic, with the properties

i s l i n e a r , and

LP(t)

of r e g u l a r i t y and

P

T - p e r i o d i c i t y assumed i n chapter 111.5. Define

F

P

and

a s there.

Sp(T) = DF (0) I-I

Assume now, t h a t

hz = 1

*

,

with

n

23

,

(xo#l,-l)

)

and denote by

So , 5 0 the eigenvectors such t h a t

We define

C0(t) = e then

co( .)

is

-ilin8 ot/T

, where h 0

So(t)co

=

2 in8 e

,

T - p e r i o d i c and s a t i s f i e s :

Now l e t us consider

c*o w

= e

-2inOot/T

-S o ( t ) c *o

where

[ ~ o ( t ) ] - l = [ S o ( t ) ] * (see Lemma 5

of 111.5) then

*

Go(.)

is

T

periodic and s a t i s f i e s

These properties lead t o a means t o c a l c u l a t e very quickly and d i r e c t l y the map

(61

, which

FE(Xo)

i s one goal here:

Fn(X ) = S (nT)Xo + B(n)(Xo.Xo) P

O

P

P

+ C(n)(Xo,Xo,Xo) + O ( l \ X d l 4) P

,

S u b h a r m o n i c b i fu r e a t i o n s

125

where

and

has an expression of t h e type w r i t t e n i n chapter 111.5,

Cp)(Xo,Xo,Xo)

b u t with

nT

i n s t e a d of

Let us c a l c u l a t e

Xo = z

so(s)x0

5,

f

T

.

Bo")(Xo,Xo) (

5

= z e

co

We have t o w r i t e

, and

2 i n 8 ,s/T

We have also t o p r o j e c t

.

(p=O)

cob)

+

e

-2inBos/T

-

5,(s)

~ ~ ~ [ s ; S o ( ~ ) X o y ~ o (ons ) X 5, o ~and

*

to , hence

it

occurs q u a n t i t i e s l i k e

We have

Now, t o c a l c u l a t e

where

f

is

B(n)(Xo,Xo) 0

T - p e r i o d i c , and

t o t h e f a c t o r considered.

w e o b t a i n i n t e g r a l s of t h e following t y p e :

B o = p/n

, and

3 , k2

k = +1 3 Writing t h e F o u r i e r s e r i e s of f :

,

according

126


f(s) =

C

kEZ

f,- e

2ink s/T

we obtain i n t e g r a l s over

,

[ O,nT]

LT2 i n s

of terms a s

P(%+ k2- k3)

e

+ nkl

.

To

obtain a non-zero c o e f f i c i e n t we have t o consider t h e terms where P(kl+k2 - k3 ) + n k = 0 because

p

, i . e . terms where % + k 2 - 3

i s prime with

Hence, i f

nf3

n

i s a multiple of

n

,

.

, Bt)(Xo,Xo)

= 0

,

and i f

n = 3 we have t o consider

k = k = k = + 1, which gives here 1 2 3 BF)(Xo,Xo) =

Exercise 2 :

3.

CY.

i2c0 +a z25,

, where

Calculate i n t h e same way

can be e a s i l y c a l c u l a t e d .

a!

.

CLn)(Xo.Xo,Xo)

Subharmonic b i f u r c a t i o n f o r an autonomous d i f f e r e n t i a l equation. Let us consider t h e autonomous d i f f e r e n t i a l equation i n R3

of t h e chapter 111.6, and assume again t h a t t h e r e e x i s t s a s o l u t i o n of (1). Then we construct t h e Poincare' map and assume t h a t i t s d e r i v a t i v e hn(o) = 1 , n

that

13 ,

A

P

A(o) # + l

a s i n chapter 11.3 h(k)

,

such

x(p)

. Fn

P

a s i n paragraph 2 , t o

c a l c u l a t e quickly the c o e f f i c i e n t s i n A. (2)(Yo,Yo)

, A. (31 (Yo,Yo,Yo) because

which i s t h e analogue of

[ O,nT]

m e must take account of the f a c t t h a t i n t h e expression of

Fn b e

Fn we have i n t e g r a l s o v e r P

i n $2.

P

has two eigenvalues

We can make t h e same c a l c u l a t i o n s f o r

in

F

T(p) - p e r i o d i c

use a time of

th

"n-

return":

[ O,nT(p)]

7,(Yo,n)

2

= nT(P) + O(l/Yol( )

.

P


4. 4.1.

Relation with the paper of ARNOLD

[I]

and comments.

V . I . ARNOLD considers i n [ 1 1 a d i f f e r e n t i a l equation i n

by r o t a t i o n s of angle

2n/n

about t h e o r i g i n .

equation i s supposed t o approximate our map the case when

n

ho

= 1

The time

Fn

P

.

too d i f f i c u l t t o j u s t i f y f o r the d i f f e r e n t i a l equation. Fp

, because of the assumption Reh 1 >

having an invariant c i r c l e a t a distance p i c t u r e i n the case

kt

= 1

for

F

12

, invariant

1 map f o r t h i s

kt

= 1

, a r e not

I n the case of the

0 we exclude t h e p o s s i b i l i t y of

O(p)

from the o r i g i n . The corresponding

seems t o be:

Fig. 2: invariant manifolds i n the case

C

up t o high order terms, i n

The pictures he gives a t Fig. 2 of [ 1 ] i n the case

map

127


128

The curves a r e supposed t o be t h e i n v a r i a n t manifolds ( t h e s t a b l e and t h e u n s t a b l e one) a s s o c i a t e d with t h e h y p e r b o l i c p e r i o d i c p o i n t s ( s e e c h a p t e r V f o r t h e i r e x i s t e n c e and computation).

X4 = 1 , t h e p i c t u r e s at F i g . 3 of

For t h e case

Nevertheless t h e r e c e n t r e s u l t s of Y.H. WAN

[11 beem

only t o be c o n j e c t u r e s .

and F. LEMAIRE [23]

b3]

give t h e

d e s c r i p t i o n of what happens i n t h e case ( i )of theorem 3 ( t h e r e i s no f i x e d p o i n t

of o r d e r

4 bifurcating

i n t h i s c a s e ) . The r e s u l t i s i n f a c t t h e same as t h e Hopf

b i f u r c a t i o n d e s c r i b e d a t c h a p t e r 111. The map has t h e form

(1)

~

~

(

=2 i ) [(l+p~,)z + a l z22 + a2

11, a21

where it i s assumed t h a t ReX1 > 0, Re a

1

"31

+ 0(lpl2

1 and

b

perty of the technical lemma of chapter I, f o r a small enough 1. The hyperbolic case.

Theorem 1. Assume

F

t o be

C1

Then there e x i s t i n a neighborhood of

3and

or

Ck'l

0 in

E

for

6 .

is locally attracting.

36 >

0

Moreover,

is

such

m2

Remark 1. We can imagine t h e s i t u a t i o n i n f i g u r e 1

Figure 1. (of course t h e continuous t r a j e c t o r i e s mean nothing! u n l e s s t h e map comes from a d i f f e r e n t i a l equation.) Remark 2 .

If

F

is linear

(F=L)

, a l l t h e r e s u l t s of theorem 1 a r e

obvious

Proof of theorem 1. Following t h e t e c h n i c a l lemma of chapter I, we can choose t h e norm i n

E

b

3

=

El@ E2

Example 2 . Let us c a l c u l a t e the quadratic p a r t s of

and

cpl

'p2

.

For t h i s we

write

R(x) = R ( 2 ) ( ~ , ~+ )O ( ~ / X ( / ~ ) where

,

R (2) i s b i l i n e a r , symmetric, bounded i n

cpj(x) = YS')(x,x!

+ O(llxl13)

We proceed by i d e n t i f i c a t i o n :

, xE

Ej

E

, and i n a s i m i l a r

way

,

145

I n v a r i a n t m a n i f o l d s and a p p l i c a t i o n s

This leads t o :

where the s e r i e s converge i n norm. dimensional, of dimension

then, t o know

cp2(2)(ui,uj)

vectors

the

,

n

E2

I n f a c t , i f f o r example

where

{u,]

vi2)

is finite

it s u f f i c e s t o know

i s a b a s i s of

E2

; and we

know t h a t (35) gives an i n v e r t i b l e l i n e a r system f o r t h e s e vectors. case when

L2

i s diagonalizable i s p a r t i c u l a r l y simple: assume

The

L u =L.u 2 j ~j

then

where we observe t h a t

h.h

l j

i s i n t h e open u n i t d i s c .

-

L1

i s i n v e r t i b l e because t h e spectrum of

I n the case when

L1

L 2 i s not diagonalizable, use

t h e Jordan decomposition t o show t h a t it i s always possible t o f i n d

v2(2)(ui,uj) 2.

f o r a good b a s i s

{uil

.

The c e n t r a l case. We s h a l l say t h a t t h e Banach space E2

e x i s t s a function

of the u n i t b a l l

well known t h a t i f

P

of c l a s s

E2(l) , and E2

C‘”:

E2 -’R

has t h e property i-

,

such t h a t

P = 1 inside a b a l l

i s f i n i t e dimensional, o r i f

space then i t has the property

(P),

.

E2(6)

(P),

outside

P = 0

,

6

>

i f there

0

.

It i s

E2 i s a r e a l H i l b e r t

’


146

!I

Theorem 2 .

(Center manif old theorem)

Assume F and

b l

Lkp2(0) = 0

that a l l its iterates

.

(P)k

i n general, which i s graph o f

of c l a s s

and

i n a neighborhood of

FE C1 ) .

( k = O if

Assume that the subspace E2

21 ,

k

.

.

F

F :if

,

nEN

x

such

are i n a certain fixed

dist(Fn(x),%)

, these

i s a point i n E

+

n+

0

.

r e s u l t s are obvious:

.

$=E2

Proof of theorem 2 . Let us choose the norm i n \/L1\\ =bl

-1

, \\L2 \ \

=b2

E=E1@ Ep

, \ \ P j \ \=

such t h a t 1

,

bib:
2

.

?ili s l o c a l l y i n v a r i a n t and l o c a l l y a t t r a c t i n g under

H

in WXE

.

Let

us show t h a t

For t h a t we consider t h e f u n c t i o n a l space which occurs i n t h e proof of theorem 2 :

The subspace of such

'p

satisfying

cp(p,O) = 0

t h i s subspace i n i n v a r i a n t under t h e map paragraph 2. of lemma 2

I n fact, if

9

i s closed i n A'

M

.

Moreover,

defined by t h e equation (9) i n

4, belongs t o t h i s subspace, we have, by d e f i n i t i o n


160

Now, f o r a f i x e d

i s a manifold

under

F

P

M

, by

p

,

construction.

Taylor expansion of We note f o r

and

Ck-l, 1

of c l a s s

P

L e t us assume t h a t

where A1

t h e graph of

[p

k

23

, l o c a l l y i n v a r i a n t and l . o c a l l y a t t r a c t i n g

This manifold i s not unique i n general. a t l e t us compute t h e terms of order 2 i n t h e

(at least i n

C2'l

1:

x€ E

[pll

a r e bounded l i n e a r o p e r a t o r s :


161

By t h e formula (20) of paragraph 2 we obtain ( o ) ~ ( L ( 0 ) ) -1 = L:o)A (L(0) 11 2 1 2

(11)

'pll-Ll

(12)

CpO2(*>*)

(01-1.

(0)

-L1 'Po2(L2

)-lc

Wl.) =

YL2

~ E L o ;E!o)) )

pp)F 02

(&0)-1. ( o ) - l . ) 2 L2

The equations (11) and (12) allow t o f i n d t h e operators

i s a s t r i c t c o n t r a c t i o n i n E (') 2 we have t o take t h e t r a c e of t h e map F

.

LL0)-'

in

ELo)

(13)

P

on

P

because of the parametrization (10) of G P : xE ELo)&+ PLo)F,[x+q(w,x)]

We need t o e x p l i c i t

l e t us assume t h a t

so, we o b t a i n f o r

G

k

c1

and M

'po2

P

. because

i s known,

, i . e . we s h a l l f i n d a map M

P

:

.

t o use t h e results of chapters II,III,IV. For t h a t ,

24 ,

xE E2( 0 )

with obvious n o t a t i o n s .

'pl1

Cklce t h e manifold M

,

and w r i t e


162

It can be u s e f u l too, t o express t h e t r a c e of t h e map

i n t h e i n v a r i a n t subspace a s s o c i a t e d with t h e p a r t for

p

P2(p)E = E2(p) 02(p)


0

F

v

by t a k i n g a parameter

, where P2(p) i s t h e p r o j e c t o r

of t h e s e p a r a t i o n of t h e spectrum of

.

A

CL

me new v a r i a b l e i s

The new map

where

x

+

GI

: E2(p) + E2(p)

i s now

i s expressed by formula (18).

Hence, f o r

x ' F E2(p)

we o b t a i n


hence

We note on (21) t h a t the d e r i v a t i v e a t of --

A

P

to

_____

Lemma 3 .

~ ~ ( p, )s o i t s spectrum i s

The manifold

space E2(p) L.-

0 of

at

0

.

M

P

02(p)

G'

P

i s e x a c t l y the r e s t r i c t i o n

.

(non-unique i n general) i s tangent t o the sub-

-__I_

Proof:

We can write an other form f o r

By construction cp'(p,O) = 0

M

P

:

, and we want t o show t h a t

I n f a c t , we have by construction

163

164


hence

.

D x l ' P ' ( ~ , O ) C 4E2(p);El(p))

where

Now (26) l e a d s t o D x , ' p ' ( p , o )

because t h e f u n c t i o n a l equation (26) i s such t h a t in

and

E1(p)

P2(p)A

E (p) 2

has a n i n v e r s e on

P

A

= 0

i s a s t r i c t contraction

P

of s p e c t r a l r a d i u s c l o s e t o 1

(analogue of equation (21) of paragraph 2 ) . Remark.

The maps

G

P

and

G' P

uniqueness of the manifold M

a r e not unique i n g e n e r a l because of t h e non-

P

.

I n f a c t t h i s t r o u b l e i s i r r e l e v a n t because

of t h e following Lemma 4 . Assume t h a t , f o r under

F

P


p

0

, b i f u r c a t e s from the f i x e d point

on t h e choice of

M

,

a set

i n a s u i t a b l e topology), and assume t h a t on any

M

P

i sdo l a t e d . i nThen E.

Proof. lpl

By assumption

.Y,, 3

the s e t

M

w

y

P

is

and it i s

'

a neighborhood

@

of

in

0

E , such t h a t f o r

small enough, we have

Hence

Fn(y ) P

P

C (9

f o r any

nEN

, and by t h e c e n t e r manifold theorem 2,

belongs t o any c e n t e r manifold (closed s e t i n manifold M'

P

to M

: E2( 0 ) +El( 0 )

cp(p,*)

does not depend on t h e choice of I

, invariant

0 and depends continuously

(more p r e c i s e l y it depends on

P

yP

CL

in

@

, t h e r e corresponds ; then

y'

w

y'

CL

belongs t o

E

close t o M

P

too.

n@).

yP

To another c e n t e r

M' i s c l o s e CL So, t h e r e i s a c o n t r a d i c t i o n

y

P

because


with t h e assumption t h a t

y

P

i s i s o l a t e d on

M

P

165

Y’ =

unless

P

yP

.

This lemma w i l l be u s e f u l t o prove the uniqueness o f t h e b i f u r c a t e d

N.B.

p e r i o d i c points o r closed curves t h a t we w i l l f i n d on the manifold Cases when

o (01 2

Assume t h a t

only contains simple eigenvalues of l o €oio’

such t h a t , i f

We note

F € Ck

,

l(p)

P

.

.

0

i s a n i s o l a t e d simple eigenvalue of

Thanks t o t h e p e r t u r b a t i o n theory, t h e r e e x i s t s Ap

A

M

A.

,

lhol = 1

simple eigenvalue of

and i f we w r i t e

*

5, , 50 t h e eigenvectors of

A

0

and A.

*

( a d j o i n t of

Ao)

such t h a t

Then = Re X1 p=o

Remark. and t h a t of

A

P

stable.

If we assume t h a t

’0 ’:

c o n s i s t s with simple i s o l a t e d eigenvalues

0 f o r a l l t h e s e eigenvalues, t h e n f o r

i s i n t h e open u n i t d i s c , hence t h e f i x e d point For

p

>

0 we a r e i n t h e hyperbolic case and

over t h e r e e x i s t s a n unstable manifold %(p)

0 0

0

t h e spectrum

i s asymptotically i s unstable.


0

More-

,

.


166

which i s tangent t o

at

E2(p)

.

0

A c a r e f u l examination of t h e proof of

theorem 1 shows t h a t the neighborhood of

t o exist is

O(p)

.

Hence

0 in

E

where

may coincide with

%(p)

%(p)

i s proved

only i n t h i s

M I-L

l i t t l e neighborhood which i s n o t s u f f i c i e n t because t h e b i f u r c a t e d c i r c l e s a r e mostly

@

O(

I

) 2 ' 1

.

Then we need t o use t h e lemma

Case of one simple eigenvalue i n

u2( 0 )

4 for

t h e uniqueness.

-

Let us make t h e following assumption

I

- __

- --

H.2. a .

0")

= {do]

Moreover,

, Lo= 21 , and ho i s a simple eigenvalue of k1 > 0 (Hopf c o n d i t i o n ) , where hl

A.

i s defined by ( 2 8 ) .

The formulas (11) and (12)g i v e us

which defines t h e operators xE

Eke'

in

(15), then t h e map

and G

P

cp

02

.

Let us p u t

becomes a map i n W

GP(x) = k o ( l + p hl)x + ao2x2 + P2a 2 1 ~ + pa 2 12x

(32)

where

cpll

X1

i s defined by ( 3 O ) , and

+

.

xc0 :

i n s t e a d of


& the form

(32), we e n t e r i n t h e frame of chapter I1 t o f i n d the fixed points,

o r points of period 2, b i f u r c a t i n g from f o r the map F

@

167

IJ.

.

0

The corresponding point i n

E

, w i l l be:

Case of two simple conjugate eigenvalues i n

u2(01

.

Let us make the following assumption

[-i

0 for

an

boundary condition on

.

pE I

H = L2(n)

I n t h i s example

We

and 8 =

$(n) n

, where

0

i s t h e eigenvalue of

X(p)

such t h a t

S [T(p)] P

.

h ( 0 ) = lo _.

ThiB assumption means t h a t , f o r

p

whereas f o r

t h i s c i r c l e i s unstable because t h e f i x e d p o i n t

0

0

of

the Poincare' map i s unstable. We know t h a t

axO

i s t h e eigenvector of

x ( 0 , ~ =) Co(p)

S [T(p)] )I

for

t h e eigenvalue 1 and t h e p e r t u r b a t i o n theory of i s o l a t e d simple eigenvalues gives eigenvalues close t o

0

k(b)

and

i(p)

Sp[T(p)I

near

ho

,

Co(p)*

by

and

xo

for

p

.

We define t h e eigenvectors

We define

Let us w r i t e

of

T( p)

- periodic

C(p)

,

C*(p)

vector functions a s previously by (26) and (29),


187

then

with

(64)

m

*

A1 = T1(Lo(0)5(O)y5 ( 0 ) )

+

s

L

0

(Ll(s)60(s),C:(s))ds

0

b y the same proof a s f o r (23) i n

The simplest choice f o r

(651

x*

= C"*(O)

Let us compute the form

§111.6. t o compute

X*

T(Yo,p)

and

pp

is

. T") Ll

:for

,

Yo€

with

I n t h e same way, we g e t m

I n the following computation we s h a l l use t h e f a c t t h a t f a c t defined i n a l l a neighborhood of

0

in

i s easy t o see from t h e r e s o l u t i o n of ( 4 9 ) .

E

7(Y0,p)

is i n

, and not only on $

.

This


188

Now l e t us note t h a t the p r o j e c t o r

struction, but

P

Po

does not commute w i t h

commutes w i t h

S [T(p)] P

operating i n t h e i n v a r i a n t subspace

use t h e technique developed i n $3, c a s e @ .

by con-

.

This remark l e a d s t o t h e f a c t t h a t t h e spectrum of spectrum of So(To)

So[To]

i s j u s t the

. A

fi

.

So, we may

For t h i s we put

(69) and f o r any X

in

E

We s h a l l use i n t h e following t h e f a c t t h a t

which r e s u l t s from t h e g e n e r a l p r o p e r t i e s of t h e eigenvectors a s s o c i a t e d t o d i f f e r e n t eigenvalues f o r t h e same operator. So, l e t us compute t h e c o e f f i c i e n t s of the map

(37) i n t h e paragraph 3 :

G

P

in

C

, given by

I n v a r i a n t m a n i f o l d s a n d applications

c2,

s,,

3

i n s t e a d of

189

to, a r e given by the same formulas a s ( 3 5 ) , (37), (38), w i t h T~ T . The c o e f f i c i e n t c,, given by (39) has t o be added t o

t h e terms m

t o get the coefficient

521 of (72).

So we a r e a b l e t o compute an eventual i n v a r i a n t c i r c l e b i f u r c a t i n g from 0

.

We r e f e r t o t h e d i s c u s s i o n o f chapter i n v a r i a n t two-dimensional t o r u s t o t h e theorem

5

2 P

E

in

111.6 t o show t h e e x i s t e n c e of an f o r the f i e l d (44) and s p e c i a l l y

6 of chapter I11 f o r t h e d e s c r i p t i o n of the flow on t h i s t o r u s .

If we f i n d p e r i o d i c p o i n t s f o r

,

Gp

t h e y correspond t o p e r i o d i c o r b i t s .

For i n s t a n c e i f we have a family of 3 p e r i o d i c p o i n t s of period 3 , t h e o r b i t X(t)

of ( 4 4 ) has a period c l o s e t o

if Yo

which i s e x a c t l y

3To

i s one of t h e p e r i o d i c p o i n t s .

The s t a b i l i t y of t h i s type of o r b i t

i s t h e same as the s t a b i l i t y of t h e p e r i o d i c p o i n t s f o r t h e map Remark. of

From

(44) with

zf C

on which a c t s

the i n i t i a l data

G

P

,

we can e a s i l y compute

G

F

X(t)

. solution


190

We have

where

2

with

L ); (

part

0")

i s given by (54) and (13),

=

Plo)So(To)P1( 0 )

,

(14), (15) and

P!J,

being t h e p r o j e c t o r a s s o c i a t e d t o t h e

, which commutes wtth So(To)

of t h e spectrum of So(To)

1

,

and

A1 = gOSl + FISo(To) where

S1

Pl

i s defined by (67) and

by (TO), and

m I

P$~)(Yo,Yo)

=

0

So(TO,s)No2[ s ; S ~ ( ~ ) Y ~ , S ds ~ ( ~ ) .Y ~ ~

P1 0

The inverses subspace

[ho-Lio)]-l and [l-L1(0)]-1 have t o b e taken i n t h e i n v a r i a n t

P1(0)E

I

Let us make t h e assumptions H.l and H.2.a of sV.3 on t h e spectrum of and l e t u s divide t h i s case i n d i f f e r e n t cases according t o t h e f a c t t h a t or

-1 i s an eigenvalue of

A.

.

corresponds t o a double eigenvalue c a r e f u l l y w h a t occurs for S$T(P)I

When

1 i s a n eigenvalue of

1 for

close t o

0

So(To)

A.

A. 1

this

, s o we have t o examine

f o r t h e eigenvalues near

1 of


191

Let us f i r s t consider t h e case when. 1 i s a double semi-simple i s o l a t e d eigenvalue o f

So(To)

.

We note

, where S1 i s given by (67). By

PoSIPo

t h e eigenvalues of

3

near

[ 191

t h e form

where cz i =1,2 a r e t h e eigenvalues of i'

But we a l r e a d y know t h e eigenvalue (75)

trace

( P ~ s ~ P> ~o )

and we obtain,regular i n

W e define

CL(t)

o*

p

, c,(t) ,

,

S,(t)

SIJ.[ T(p)

PoSIPo

t o get

h,

poslpo

, eigenvalues and eigenvectors such t h a t :

*

a s i n (62) (with

, C,(t)

, noting t h a t

c(0)

co(p)

We have

with

are of

, hence t h e Hopf condition w i l l be

0

s t i l l given by (64), and

X1

>

0 because of

(75).

To

u ( o ) = 0).

compute e x p l i c i t e l y t h e condition ( 7 5 ) , we only need a b a s i s of

we diagonalize

,

1

t h e p e r t u r b a t i o n theory l + p c z i + o(p)

So(To)

Then t h e Hopf condition has t o be expressed

a s s o c i a t e d with t h i s eigenvalue. on t h e two dimensional operator

t h e p r o j e c t o r commuting with

Po

PoE

.

Then

is a l r e a d y known.

*


192

I n f a c t , t h i s case i s not t e c h n i c a l l y d i f f e r e n t from t h e case when and

-1 a r e simple eigenvalues of

So(To)

1

, s o we group these t w o cases

i n t h e assumption: - __ HI.2.a

The spectrum of

SOITo]

d i s c and a p a r t

)D ':

i s t h e union of a p a r t on t h e u n i t c i r c l e .

0")

1

i n the open u n i t

Moreover we have one o f t h e

following s i t u a t i o n s :

(i)

, t h e two eigenvalues a r e simple, and hl

{l]U{-l]

0 : 0) =

i s p o s i t i v e (Hopf condition) where

(78)

i s t h e eigenvalue of (ii)

____- _-

Al

0(!471

= 41+Phl+

X(,)

2

=

, equal t o -1 f o r

SP[T(p)]

{l]

p=O

;

, t h e eigenvalue 1 i s double, semi-simple and

> o i n (77). X o = A 0) i n both cases ( i )and (ii)and we keep (76) f o r t h e

We note

5 ( t ) and 0

, where

1

i s defined by (88).

I n t h i s case, the c l a s s i c a l theory says t h a t we can f i n d vectors

, Go* , C1*

So, we choose

condition.

such t h a t

* Sly

X =

5, ,

f o r the Poincard map.

For t h a t , we write a s before

L e t us first compute the Hopf


A

(83)

195

= A o + p A 1 + O ( p 2) P

with

Hence, t h e simple eigenvalue close t o

with

0

satisfies

h(p)

of

e t

1 wh n

p

is


196

because c**

pox =

and

A:Co*

x* - 5 1

*

* *

*

0

*

Po

(X $ 5 (0)) f o r any X E E

= [So(To)l

5O*

=

CO*

For t h e computations l e t us introduce t h e vector functions,

C0*(t)

, cl*(t)

(

f(t)

,

, c 0 ( t ) which a l l a r e To - p e r i o d i c and s a t i s f y

S"o(t,O)S1* = 5l* ( t )

-8

G0*(t)

.

0

Let us note t h a t

co(t)

is a

To - p e r i o d i c s o l u t i o n of

(For a g e n e r a l r e s u l t about t h e case of a non-semi-simple i s o l a t e d Floquet exponent, s e e [ 141 ) .

Now ( 8 5 ) l e a d s t o m

where t h e d i f f e r e n c e with (64) i s obvious. Let us now compute t h e time i n t h e hyperplance

(Yo,5 I*)

=0

r(Yo,p)

.

of f i r s t r e t u r n of t h e t r a j e c t o r i e s

By (51), ( 5 2 ) , (53) we have

197

Now we define a s usual L ( O ) 1 subspace

a s the r e s t r i c t i o n of

i n the

of the spectrum, and we use

associated t o t h e p a r t ":JC

Pio)E

So(To)

the f a c t t h a t

t o show t h a t Our map

with

hl

F'~o)AoF'~o) = P ~ o ) S o ( T o ) P P ) G

in R

IJ.

defined by

.

i s now expressed by

(88), and

rn

I n t h i s case, a s i n the case of assunption H'.2.ay any b i f u r c a t e d f i x e d point of

G

12

from X o ( * , g )

corresponds t o the b i f u r c a t i o n of a periodic s o l u t i o n of (44)

.

which i s c l o s e t o

The period of t h e new b i f u r c a t e d s o l u t i o n i s To

and d i f f e r e n t from

r e s u l t s on the fixed points of

G

I.L

T(g)

i n general.

7(Y0,p)

The s t a b i l i t y

apply on the bifurcated periodic o r b i t s .


198

The s o l u t i o n

X(t)

of (44) i s obtained from

xER

by t h e formulas


199

Comments on Chapter V. A systematic study of the p r o p e r t y

(P)& ( s e e theorem 2 ) f o r a Banach

space can be found i n [ 21. The proof of the c e n t e r manifold theorem (theorem 2 )

i s mainly the one of [22] [ 2 5 ] , Lemma 3 answers a q u e s t i o n of H. Weinberger. Paragraph

4 provides

e x p l i c i t formulas t o g e t t h e p r i n c i p a l p a r t of a

b i f u r c a t e d two dimensional t o r u s or closed t r a j e c t o r y f o r a wide c l a s s of d i f f e r e n t i a l equations and m y even be u s e f u l i n a f i n i t e dimensional space (example i n [ 3 ] ) . The s t u d y made i n b . 4 . 2 comes from [lb] f o r t h e cases when t h e assumptions HI.2.a o r HI.2.b a r e r e a l i z e d .

The a n a l y t i c a l study

of t h e consequences of t h e assumption H’ .2.c (1 i s non-semi-simple of

eigenvalue

s ~ ( T ~ )i s) new.

P e r s i s t e n c e of f i x e d p o i n t s (corresponding t o closed o r b i t s f o r an autonomous d i f f e r e n t i a l equation). I n all paragraph 3 we assumed t h e e x i s t e n c e of a t r i v i a l f i x e d p o i n t f o r t h e map F

P

. In

f a c t t h e assumptions, done a t p = 0 , a r e s u f f i c i e n t t o o b t a i n a l l

t h e r e s u l t s , provided t h a t 1 i s not i n t h e spectrum of Dx Fo(0). The p e r s i s t e n c e of a f i x e d point for p

#

0 i s obviously obtained v i a t h e i m p l i c i t f u n c t i o n theorem.

To simplify t h e study h e r e , we assumed t h a t it i s t h e o r i g i n . If it i s not t h e case, and i f t h e p e r s i s t i n g f i x e d p o i n t has t o be computed, t h e formulas have t o be modified and t h i s i s l e f t as an e x e r c i s e f o r t h e r e a d e r . This a p p l i e s i n t h e context of Hopf b i f u r c a t i o n f o r maps, because t h e only eigenvalues of moduli 1 a r e A

0’

-X

0

#

1 . In t h e s p e c i a l resonant case where A.

= 1

i s eigenvalue of D F ( 0 ) , t h e assumptions a t P = 0 , given h e r e , a r e i n g e n e r a l x o

not s u f f i c i e n t f o r t h e f i x e d point t o e x i s t and vary smoothly when u#O ( s e e @bid

for t h e e x p l i c i t conditions i n t h i s c a s e ) .


VI.

OF AN I N V A R I A N T CIRCLE I N T O AN I N V A R I A N T

BIFURCATION

2

-

TORUS

FOR A ONE PARAMETER F A M I L Y OF NAPS

T h i s C h a p t e r d e s c r i b e s a j o i n t work

*

w i t h A l a i n CHENCINER, a n d i s

t h e n a t u r a l c o n t i n u a t i o n of t h e Hopf b i f u r c a t i o n s t u d i e d i n C h a p t e r 111. P o s s i b l e a p p l i c a t i o n s may be, a s i n d i c a t e d i n

V.4.

a wide class of pro-

blems modelled by p a r t i a l d i f f e r e n t i a l e q u a t i o n s , f o r i n s t a n c e f l u i d mec h a n i c s problems which use t h e Navier-Stokes

e q u a t i o n s . The i d e a h e r e

c o n s i s t s t o start w i t h a f a m i l y of d i f f e r e n t i a l e q u a t i o n s i n v a r i a n t f a m i l y of

see u n d e r 3

-

2

-

tori

what c o n d i t i o n s t h i s

. It i s

torus

TIJ,

2

, smooth - torus

h a v i n g an

Ep

i n t h e parameter

p

, and

to

bifurcates i n t o an invariant

t h e aim of what f o l l o w s t o s u g g e s t t h a t t h i s l a s t b i -

f u r c a t i o n is n o t a g e n e r i c phenomenon. We s h a l l n o t g i v e a l l t h e d e t a i l s

of t h e p r o o f s which t h e i n t e r e s s e d r e a d e r w i l l f i n d i n t h e g e n e r a l i z a t i o n o f t h e b i f u r c a t i o n from a rus for

*

n

>2

.

T"

[4 ]

torus to a

also with ,."+I

to-

T h i s i s a l s o summerized by A. CHENCINER i n a CIME c o n f e r e n c e on dynamica1 s y s t e m s (1978)

201


202

1

.

INlRODUCTION

- DEFINITIONS

L e t u s start w i t h a f a m i l y of s t a b l e i n v a r i a n t m i l y of d i f f e r e n t i a l triction

* , and equations

- tori

2

f o r a fa-

make t h e a s s u m p t i o n t h a t t h e res-

of t h e flow t o t h e s e t o r i a d m i t s c r o s s - s e c t i o n s ,

i.e. a f a m i l y

of t r a n s v e r s e c l o s e d c u r v e s on which t h e P o i n c a r g r e t u r n maps may b e def i n e d . We e x t e n d t h e s e P o i n c a r 6 maps t o g e t c r o s s - s e c t i o n s of t h e f l o w i n a neighbourhood o f t h e

- tori.

2

L e t u s n o t e t h a t t h i s a s s u m p t i o n is rea-

l i s t i c i f we c o n s i d e r t h e s i t u a t i o n f o r

t i o n of a c l o s e d o r b i t t o a

2

- torus

close t o a point of bifurca-

p

(see

,

111.6)

T h i s l e a d s t o t h e f o l l o w i n g problem (compare w i t h where t h e b i f u r c a t i o n from a c l o s e d o r b i t t o a

- mapping , where

Ck

0

i n t h e Banach s p a c e

T1 E

:

i~

( i f a n y ) of

0

i n a neighbourhood of

F,

-

@

V.3.

t o r u s is reduced t o

let

m i t s a n i n v a r i a n t circle g e t t i n g close t o s t a b l e when

-

or

1 1 F : T x "v T x E be p i s t h e circle and 'J is a neighbourhood o f

t h e Hopf b i f u r c a t i o n f o r maps)

a

2

111.1

1

when

M

F

P goes t o

ad0

. What i s t h e new attractor

x 0

for

p

,

> 0 small ?

T h i s problem o p e n s t h e q u e s t i o n s :

i s it r e a s o n a b l e t o e x p e c t t h e p e r s i s t e n c e of a n i n v a r i a n t cir-

(i)

cle under

Fp

(ii]

?

if y e s , how t o s t u d y t h e s t a b i l i t y of t h e i n v a r i a n t circle and

t o c h a r a c t e r i z e a change of s t a b i l i t y when w p

We s h a l l see i n close t o

0

, if

3

5

crosses

0 ?

a t h e o r e m of p e r s i s t e n c e of s u c h

2

-

tori for

we o n l y assume t h e e x i s t e n c e of one t o r u s f o r p = 0 .

H i g h e r order bifurcations

.

NOTATIONS

T1

The c i r c l e

T1

map from

203

The r o t a t i o n

R/z

w i l l be i d e n t i f i e d w i t h

to

T1

R,

i s d e f i n e d by

and, f o r formulas,

w i l l be l i f t e d t o a map f r o m

R,(B)

+

= 3'

.

u)

w

as i n

to

(0, x ) E T1 x

If

a 111.3.

71-

,

we w r i t e F p ,

(11

For any (2

F(e,

XI

t

(0, x )

1

T

1

x E

g(e]

f(e,

=

0, 0)

III, +(e,

XI

TIx

t

x,

E

l e t us s e t

Go(B, x )

where

(f(e,

!-J =

XI

=

,

(g(81,

Ao(e)

=

E T1 x

Ao(e)X)

@(e,

Dx

,

E

.

0, 0)

We make t h e f o l l o w i n g assumptions :

F is (31 DEFINITION

1

Ck

,

k large enough,

@(e,

0,

o)

.

For

=

o

.P,I

k

spectrum o f t h e l i n e a r Qo

:

is a Ck diffeomorphism,

g

(i.e.

- 1 , the

map

Ca(T1;

E)

T

R

-

1

x 0

)

.

.l+1 1 C ( T ; E)

,

i n v a r i a n t under

spectograph o f

-

F

Fa i s t h e

A 1 C (T ; E)

defined by

14

1

(uox)(e) graph (do X)

i.e.

REMPRK

:

t h e formula

if

=

=

A,

[ g - ~ ( e ~ ]x [g-l(e)]

Go(graph X )

, for

any

9

A

8 i s a s m a l l enough neighbourhood o f

1

C (T ; E l

X

0

in

.


204

d e f i n e s a map t i a b l e at

, butthe

0

. T h i s map i s n o t d i f f e r e n -

(T ; E ]

C "'

: 8

5

composition

d'

8

CG1(T';

0

Ca(T';

E)

i s d i f f e r e n t i a b l e and i t s d e r i v a t i v e a t

[4 ]

-

is Qo

0

2

.

Assume t h a t t h e r o t a t i o n number

Lo = p e

t i o n a l ; an e i g e n v a l u e if for all

z ,

q

2

no -t q

2 i n sh

u0

of

ffo

.

f z

9 11.1.1

(see

of

3o a t a n y p o i n t )

f o r t h e c o m p u t a t i o n of t h e d e r i v a t i v e of

DEFINITION

E]

of

UI

i s irra-

g

is c a l l e d

"

non real

The i m p o r t a n c e of t h i s n o t i o n is due t o t h e f o l l o w i n g c o n s e q u e n c e s : t h a n k s t o t h e Denjoy theorem ergodic rotation l u e of

ffo

,

is

g

Go

"JO

l o

t h a t t h e whole circle of r a d i u s The c o n d i t i o n

2 Qo

+

.

I n f a c t , one c a n show (see e i g e n v e c t o r of

in

for the cigenvalue

ff

L e t u s c o n s i d e r t h e 'C

1

family

11.3.5

are l i n e a r l y i n d e p e n d e n t i n

(6

I'

f o r all.

5,

=

{

(e,

E

for a l l

q €

E

a.

means t h a t

shows

(closed].

ho

and

of e i g e n v a l u e s on t h e circle.

[4 1 ho

) that

, then

9t T

- s u b b u n d l e of X]

z

, which

p

i s i n t h e s p e c t r u m of

p

z

j?

q w0

d o n o t b e l o n g t o t h e same

A.

t o an

a

A l l t h e s e e i g e n v a l u e s are d e n s e on t h e c i r c l e o f r a d i u s

-

- conjugate

and i t is n o t h a r d t o see t h a t if h is e i g e n v a 2 i n n UI O i s e i g e n v a l u e of for all n E z

R

, then

(see 111.3)

T

.

1 1

x E

T1 x E ; x = z

i f we n o t e

Xo(0)

and

Xo

an

xo(0)

:

xo(e) + ? T o ( e ) 3

.

Higher order b i f u r c a t i o n s

5,

Then

-

C'

is

(8,

.

2

, and

Go

under

Go

conjugate t o t h e map

[71

R2

where

, invariant

T1 x F?

-

i s isomorphic t o

205

4

i s i d e n t i f i e d with

( d o ) , lo 4

.

c

WIN THEOREM AND COMNENTS We make t h e f o l l o w i n g assumptions :

I/

g

Ca

i s

-

conjugate t o t h e i r r a t i o n a l r o t a t i o n

Rwo

1

9

l a r g e enough 2/ CJ

Fo

of

i s contained i n a d i s c centered a t

1

and i s

U

, where

o2

r a d i u s l e s s t h a n one,

c o i n c i d e s w i t h t h e u n i t c i r c l e . Moreover, one assumes t h a t o, L 2irr uo generated " by a couple o f " non r e a l " eigenvalues h = e 9 0

S,

3/

> 0

Ca(T1; E) = & & ' &2

sense t h a t t h e decomposition

of

r Ci0

, V P

1

i n v a r i a n t subspaces r e l a t i v e t o

r'(5) , t h e

bundle

&

, of

0

o1

L

closed Q0 =

i s an u n i o n

0"

-l o , i n t h e

i

- spectrograph

1

The

subspace o f

1

T

+

x E

E

6

0

Z

for

1

sections o f the

described i n

q w0

>

c

C'

o

1

r =

.

I, 2 ,

3, 4

and

Go

,

o2

,

into

, satisfies

- invariant q

EZ

sub-

;

such t h a t 3

v q

t Z \ { O ]

,

for

Then, one has, i n general, t h e f o l l o w i n g conclusions

r = l

:

and

r = 3

Bifurcation o f Maps and Applications

206

For small

I/

i n v a r i a n t u n d er

11.11

,

F P

*

t h e r e e x i s t s a circle

close t o

and d e p e n d i n g c o n t i n u o u s l y on

of t h i s c i r c l e c h a n g e s when

, an d

0

crosses

p

P

T

1

,

x 0

, The s t a b i l i t y

d e p e n d s on t h e s i g n of

a c e r t a i n number which w e assume t o b e non zero. F o r small

2/

11.~1

,

>0

or

p E

+ w0) -

A',

, has

Irj = 1 or

3

y ( e ~= a

,

~ ( 0 1

unique s o l u t i o n

,

(see Lemma 7

e y

.

y2

R2

L e t us note t h a t i f we i d e n t i f y

of

(17)

may be found u s i n g Lemma 1

some d i f f e r e n t i a b i l i t y w i t h r e s p e c t t o Now, we o b t a i n a new map ned as i n

(13)

with with

.

b2

: TI x E x F?

.

111.6)

of

C

1

CW1-"

in

[8 1

The proof of such a lemma i s analogous t o t h a t o f VII

E T'

Chapter

then the s o l u t i o n

r = 1

-

, and T1

we l o o s e

x E x

8

defi-

:

(we suppress t h e primes)

b u t we have now

D @ . ( 0 , 0 , 0 , O] I J . 1

(201 and t h e m a t r i x

(15)

=

0

is now constant

:

,

i =

1, 2 ,

So, we may use t h e center m a n i f o l d theorem because we have here a = 1, tisfied.

f3 = 1

(see t h e assumptions

There e x i s t s a sub-manifold

in of

5

3

]

:

b

< 1,

and a l l assumptions a r e sa-

T 1 x E x I? which i s the graph

21 7

H i g h e r order bifurcations

(p2

: T1 x R3

pl

E E

of a map v p r

5 1

s

-

E

, such

(x,, b ] E T(0)c

that for

,

pi3

and

and t h e q u a d r a t i c terms i n

of

(x2, p )

cp2

(3.7).

may b e found by u s i n g

Because of t h e p r o p e r t i e s o f t h i s c e n t e r m a n i f o l d , w e can r e d u c e t h e study of o u r p r o b l em i n t o i t s trace on t h e c e n t e r m a n i f o l d which c o n t a i n s

i

, hence

.

k?

a l l the recurrence o f

So, l e t us r e p l a c e

we o b t a i n now a new map

fP

x

1

T1 x F?

'

by

in

cp2(e, x 2 , p )

-

T

1

x I?

defined

by (23)

PJ~,

~ 2 1=

F(e,

J, m,(e,

( ~ ~ ( x2, 0 , d,x2,

( ~ ~ ( x2' 0 , b1,x2,

Note t h a t w e m i g h t e x p l i c i t a t e t h e T a y l o r e x p a n s i o n of

close t o

5

.

0

F

P

for

b & O

.

Identifying

Ff Fb

with :

T

1

6:

x 7

, we

-

OF INVARIANT CIRCLES

h a v e now t o s t u d y a map T

1

x

C

(

"y'

n ei g h b o u r h o o d of in

such t h a t

(x2, p )

(left t o t h e reader).

PROOF OF THE WIN THEOREM. STEP 2.PERSISTENCE

FW

IJI.

A A

(new n o t a t i o n s ]

:

C

0

1


21 8

when it d o e s n o t a p p l i e s :

We s h a l l need a lemma c o m p l e t i n g t h e Lemma 1

2

LEW

,

Let

wo

f 114

,

TI1 -

gl + TI)

, and

r E N

r

no +

uo

, then

f Z

Z

t h e equation (4

1

-

where 9

Y ( B + w0r

r = 0

CI

E Cm(T1; C )

7)

y(0,

and

7>

0

Cm(T1; C)

, has

.

=

491

9 E T’

1

,

an unique s o l u t i o n

Moreover, i f

, then

cu(0) d e = 0

JT1

.

7 - 0

in

and

TI1

Y(9,

y(.,

qii

r f 0 Cm

=

, or

if when

.(I)

T h i s Lemma d o e s n o t a s k a n y d i o p h a n t i n e c o n d i t i o n s a s i n Lemma 1, we pay t h i s by a d d i n g a term

9 y ( 9 , TI)

= o(1)

The p r o o f o f a more g e n e r a l r e s u l t i s i n

V.2

i n s t e a d of

0 in

(4.18)

.

of

Now we may p r o v e t h e f o l l o w i n g :

LEMMA

3

r = l (51

.

Under t h e a s s u m p t i o n s

,

I/

,

2/ and 3 /

with

only

t h e n if

Re

j

TI

t h e r e e x i s t s far

A1b) p

#

,

0

close t o

0

,

K

h,(e)

d e f i n e d by 1

[3)

circle close to T x 0

,

H i g h e r order b i f u r c a t i o n s

i n v a r i a n t under the integral

F

, repelling

l~ < 0

and which d e p e n d s c o n t i n u o u s l y on

P

(5)

21 9

.

p

If

is p o s i t i v e , t h i s circle is a t t r a c t i v e f o r for

p > 0

(if (5)

i s n e g a t i v e t h e co n cl u -

s i o n s are r e v e r s e d 1. T h i s lemma is more p r e c i s e t h a n t h e a s s e r t i o n

rem, and s a y s t h a t t h e i n v a r i a n t circle i s r e g u l a r l y p e r t u r b e d when

is c l o s e

1.1

T

1

x 0

to

0

I/

o f t h e main t h e o -

which e x i s t s for p = 0 ,

.

T h i s h a s t o be more

c o m p l i c a t e d t h a n i n t h e Hopf b i f u r c a t i o n , where t h e e x i x t e n c e o f a f i x e d p o i n t close t o

0

for

t he or em , from t h e f a c t t h a t

1

P

follows, v i a the implicit function

F (0) = 0

and

Dx Fo(0)

does n o t admit

as a n e i g e n v a l u e .

PROOF OF L E W

(i)

where

c ( 8) in

F

5

A

3

,

ch an g e of v a r i a b l e s o f t h e f o r m

i s t h e s o l u t i o n of an e q u a t i o n

i n s t e a d of I(8, z , p)

.

a( 8 )

,

(4.18)

with

r

=

1

l e a d s t o a new map w i t h o u t t h e term

and 2

1

c(9)

O f c o u r s e , w e may d o a n a l o g o u s c h a n g e s up t o t h e or-

d e r w e w i s h . So, a f t e r t h e s e c h a n g e s , t h e map

s i m i l a r f o r m w i t h now

(I),

(21,

(3)

has a


220

L e t u s em p h a s i z e t h a t we o n l y u s e d t h e d i o p h a n t i n e c o n d i t i o n with

r = 1

.

With o u r c h o i c e

7

=

1111

, and

t h e lemma 3

new map t a k e s t h e form ( s u p p r e s s t h e p r i m e s )

(iii) I n t h e same way, i f tion (111

B(e, 7 )

, we

3/

can p r o v e t h a t t h e

:

is t h e s o l u t i o n of t h e equa-


put

i n t o a n a n a l o g o u s form, b u t w i t h

F

P

f(e, z ,

(13)

=

e+

wo

+p

z

=

p

(14)

and assume

O1

I

(101

r e p l a c e d by :

+ o[i)z + o(117 + 0(1zl23 +

s 5 4

and

Z1

Fp

then

I

are

o[

s-2

z ' ,

1

(2'1

becomes

( s ~ i p p r e s st h e p r i m e s )

z

i s i n v a r i a n t u n d e r t h e map

, because

Re A

(151

f 0

1

for

O1

p >0

, Z1

which are

or f o r

of

o(

I]

So, lemma 3

, and

p < 0 and

.For

,

9

E

, one

has

t h a t we may use t h e same method as

s t r e n g t h of o r d e r

.

P

,

T1

l e a d s t o a contraction if

]

( d i l a t a t i o n if p R e h,, < 0

terms

z*[B, b )

=

F

such t h a t

z : T1 --C

(161

:

.

1~ 1 )

*

t o l o o k f o r t h e g r a p h of

111.1

.

1

To f i n d t h e p e r s i s t e n t i n v a r i a n t circle u n d e r t h e map

in

ollpl)

Let u s pose

(iv)

where

w

in

f

221

The proof of

~1 Re A

, larger

p

111.1

I

>

0

than t h e

applies separately

l e a d s t o t h e s t a b i l i t y r e s u l t too,

I/ of t h e main theorem i s p r o v e d .

part

L e t u s j u s t r e m a r k t h a t t h e i n v a r i a n t circle is s u c h t h a t it is of order t h e i n i t i a l map

f o r t h e map on t h e form (4.9.)

)

.

(2)

-

(7)

(of o r d e r

IJ.

for


222

6

,

PROOF OF

THE MAIN THEOREM

,

L e t us s t a r t a g a i n w i t h t h e form

(5.2)

,

(5.3)

STEP 3

rrap

:

.

BIFURCATION

F : T1 x P

v

-T

. 1x G

i n the

where

The i d e a c o n s i s t s ,

as i n

111.1

, of

f i n d i n g a "normal form"

t r u n c a t i o n o f which l e a d i n g t o a b i f u r c a t e d i n v a r i a n t

2-torus

.

,

a

The c o n c l u s i o n o f t h e theorem l e a d s t o t h e n a t u r a l change of s c a l e

From now on we assume

P

>

0

( t h e case

Bifurcation of maps and applications, Volume 36 (North-Holland Mathematics Studies)

Bifurcation Theory and Applications

Bifurcation theory and applications

Hopf bifurcation and its applications

The Hopf Bifurcation and Its Applications

Holomorphic Maps and Invariant Distances (North-Holland Mathematics Studies, 40)

Comprehensive Chemical Kinetics, Volume 36: Photokinetics : Theoretical Fundamentals and Applications

Horticultural Reviews (Volume 36)

Theory of Plates, Volume Volume II (Studies in Mathematics and its Applications)

Singularities of Differentiable Maps (Monographs in Mathematics)

Geometry of Chemical Graphs: Polycycles and Two-faced Maps (Encyclopedia of Mathematics and its Applications)

Design Theory: Volume 2 (Encyclopedia of Mathematics and its Applications)

Hilbert Transforms: Volume 1 (Encyclopedia of Mathematics and its Applications)

Graph Theory and Applications (Annals of discrete mathematics, Volume 38)

Wavelets Mathematics and Applications

Mathematical Methods of Game and Economic Theory (Studies in Mathematics and Its Applications Volume 7)

Bifurcation theory an introduction with applications

Applications of combinatorial mathematics

Construction and Applications of Conformal Maps, Proceedings of a Symposium

Construction and Applications of Conformal Maps, Proceedings of a Symposium

The Alkaloids: Chemistry and Pharmacology, Volume 36

International Review of Neurobiology Volume 36

Aboriginal History Journal; Volume 36

Advances in Immunology Volume 36

Advances in Agronomy, Volume 36

Progress in Optics, Volume 36

Advances in Parasitology Volume 36

Local bifurcation and symmetry

Advances in Catalysis, Volume 36

Mathematical Elasticity, Volume 2: Theory of Plates (Studies in Mathematics and its Applications)

Local Bifurcation and Symmetry (Research Notes in Mathematics Series)

Bifurcation of maps and applications, Volume 36 (North-Holland Mathematics Studies)

Bifurcation Theory and Applications

Bifurcation theory and applications

Hopf bifurcation and its applications

The Hopf Bifurcation and Its Applications

Holomorphic Maps and Invariant Distances (North-Holland Mathematics Studies, 40)

Comprehensive Chemical Kinetics, Volume 36: Photokinetics : Theoretical Fundamentals and Applications

Horticultural Reviews (Volume 36)

Theory of Plates, Volume Volume II (Studies in Mathematics and its Applications)

Singularities of Differentiable Maps (Monographs in Mathematics)

Geometry of Chemical Graphs: Polycycles and Two-faced Maps (Encyclopedia of Mathematics and its Applications)

Design Theory: Volume 2 (Encyclopedia of Mathematics and its Applications)

Hilbert Transforms: Volume 1 (Encyclopedia of Mathematics and its Applications)

Graph Theory and Applications (Annals of discrete mathematics, Volume 38)

Wavelets Mathematics and Applications

Mathematical Methods of Game and Economic Theory (Studies in Mathematics and Its Applications Volume 7)

Bifurcation theory an introduction with applications

Applications of combinatorial mathematics

Construction and Applications of Conformal Maps, Proceedings of a Symposium

Construction and Applications of Conformal Maps, Proceedings of a Symposium

The Alkaloids: Chemistry and Pharmacology, Volume 36

International Review of Neurobiology Volume 36

Aboriginal History Journal; Volume 36

Advances in Immunology Volume 36

Advances in Agronomy, Volume 36

Progress in Optics, Volume 36

Advances in Parasitology Volume 36

Local bifurcation and symmetry

Advances in Catalysis, Volume 36

Mathematical Elasticity, Volume 2: Theory of Plates (Studies in Mathematics and its Applications)

Local Bifurcation and Symmetry (Research Notes in Mathematics Series)

Recommend Documents