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BIFURCATION OF MAPS AND APPLICATIONS

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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 k

.

Now

i s exponentially s t a b l e .

Example 2 .

where

, equivalent t o t h e given one, such t h a t

< 1 (We s h a l l see i n t h e t e c h n i c a l lemma 1 how t o choose t h e norm).

~ ~ F ( 5x )(k+e)(jx/l ~ ~

Hence

E

E = Rn

IlN(X)ll

5

, l e t us consider t h e d i f f e r e n t i a l equation

~ l l X 1 1 ~f o r

i s regular enough.

llxll 5

6 , and A

i s a l i n e a r operator, while

Stability of instability of a fixed point

Consider the map

F : X H X(T) 0

s o l u t i o n of (l), s a t i s f y i n g

It i s known t h a t 5

*

If

(5

for

2.

1

Let

llX(t)

A

Re u

i s such t h a t

F and ---

X(O) =

i s the

.

x

eAT i s

eTo where

( f i n i t e number of eigenvalues of t o t a l m u l t i p l i c i t i e s

<

0

, then the condition of theorem 1 i s f u l f i l l e d

Proof due t o K. Kirchggssner and J. Scheurle [20].

F :E+E be of c l a s s

sup Ih A€ 9

C1

i n a neighborhood of

L = F ' (0) be of the form

< inf

hE a2

1x1

Then t h e r e e x i s t s a doublecone Vx'xE

, where X ( t )

/ / approaches zero exponentially.

Let the spectrum of

that

0

F ' ( 0 ) = eAT and t h a t the spectrum of

i s t h e spectrum of

n)

defined near

3

and K

i n f Ihl h€ a2 C

E

(T

>1

and a b a l l

= u

0

,

u

u2

1

satisfy

F(0) = 0

where

. B

S

centered a t

0

such

gsn k 3 n E N such t h a t I I F " ( ~ )> I Is

(we write

R

=

K\{o])

Figure 1. Proof of t h i s i n s t a b i l i t y result. Let us note

P1 and P2 the p r o j e c t o r s , which commute with L

associated with the p a r t s

ul

and

a2

of the spectrum

u

.

,

Then, we s h a l l

I

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

4

, equivalent

see i n t h e t e c h n i c a l lemma 1 that we can choose a norm i n E t o t h e previous one, such t h a t

llxll

where

sup A € u1

Cur cone V6

>

0

IIPILxll

+

>

< a - p < a < i n f Ihl

0

lP211 =

=

and

A € o2

xE

5

, llP1"

llP2"ll

i s defined by:

3s

Let us take (2)

1x1

K

,

= IIP1Xli

{xE E;IiP1xll

5

such t h a t f o r

/IyI(5 s

,

5

then

(1 -~)qlIP,Lxll

IIPIL"II

,

and

a

LxE K

I

>1 ,

p

>0

.

>0

.

NOW,

(a-P)q11P2xl/

,

hence

qllP2xli3

,

g

,

(a-P)/lPlxl/ 0

1

5

( i n t e r i o r of

K

).

Moreover

W e assume t h a t

q

i s small enough t o have

We s h a l l now show t h a t i f

3a'

>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

<

s

.

.

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

6

Example 3.

E = %In

spectrum of

and

is defined as i n example 2.

F

We assume that t h e

contains some eigenvalues of p o s i t i v e r e a l p a r t .

A

s a t i s f i e s t h e required property f o r values a r e i s o l a t e d .

Hence

Hence

AT e

i n the theorem 2 because t h e eigen-

L

0 i s Lyapunov u n s t a b l e .

Technical lemma 1.

, assume

Given LE $E) L : u = o U o2 1

o of

b = P1 Then V E

, with <

sup

hE u1

t h e following decomposition of t h e spectrum

.

i n f Ihl = a A € o2

P2

>

, we can choose a norm i n E , equivalent t o t h e given one,

0

a r e t h e commuting p r o j e c t o r s r e l a t e d with

crl

and

u2

and

such t h a t IIXII

Proof: -

lILPlxll

5

(b+E) llP1xll

IILP~xII

2

(a-e)llP2xll

L1 = L

x1-

1%

s€(O,a)

111 XlIll

,

,

= P2E

,

-

P2 =I P1

.

L = L 2 lE2

, we =

= l\P211 = 1

.

, E2

El = PIE

We w r i t e

For a given

, /IP1ll

= /IPLXIl + llP2xll

f i r s t d e f i n e a new norm on

n IIL1xlll

SUP 7 n _ > 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

<

*In the case when

0

x = x(s)

,

p

> 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

<

,

then the

,

0

and

I

F

i s a n a l y t i c near

0, p

and x

are a n a l y t i c functions.

,

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

10

Remark.

We consider, f o r t h e s t a b i l i t y of a b i f u r c a t e d f i x e d p o i n t , o n l y

those sides of

p = 0

, where i t e x i s t s .

Proof. The f i x e d points of t h e map F

LL

a r e given by

(1) x = F (x) = h ( p ) x + x h(+,x) P

where

h and hE C k - l

dividing by

x

,

ah

because that

k

x=s

.

-(O,O) ap

.

>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

Bifurcation of Maps and Applications

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

.

0

If

, F (x.) k

.

,

p(0) = 0 J

= x

j'

,

The f i x e d point

p(-s)

j#j'

0

= p(s)

.

,

The

i s stable

keeps a constant s i g n f o r

p'(s)

then the bifurcated f i x e d points a r e s t a b l e * i f

.

I n the case when

F

i s a n a l y t i c near

0

,

a r e a n a l y t i c functions.

j

Proof. I n t h i s case we cannot have any f i x e d point b i f u r c a t i n g from t h e f i x e d point

0

.

I n f a c t the equation

(7)

x = h(p)x + x h(p,x)

with

h ( ~ =) - l + h ' ( O ) p + o(p)

,

has nea

0

,

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

a s can e a s i l y be seen using t h e i m p l i c i t function theorem. holds f o r the equation

n

If

n x = F (x) P

if

The same reason

i s odd, because

n

i s even, we have t h e s i t u a t i o n of theorem 1 f o r

Fn

P

.

We r e s t r i c t

ourselves t o looking f o r f i x e d points of order 2 , i . e . fixed p o i n t s of Now l e t f i x e d point of

*We

x1

be a fixed point of

fP

=o ,

F2 , then x2 = F (x,) b

P

8P

is also a

:

say t h a t a periodic point of period

a s t a b l e fixed point f o r

Fn

.

n

for

F

is stable iff this pointie

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

c(x,) If

x1

= F&$(x,)l

=

.

FP(xr) = x2

i s a b i f u r c a t e d fixed p o i n t , hence near F (x ) P

1

#

, because F (x )

x1

P

1

By theorem 1, we o b t a i n a unique branch Hence, t o a fixed

p

13

, we have

0

.

-xl (p(x) , x )

of f i x e d points of

t h e r e corresponds an even number of d i f f e r e n t x

2

F

P

.

,

and we have P[ Fp(X)(XI I = '&(XI

The problem i s now t o f i n d a new parametrization such t h a t

p(-s) = p ( s )

of the branch

(p(s),x(s))

and

F p ( s ) [ x ( s ) I = x(-s)

*

To do t h i s , l e t us consider the following system

which w i l l give two f i x e d points

x +y

, x - y f o r t h e same

p

.

Adding

t h e two equations i n ( 9 ) , we f i n d

This equation i s solvable with respect t o (k(0)= -1)

.

Hence we f i n d x = x ( p , y )

x(p,y) = x(p,-y)

x

for

of c l a s s

thanks t o t h e form of (10).

!J

and

y

near

0

Ck and which s a t i s f i e s

Moreover, we have

14

where

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

o(1)

3

0 when

IyI

and

I pi *

0

.

Now, the f i r s t equation i n (9) gives us:

which i s of the form

Writing fixed point

x(p,y) = y <(p,y) 0

, we

we have

fie

Ck-l

,

and eliminating the

obtain

which leads t o a unique solution i n

Ck-'

:

by t h e i m p l i c i t function theorem.

Now the second equation i n (9) i s automatically s a t i s f i e d .

Then

p(-y) = p(y) If we define

because of the uniqueness of

p

.

Hence

Bifurcation o f f i x e d p o i n t s i n W

and

x.E C k - l

for

J

y

near

.

O ( j =1,2)

To study the s t a b i l i t y of t h e s e f i x e d p o i n t s , we can use the result of

theorem 1.

When F i s a n a l y t i c , the r e s u l t i s straightforward from the i m p l i c i t function theorem i n t h e a n a l y t i c a l case.

This leads t o a p r a c t i c a l mean t o

obtain the bifurcated branch: Let us assume

F

X(E) = e + m

P(E)

=

c

k=l

t o be a n a l y t i c .

' m

"2ke

Hence

2k

k=l

P2ke

2k

h(p,x) = hOlx +

m

k+&= 2

hkL

'k &

Let us write

then by i d e n t i f i c a t i o n of c o e f f i c i e n t s we obtain: 2

order

e

order

c3

2x2 - ha 0 = k ' ( 0 ) p 2 + 2h

x +h 01 2 a2

15

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

16

order

E~~

order

E

Exercise

2x

2k

2k+l

= known function of

(x2,.

. . ,x2k-2~~2L2""''2k-2

0 = A'( 0) p2k+ 2h01x2k+known function of ( x 2'

1. Express

X ( E ) and

the given c o e f f i c i e n t s

Lk,hkL

p(e)

.

up t o t h e order

-

'X2k-2'

c4

k2¶*

-

* I

%k-2)'

i n function of

What r e g u l a r i t y i s needed f o r

F

, for

t h i s t o be t r u e ? Exercise 2 .

L e t us consider the following d i f f e r e n t i a l equation i n R :

where we assume

t

.

L and

n

regular enough i n

p,t,x

and

T-periodic i n

We assume

The solution of the Cauchy problem f o r (l), defines a map i n R

for

xo near Let us put

0 and

[tl

s,(t)

= e

5

F :x~-X(X~,~,T) P

T

:

. , and h ( p ) = sp(T) , and consider

.

Bifurcation of fixed points in R

Show t h a t

DF (0) = k ( p ) 12

calculate

a ( p)

A(0) = 1

,

, and

17

writing

, b ( p ) a s f u n c t i o n a l s of the c o e f f i c i e n t s i n

calculate

#

and assume i t

X'(0)

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

E

0

.

Assume

(2).

Then c a l c u l a t e t h e

f o r the bifurcated solution.

This b i f u r c a t e d f i x e d p o i n t corresponds t o t h e b i f u r c a t i o n of a

T-periodic

s o l u t i o n of (1). Remark on examples. 1) To o b t a i n from d i f f e r e n t i a l systems w i t h

p o s s i b i l i t y f o r one eigenvalue of

22

a space of dimension

.

DF (0)

P

T-periodic coefficients, the

t o be

-1

, we have t o consider

But, t o u s e t h e frame of c h a p t e r 11, we have

t o reduce t h e dimension by a n o t h e r mean, which w i l l b e given i n c h a p t e r V (see t h e a p p l i c a t i o n s of t h e c e n t e r manifold theorem). 2)

Let us consider a n autonomous d i f f e r e n t i a l equation i n Rn

t h e r e e x i s t s an i n v a r i a n t c y c l e f o r t h e values of t h e parameter interval.

,

such t h a t i n an

p

We can reduce t h e problem of t h e behavior of t h e t r a j e c t o r i e s near

t h i s c y c l e , t o t h e s t u d y of a map which has a f i x e d p o i n t .

It i s t h e so-

c a l l e d Poincarg map.

3.

The Poincarg map

-

orbital stability.

3.1. Let us consider t h e following d i f f e r e n t i a l equation i n Rn (1) that

dx

= Gw(,(X)

:

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

twXo(t,p)

is a

T ( k ) - p e r i o d i c s o l u t i o n of (1).

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

18

Let us put

(2)

%

=

,

X = Xo(t,p) + Y

,

Lp(t)Y + $(t,Y)

a non-linear one, which are

then

where

T(p)

(1) becomes

\(t)

- periodic

d s (t) = dt

P

Lp(t) S J t )

We have now a map i n En: Xot h e cycle

P

t

P

(0) = 1

X(XO,pL,t)

, and such t h a t

:

.

, which i s d e f i n e d f o r Xo

near

Xo(.,~) , such t h a t

and f o r bounded times: for a l l

, s

in

S (t)

Let us define the fundamental matrix

(3)

i s a l i n e a r o p e r a t o r and % ( t , . )

t € [ -T,T]

i n a neighborhood of

n map i n R : Yo*v(Yo,p,t) construction, we have

,

T

i s choosen l a r g e enough t o be

0

.

The system (2) l e a d s us t o another

defined for Yo

near

0

,

and

ta-T,T]

> T(p)

.

By

B i f u r c a t i o n o f f i x e d points i n R

19

and i t can b e e a s i l y shown t h a t

L e t us note t h a t 1 i s an eigenvalue of axO

-(O,p) at

S [ T(p) ] P

,

t h e eigenvector being

, t h i s being due t o t h e invariance of (1)w i t h r e s p e c t t o t r a n s -

l a t i o n s i n time.

I n f a c t we have

which g i v e s

hence

and because of the p e r i o d i c i t y

Now t h e map derivative a t

0 as a f i x e d p o i n t , has a

which has

Yo-')fYo,p,T(p)]

0 w i t h t h e eigenvalue 1. I n f a c t we don't u s e t h i s map t o

study t h e behavior o f t h e t r a j e c t o r i e s near t h e c y c l e

Xo(*,p)

, b u t we

c o n s t r u c t t h e s o - c a l l e d P o i n c a d - map.

We s t a r t w i t h i n i t i a l d a t a

a hyperplane t r a n s v e r s e t o

,

a

Xo(O,p)

SIL[ T(p)]

PY = 0 P

, onto t h e eigenvector

, where

in

near t h e o r i g i n , and look f o r t h e

f i r s t r e t u r n of t h e t r a j e c t o r y on t h i s hyperplane near t h e o r i g i n . choose t h e hyperplane

Yo

P

P

Let us

i s t h e p r o j e c t o r commuting w i t h

a a t Xo (0, p) .

(We assume t h a t t h e eigenvalue

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

20

1 i s such t h a t t h e r e e x i s t s

a

space t o

Xo( O,,)

1 i s a semi-simple eigenvalue)

; t h i s i s t r u e if

The Poincare‘ map w i l l be

where

a ( n-1)-dimensional i n v a r i a n t supplementary

Yo*

such t h a t

FL(Yo)

has t o be determined such t h a t

T

.

P F (Y ) = 0

PP

0

which i s i n f a c t a one dimensional equation, of t h e form We note

where

S

* P

P Y =

P

*

(Y,5 ) C

P P

i s the adjoint of

S

,

Hence because of t h e r e g u l a r i t y of and leads us t o

Now

T

= ?(Y 0, p )

5

where

.

P

=

a

Xo(O,P)

We t h e n have

f(Yo,p,r) = 0

.

,

; then

f

, t h e i m p l i c i t f u n c t i o n theorem a p p l i e s

s o l u t i o n of

(7), such

that

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 R

The Poincarg map i s now

21

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

22

Toexplicitate

F

[c:), . . . ,c(n-l) P

} and a d u a l one

*

P Y = O

P

c1

i n t h e hyperplane, we have t o choose a b a s i s (n-1)

.

Exercise 3.

T(p)

for

t h e Poinear; map i n a one-dimensional space. i s an eigenvalue o f 0

adjoint.

.

i n t h e d u a l subspace

2 Let us consider a n automous d i f f e r e n t i a l equation i n pi

w i t h a n i n v a r i a n t cycle of period

near

*3

p

near

.

0

L e t us d e f i n e

For t h i s we assume t h a t

1

Jt

which i s semi-simple f o r a l l values of

S [ T( p ) ]

P

The t r a n s v e r s e eigenvector i s noted

G(l) P

and

p

5,('I* f o r t h e

We. have

Express t h e f i r s t terms of t h e one-dimensional Poincare' map as f u n c t i o n a l s of t h e c o e f f i c i e n t s of t h e d i f f e r e n t i a l equation ( s e e (2) f o r t h e n o t a t i o n s ) . A s i m p l i f i c a t i o n on these f u n c t i o n a l s w i l l be given a t t h e end of chapter I V .

I n t h i s e x e r c i s e , it i s not p o s s i b l e t o have determinant of

3.2.

S (T(p)) P

Orbital s t a b i l i t y

0

- limit

Let us assume t h a t know t h a t t h e f i x e d point

X(0)

>

is

%F

P

(we need a higher dimension f o r t h a t ) .

phase.

(0)

0

has a s p e c t r a l r a d i u s

i s asymptotically s t a b l e .

i s c l o s e enough t o t h e closed curve

tends t o the closed o r b i t

%

h ( 0 ) = -1 , because t h e

Xo(.,p)

<1

.

Then, we

This means t h a t i f

then the trajectory

:

I n f a c t , we have a more p r e c i s e r e s u l t :

*

When

1 i s not semi-simple f o r

P =

0

, see t h e computation i n gV.4.2.

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

23

Theorem ( l i m i t phase) Assume t h a t

X b

a t u b u l a r neighborhood

2 a " l i m i t phase"

then

<

has a s p e c t r a l r a d i u s

D F (0)

of t h e closed o r b i t

r

1

1 , then there exists

such t h a t i f

X(0)E

n

such that

6

I

i-

exponentially, where ..

X(t)

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 (1).

-

Proof.

Without loss of g e n e r a l i t y , we can assume t h a t t h e i n i t i a l d a t a

X(0)

verifies :

By assumption, we can choose a s u i t a b l e norm i n R and

k
n

such t h a t

3 7 >0

such t h a t

L e t us note

then, t h e time of

n s r e t u r n of t h e t r a j e c t o r y on t h e plane

P Y = 0 is c1

equal t o

Let us show t h a t t h e r e e x i s t s

6

, function of Yo

ard

p

, such t h a t

,

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

24

(18)

T(n)-

and 3 K

>

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) .

This Page Intentionally Left Blank

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

Bifurcation of Maps and Applications

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 )

Let us assume t h a t

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 th

o r i g i n i s unstable. We assume now

H.3

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

.

Let us prove t h e following. ~~

(X)

See a t paragraph

~

8 t h e proof t h a t any i n v a r i a n t b i f u r c a t e d closed curve

l i e s in t h e annulus 1x1 4 1 .

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

2

33

Theorem 1.

L e t us assume

t o be of class

F

H.l , H.2 hold w i t h

assumptions

holds ( s o t h a t

0

f o r t h e map

i n R2

=

H.3'

, b i f u r c a t i n g from 0

holds (so t h a t

neighborhood of t h e map

For

p

F

P

>

0

p = 0

1,2,3,4

.

Then, i f

H.3

) , there exists a right

p = 0

exchange of t h e s t a b i l i t i e s of t h e f i x e d p o i n t If

, k 2 6 , and t h e

0

, i n which t h e r e i s 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

p = 0

k

near

:k f 1 , n

is attractive for

neighborhood of F

Ck

i s repelling f o r

0

.

I n t h i s case t h e r e i s a n

0 and t h e i n v a r i a n t c i r c l e . p = 0

) , there exists 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 t h e f i x e d point

0

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

i s unstable.

Moreover, i n a s u i t a b l e system of coordinates corresponding t o t h e normal form, t h e i n v a r i a n t c i r c l e can be expressed a s :

I

with

and

E

small enough

(6. J

>

o

,j

= 1,2)

.

Remark 1. Except f o r t h e r e g u l a r i t y r e s u l t i n

p

, t h i s theorem

was proved

independently by R . J . SACKER [28] and by D . RUELLE and F. TAKENS 1271 Remark 2.

The case

0

= 0

, will be s t u d i e d

i n theorem 2.

a

34

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

Proof of Theorem 1. lSts t e p : we assume f3

, and a > 0 , and just prove t h e e x i s t e n c e of t h e i n v a r i a n t c i r c l e

= 0

yIL

.

After t h e change of v a r i a b l e s (16), our map t a k e s t h e form X = (1-2pRe kl)x + p3'2X

where

k z # 1 , s o t h a t we may t a k e

XI and

1

(x,(p,p)

*1E $ - 5 J

Let us consider t h e following complete metric space

with , d i s t ( u , u ' ) = l/u-u'I/ = sup lu(cp) -u'(cp)I

.

cpE T1 Let us s t a r t from uE U x = u(q)

that i f

we have

.

X =

We then show t h a t we can define

G(Q) .

Thus we w i l l have a map

and we will show t h a t it i s a s t r i c t c o n t r a c t i o n i n

a unique s o l u t i o n gives X1

*

X = u (0)

and

us d e f i n e

*

u

.

Il i n C 0 ' l

in U

*

of u = $u*)

U

(x,cp)

9: U+U

,

, so that there exists

*

, which means t h a t x = u (cp)

We f i r s t do the proof f o r a f i x e d with r e s p e c t t o

;I€U such

p

>

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

<

1

Iu(cp+l) = fiU(cp) + 1

.

We assume

that

2p>/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

Bifurcation of Maps and Applications

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) -'<- 1

vj We have

-1 = iPu 0) j

*

r e a l i z e d if

We then have defined a map 3 : uH

Let us show t h a t it i s a c o n t r a c t i o n :

where

, which i s

6

p

in

is U

.

1

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

enough.

*

3: u

2

37

We can conclude the existence of.a unique fixed point i n =

U

for

.

Z(U*)

2nd = s t e p : Regularity r e s u l t . Now, l e t us assume uniformly i n

X1 and

.

pE ( 0 , 8 )

the sup i s taken f o r a l l

in

CL71

We again define pE ( 0 , s )

, 1x1 5

with r e s p e c t t o

(x,cp)

P=s~p(llX~il~,~,IlS~ll,,~ , where 1 . 1 , cpE T

The new function space i s now: UL = cu : T l X R + R ; l!ull.e,l with

)t

dist(u,u') =

sup 'pE T1

5 13

lu(cp,p) -u'(cp,p)I

.

We want t o see t h a t t h e

ii.E (P11P2)

map

3: u w

in

UL

and

Q

defined before f o r

uniformly i n

>

pLE (pl,p2f

*This

small enough, i s i n f a c t a contraction

where

p1 = h16

,

p2 = h 2 c

0 small enough.

I n f a c t , we have f o r

where

p

-1 cp = Iu (cp)

-___---

cp

_> cp'

leads t o

metric space i s complete because of t h e Ascoli theorem.

with

h2 > h1 > 0

38

Bifurcation of Maps and Applications

Let us show t h a t

I;(*)

u,

GE

I -< 1-2C

if uE UL

:

- 1 for c small enough <

61ReA1+ 0(P3/‘)

D G ( # ) = (1-2p62ehl)Du. Dr;1+$/2[DlXl.Ilu

(25)

D2G(*) = ( 1 - 2 p 0 e A l ) [ D 2 ~ ( D I ; 1 ) 2 + D u . D

-

2..

\ID u / \ ~5 , 1~ 26

then

fjl

u,

, uniformly

U ,

for

c

PE (pl, p2)

y

uE 2

Re A1 + O ( C ~ / ~ < ) 1 for

It i s now easy t o see t h a t f o r

GE

+D2X1]D@U -1

~~D~~5 \ o1 , l- 2 c 61Rei1 + O ( C ~ / < ~ )1 f o r

then

NOW,

and

small enough.

c

1

] + O(E‘/*)

uE

small enough, and

.

.

9 : UHG

y

and

UE U,

E

,

small enough.

we have

i s a contraction i n

, because we saw i n ( 2 3 ) t h a t

(26) Then, because of t h e completeness of the space, t h e r e i s a unique f i x e d point

u

*

*

, the convergence of the i t e r a t i v e process being uniform i n

= S(u )

PE (P14*)

a

Hence, i f in

CL,l

p”

X1( , a

0

a s functions of

fixed point

* u

such t h a t

,

and

p)

(x,cp) pH

al( , by

* u (*,p)

For more r e g u l a r i t y , l e t us assume t h a t to

(x,cp,T)

where we note

7

= Afor

*, *

,p) a r e continuous, taking values

the uniform convergence, we have a i s continous, taking values i n

X1 and commodity.

are

CL’l

We l o o k f o r

(J,

with respect

-

Hopf b i f u r c a t i o n in W !I?Y [ 11,$1 ;R)

uE C"'(

, where 'dl = (

2

39

~ 6 ~ )

enough. For i n s t a n c e , l e t us consider t h e f u n c t i o n space

l€ (0,fj1I2) , xE We a l r e a d y know t h a t f o r

.

1

[-1,1], cpE T

uE U;

, and

G

small enough, we have

GE

\b .

Now we e a s i l y o b t a i n

ac

-(@,TI)

(27) where

a7

cp

= 1 1-2v2&e hl + 0($)1ii$(cp,~

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

function of

au (u,- cp,T)) acp'

.

Moreover (25) g i v e s us

For

u€

'J1

('3,v)

+

by

aU

function of ( u 9 z , c p , ~ )

(19) and

we can show t h a t

O(3)

is a

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

40

and (27) leads t o

K1 , K2 u

i n (28) and (29) a r e constants which remain bounded f o r

(instead o f

i n U1

on M o

O(E 3'2)

and

O ( e3'2)

U; ) , whereas K' and

i n (30) depend a l s o

.

Now, l e t us f i r s t take

1+2etj1&le al+o(G 3/2)

e

small enough such a s t h e c o e f f i c i e n t s

i n (28) and (29) are

<

1

.

Then t h e r e e x i s t s

Mo

l a r g e enough such t h a t (28) and ( 2 9 ) gives

Mo

being fixed, we can now choose

-

1 2 e ?j1Re

Now

A1+

O(e3/*)

, perhaps smaller, so t h a t

< 1 i n (30) , and

5

l a r g e enough t o have

6E U'

1 '

The map u

-t

6

U;

topology i n

i s a contraction i n

.

*

U; , by

( 2 6 ) , i f we choose t h e

Co

This space i s complete, hence t h i s ensures t h e existence

of a unique fixed point i n

(34)

E

u = Z(U*)

*

:

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

For

L

>1

)

Ip ,

pi

constants

and Iv$

)

.

< pl

progressive a s f o r

! a2 ~ IM,l

a"; a$a$

we use an equation s i m i l a r t o (27) f o r

where t h e non-written terms a r e f u n c t i o n of lp'l

41

5

ap'u P; P;

.

For i n s t a n c e f o r

.

9 1

with

av acp

4, = 2

, we f i r s t know t h a t

We w i l l f i n d s u c c e s s i v e l y t h e L i p s c h i t z

, M4 such t h a t :

This gives t h e r e s u l t of theorem 1, f o r a k F cC

s t a b i l i t y r e s u l t , because i f

>

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

,

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

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 <

0

.

We s t a r t w i t h

t h e map i n t h e form (17) or ( 1 7 ' ) , which we r e w r i t e a s :

f(cp+1/5)

where

= f ( Q j i s of c l a s s

cF0 ,

and

~ O ~ E. Z

We make a change of v a r i a b l e s :

with

g

Then

2

of period =

x +g(@)

Let us assume

we have

provided t h a t

1/5

.

s a t i s f i e s t h e equation

g

t o be

Cm

.

Then because

Bo

i s a m u l t i p l e of 1/5

,

44

Bifurcation of Maps and Applications

t h a t i s t o say

It i s then always p o s s i b l e t o f i n d a s o l u t i o n

@;€ Cm

of p e r i o d l/5

of

Hence t h e new map i s of t h e form (18) with t h e same r e g u l a r i t y as

(ItO).

before.

This ends t h e proof of theorem 1.

2. Non-standard Hopf-Bifurcation. We now i n v e s t i g a t e what happens when and ( 1 2 ) of l., i s

0

.

0

, defined by t h e f o r m l a s (15)

To s i m p l i f y t h e study, l e t us assume t h a t

f o r s u f f i c i e n t l y many numbers

.

n

An

0

#

1

Then we have

L e m 2. Let

F be of c l a s s

#

hold, and l e t

Ck

1, n=1,

, k 2 2p + 3 , l e t t h e assumptions H . l , H.2

...,2 p + 3 .

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

p-dependent change of coordinates, such t h a t

where and

%p+3(z,z'p)

z = r e2irrCP i n a neighborhood of

Proof.

L.e

= r 2P+3 R&,+3(r,v,~)

with 0

.

F

P

k-2p-2

C

i s p u t i n t o t h e form

of c l a s s R' 2~+3

$-2~-3

>

This i s a d i r e c t consequence of t h e p o s s i b i l i t y of f i n d i n g a s o l u t i o n

of t h e equation

(2) suchthat

I

%e

-

- ?9e

5' = 0 9E

-

(A -1qXp)y

,

for

4r

,

q+L<2p+2

because of t h e condition on

-

, q#e+l

.

This can be done

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

Let us now w r i t e t h e map F

{

(3)

P

+

i n p o l a r coordinates a s before, we have:

o ( ~P ~ 3 +r I P ~ 2 r 3+ I &+3

where t h e choosen orders f o r the w r i t i n g of

.

a2q+l # O(qL1)

the f i r s t

45

R

and

8

Theorem 1 t r e a t s t h e case

+

.4q

+

3)

a r e determined by Q: # O

3

.

L e t us

show t h e following. Theorem 2 . Assume

F

# I,

hold, and l e t for

p = 0

k C

t o be of c l a s s

,

k

-.

2

.

n =1,2,. ,4q+3

of t h e non-linear terms i n

4.9 + 4

r

, l e t t h e assumptions H . l , H.2

If t h e first non-zero c o e f f i c i e n t

i n t h e normal form of

R

i s Q: 2q+l

then Q : ~ < ~ 0 + ~,

(i) i f

neighborhood of for

F

for

p

(ii) i f

0

P

0

and t h e r e exists a r i g h t

0

.

The f i x e d point

0 loses i t s attractivity

.

a2q+l >

neighborhood of F

P = 0

, i n which t h e r e i s an i n v a r i a n t a t t r a c t i v e circle

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

P

>

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 :

'

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

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

.

Bifurcation of Maps and Applications

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.

,

Bifurcation of Maps and Applications

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

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

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

Bifurcation of Maps and Applications

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

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

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]

Let us assume t h a t

(33)

<

ICI

2 1/2

then t h e equation The case when

,

(A*+ B )

1 CI

02p+2 (cp) = 0

> (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

57

, the periodic points a r e

r e p e l l i n g , because t h e i n v a r i a n t c i r c l e i s i t s e l f r e p e l l i n g , t h e “ a t t r a c t i v e “

% .

family i s only a t t r a c t i v e on and t h e assumptions of theorem

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

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

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

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 <

8 = 0 ,

, we have

el

if

if

3

1

P(p)

... if el+

.

( s e e theorem 4)

B1 - 0 2 = O

For i n s t a n c e f o r

=

up t o the order

3

h20p+4 = 1 , t h e result f o r

I n t h e case when i n powers of

e

, p _>

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

Bifurcation of Maps and Applications

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

Hopf bifurcation i n R

where

xo(cp)

!d = cp+zl(d

(45)

where

has t h e form

-

k2

-

,

:Plh,(cp,~)x

.

,

-

-

, k3 , k6 , hl , h 4 ,

Let us assume t h a t

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)

a r e a l l of the form

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

Bifurcation of Maps and Applications

2 ,

where

If

5 ,$ ,6, ,

P =1

hd

a r e a l l of the form

(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

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

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

Hopf bifurcation i n R

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)

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

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

Bifurcation of Maps and Applications

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

Hopf bifurcation i n R

(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

Hopf bifurcation i n R

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

Bifurcation of Maps and Applications

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

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

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

Bifurcation of Maps and Applications

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 ) , <(p)

* -* 5 (p) , 5 (p) = 1

t h e eigenvectors of

.

we j u s t d e f i n e

z

by

and take the s c a l a r product:

expression which w i l l be now w r i t t e n of t h e c o e f f i c i e n t s of I

where x

a(p)

F

P

(2)

Fp(e)

.

To h e l p t h e e x p l i c i t computation

, we d e f i n e t h e p e r i o d i c vector f u n c t i o n s

i s t h e Floquet exponant*such t h a t

see paragraph 6, t h e computation of t h e expansion of

X(p)

.

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

We s e e t h a t

So, we have

and

5

P

and

*

5,

2

a r e r e s p e c t i v e l y T - p e r i o d i c s o l u t i o n s of

81

Bifurcation of Maps and Applications

82

The l a s t expression can be more e x p l i c i t i n R2 and

N(2)[T;<,(T),t&T)1 P

NP (2)[7;C,(~).5,('r)1

,

i f we decompose

on the b a s i s

C,(T)

,tJL(7)

.

I n f a c t , we have

and

So, we d e f i n e

and, f o r instance

s o the computation of (17) i s straightforward. Let us assume t h a t we have a b i f u r c a t e d i n v a r i a n t closed curve t h e map

F

JL

i n R2

.

The r e s t r i c t i o n of t h i s map t o

l i f t a s a diffeomorphisrn on R

where

gJL

is Z

y

P

yIL

i s such t h a t i t s

has t h e form

periodic (see paragraph 3 ) .

for

Now, i f we i d e n t i f y

(cp,l)

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

with

, where cpE T1 , and FP(rp)

(Fp(q).O)

quotient of

2

83

= f (cp)

P

mod 1

, we o b t a i n t h e

T h [ 0,1] by t h i s equivalence r e l a t i o n , which i s a t o r u s

8 .

I n f a c t we have t h e diagram

I T1

li.

1

1=

yp

diffeomorphism between

T1

where

{Xo(cp);cpE

T1

_____9

T

i s t h e i n v a r i a n t closed curve, and and

y

, and where 7

P

P

Xo

is a

i s isotopic t o a

r o t a t i o n because of t h e form of (20). L e t us consider

such t h a t

Y(cp,O)

Thus t h e function the torus

.

T2

=

*

t h e s o l u t i o n of (1)

Xo(cp)

Y(cp,s)

.

By c o n s t r u c t i o n

r e s p e c t s t h e equivalence r e l a t i o n used t o d e f i n e

This shows t h a t t h e s e t

3c

We shall very o f t e n f o r g e t t h e d i s t i n c t i o n between t h e l i f t projection

7

; s o we s h a l l w r i t e

(PE T1

or

q$W

f

and t h e

i n t h e same way.

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

84

8

i s diffeomorphic t o a torus

6.

, and i s i n v a r i a n t by (1).

(Theorem of Bohl)

7

I f t h e diffeormorphism a rotation

R

Po

such t h a t

of PO

T1 i s , f o r a

po

#

0

, conjugate t o

, t h e n t h e s o l u t i o n s of (1) on t h e t o r u s

pot Q

'

a r e p s i p e r i o d i c , w i t h two p e r i o d s .

-

L.--

See paragraph 3 f o r t h e conditions on

Remark.

?

needed t o s a t i s f y t h e

assumption of t h e lemma. We have a homeomorphism

of

such t h a t

T1

We now d e f i n e

where

Y

was defined by (21) and

5

h = I d + Z p e r i o d i c i s t h e l i f t of

Then w(u,v+l) = Y[h -1( v - p o u + l ) , u ] = Y[h-l(v-po~),u] = W(u,v) w(uf1.v) = Y[h-l(v-p0u =

- P,),u+l]

Y[h-dRp (v-pou-Po),u]

= Y[f

oh

PO

= W(U,V)

,

-1 (v-p,u-Po),u]

.

0

So the s o l u t i o n s of (1) a r e of t h e form

t

(24)

X(XO(CP),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

Bifurcation of Maps and Applications

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),<

$4

(p)) = 1

P Y = 0 i s defined i f

P

i(p)

,

thanks t o t h e p e r t u r b a t i o n theory. t h e eigenvector of

c(p)

such that

\Xo\ =1

,

, 1 i s a sim2le eigenvalue of S [ T ( p ) ]

and t h e r e a r e two o t h e r simple eigenvalues functions of

KO

and

.

21

i n a neighborhood of

p

Xo

YER3

.

*

i(p)

Let us n o t e

,

Then t h e p r o j e c t o r 1 - P

, by

which a r e r e g u l a r

CL

on t h e plane

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

ip

R

2

(5)

Let us w r i t e the Poincard map on t h i s plane F (Y ) = A Y +A(2)(Yo,Yo) +A(3)(~o,Yo,Yo)+ o ( l l ~ 4 ~) J I

(6)

F

O

Now Yo = z 5(p) +

P O

<(p)

P

P

,

.

and we use t h e expressions given i n 11.3 t o o b t a i n

As i n paragraph 5, we can d e f i n e t h e p e r i o d i c vector f u n c t i o n s

where

e

We r e c a l l t h a t w e defined i n chapter 11.3 the eigenvector

5

P

related

t o t h e eigenvalue 1, s o we can d e f i n e too, t h e p e r i o d i c v e c t o r functions

Then we o b t a i n f i r s t (notations of 11.3):

88

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

All these q u a n t i t i e s a r e easy t o compute for expansion of

k(p)

expansion of

S [ T(p)] P

; this

p = 0

.

Now we need the

i s given by the perturbation theory once known the

.

F i r s t , we have, by a fixed point argument :

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

Let us define

then

m

2

89

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

90

Now we can compute

with

These formulas can be used i n t h e non-autonomous case of paragraph 5 ,

Remark. by doing

T1=T2

=O

.

Let us assume t h a t the necessary assumptions t o g e t an i n v a r i a n t closed curve

yP

restriction

where

gP

f o r t h e map FP f

of the map t o

P

- periodic

is Z

The projection

i n the plane

T~ of

f

P

y

P

P Y = 0 P

, are realized.

The

i s such t h a t

(see paragraph 3 ) .

We parametrize

as a map from T~

= d onto ~ T-l

by

yP

tpER

is a

diffeomorphism. The quotient of (FP('p),O)

(here

1 T X [ 0,1] b y the i d e n t i f i c a t i o n of

cpE T

1

) , leads t o a torus

The s o l u t i o n of (1) such t h a t

(cp,l)

l?a s i n paragraph

X l t = o = Xo(rp)E yP

with

5.

, can be w r i t t e n

.

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

Let us d e f i n e a vector valued f u n c t i o n

where

$ € C"(B)

for a l l

p

11

i s such t h a t

, and where

$(O) = 0 7(Xo(rp),p)

t r a j e c t o r y s t a r t i n g a t t h e point plane

P Y = 0 k

Y(rp,s)

Xo(cp)

2

for

91

, sE[O,l]

cpER

, $(l) = 1 , $ (P) ( 0 )

=

by

= 0

$(')(1)

i s t h e time of f i r s t r e t u r n of a for

t

= 0

, and r e t u r n i n g on t h e

( s e e chapteS 1 1 . 3 ) .

Thanks t o the f a c t t h a t

t h e map onto at

SH s

To + ( T ( X ~ ( V ) ) p) , -To) $(s) i s a

[ 0, T ( X o ( ( P ) , p) 1

s = O

and

s = l

Cm

diffeomorphism from

[ O,l]

whose d e r i v a t i v e s (of a l l p o s i t i v e o r d e r s ) a r e e q u a l

.

Now, we have by c o n s t r u c t i o n

Therefore we can extend t h e d e f i n i t i o n of

Y

for a l l

sER

and we o b t a i n a r e g u l a r vector valued f u n c t i o n defined on R2 t h e equivalence r e l a t i o n used t o d e f i n e t h a t o r u s function

$(s)

T2

.

,

using

which r e s p e c t s

W e introduced t h e

t o make t h e parametrization of t h e t o r u s d i f f e r e n t i a b l e .

have j u s t shown t h a t t h e s e t

We

92

Bifurcation of Maps and Applications

i s diffeomorphic t o

!?

, and i s i n v a r i a n t b y (1).

Let us assume t h a t t h e diffeomorphism

R

jugated t o a r o t a t i o n

with

Po$ Q

PO

.

of

1 ,

T

,

po# 0

PO

Then a s i n lemma

i s con-

6 (see

paragraph 5 ) we can d e f i n e t h e vector f u n c t i o n

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

Any s o l u t i o n

X(t)

of

(1) on t h e i n v a r i a n t t o r u s

X2 has t h e form iL

with

v ( t ) = pou(t) + h(cp)

diffeomorphism of

R

,

and

h Z -periodic.

The f u n c t i o n

b u t i t i s not i n g e n e r a l of t h e form k t

.

u

So the

formula (32) does n o t g i v e us a q u a s i p e r i o d i c s o l u t i o n on the t o r u s .

L e t us consider t h e equation (1) on t h e t o r u s e x p l i c i t e l y t h e system

(I=

we j u s t have t o s o l v e t h e system:

dt

(34)

as

du

'0

[h-l(v-p0u),u]*

= G PJ(Y[h-'(v-p0u),u])

52 iL

.

is a

To o b t a i n

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

By c o n s t r u c t i o n we may assume

2

93

6 i s ZZ2-periodic (;sing (28) and f

6(u,v)

#

0 on B2

, because

i n a neighborhood of t h e closed curve new parametrization

(a,b)

Gp(X)

t++ Xo(t,p)

= h - h p 0 h) WO

#

which i s

0 on T20

.

and

w

We wish t o f i n d a

of t h e t o r u s such t h a t t h e s o l u t i o n of (1) on

2

TCL w i l l b e

where

k

i s a c o n s t a n t t o be determined.

The v e c t o r f u n c t i o n

Z

w i l l be

determined by

Now we a r e looking f o r a diffeomorphism of B2

where

m

and

p

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

Let us introduce new v a r i a b l e s i n s t e a d of

i t i s c l e a r that t h e map us d e f i n e

of t h e form

(U,V)M

(7,s)

(u,v)

:

i s a diffeomorphism of

pi2

.

Let

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

94

then (33), (34) becomes:

I /

= 0

,

2

=

"U(5,v)

,

dt

We can define too

and because of (35) we have:

Moreover because of t h e Z2 p e r i o d i c i t y of

m , p . we obtain

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

95

So, by c o n s t r u c t i o n

and because of t h e c o n t i n u i t y of

&(!)

p

and

= c o n s t a n t , t h a t we can pose t o be

t h e d e f i n i t i o n (37) o f

Now, because of satisfying

b

m

and

pot Q

, t h i s leads t o

0 a f t e r a n eventual t r a n s l a t i o n i n

.

p(u,v) = pom(u,v)

, we have j u s t t o determine E(~,TI)

(43), and da = k g i v e s t h e equation: dt

hence

(44)

aii

where t h e unknown a r e

(45)

+&

= -1

fii(597)

=

-11

k and fii +

JV

k

'

. d5

This equation l e a d s t o

+

r(5)

,

0

and the r e l a t i o n s (43) g i v e :

A necessary condition of s o l v a b i l i t y of (46) i s t h a t t h e mean value of t h e

second member i s t h e same as f o r t h e f i r s t member.

Hence

k

i s determined by

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

96

This i s not s u f f i c i e n t i n general t o g e t possible the r e s o l u t i o n of

(46),

which i s a problem of the form

where

s

i s a known Z

periodic function of mean value

0

.

L e t us give

the important Lemma 7.

,r

(small d i v i s o r condition).

Let us assume s 1 s(()dz = 0 , k 2 2

t o be a

, and _7

Ck Z - p e r i o d i c function and assume t h a t c

>

0 and

c > 0 such t h a t

Then there e x i s t s a unique continuous s o l u t i o n

(48) and

' ( 5 - Po) - r ( 5 ) = s ( T ) k-1-E (= Ck-2,1-~ rE C )

r

of the equation

J

*

The proof of such a lemma can be found i n [ 8 ] , b u t a simpler r e s u l t i s

obtain by mean of Fourier s e r i e s , where it i s easy t o see, thanks t o (49) that

(50)

r E Ck-3

.

I n Pact, i f we write

-2irrnpo

r n = (e

-l)-lsn

,

Hopf bifurcation i n R

2

97

and (49) l e a d s t o

f o r a good

.

K > 0

Let us now assume t h a t t h e r o t a t i o n number can f i n d a s o l u t i o n

Po

s a t i s f i e s ( 4 9 ) , t h e n we

of (48) and t h i s l e a d s t o t h e knowledge of

r

fi(5,T)

If we v e r i f y t h a t (37) i s a diffeomorphism, t h i s w i l l be t h e s o l u t i o n

by (45).

of our problem, because (45) i s equivalent t o (35). We have i n R

2

and t h e jacobian of t h i s map i s

by (44) and (47). i n R2

, which

So t h e map

(u,v)l-+ (a,b)

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

i s a r e g u l a r change of v a r i a b l e s

T2 because of t h e r e l a t i o n s (43).

We have proved Theorem '7.

(Kolmogorov)

Assume t h a t for

.

po#o

, t h e r o t a t i o n number Po$

on t h e i n v a r i a n t b i f u r c a t i n g curve

yP

Q

of t h e diffeomorphism,

, s a t i s f i e s t h e small d i v i s o r c o n d i t i o n

(49), and assume t h a t t h i s diffeomorphism i s conjugated t o a r o t a t i o n t h e s o l u t i o n s of (1) on t h e i n v a r i a n t t o r u s

*

,

then

a r e quasi-periodic w i t h two T2 P

periods.

*see,

f o r t h i s condition, t h e Denjoy theorem and t h e t h e s i s of M. Herman f o r a r e g u l a r conjugation [ 8 I .

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

98

Remark 1.

It i s a c l a s s i c a l r e s u l t (see [ 8 ] ) t h a t we have (49) f o r almost a l l

Po

(Lebesgue measure). Remark 2 . When (49) i s not v e r i f i e d it may be impossible t o f i n d a continuous s o l u t i o n of (48), hence t h e s o l u t i o n s of (1) need not be q u a s i p e r i o d i c i n such cases (see a counter-example i n [ 3 1 ] ) . Remark 3 . I n t h e case when t h e s o l u t i o n i s q u a s i periodic on t h e t o r u s , t h e two periods a r e

l/k

and

where

l/pok

k

i s determined by (47). Moreover,

thanks t o (40) we have

because i f we consider a t

-1

t = 0 , q 0 = h ([-Po)

we obtain a t

5=-1

3c

and t h i s period i s t h e mean value of t h e time 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 on t h e t o r u s .

So,

l/k

is close t o

To

.

Comments. A s i n t h e case of paragraph

t h e r o t a t i o n number of the map

5, we have a continuous function

?

P

of

T1 such t h a t

gives a c i r c l e reduced t o a point!)

*It

-__(_-.-I___

l . _ _

--

~

--

p(o) =

? PO

(but

for p = o

~-

i s a mean value with t h e unique p r o b a b i l i t y measure on

under

e0

P(p)

(image of t h e Lebesque measure by

h-l

)

.

T1

invariant

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

(i) I n the case when

2

99

4

and t h e conditions of theorem

B o = p/qE Q

(see

paragraph 3) a r e r e a l i z e d , t h e r e a r e two one s i d e d 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 periods of which a r e

near

.

qTo

(hly one of t h e s e p e r i o d i c s o l u t i o n s 0

.

i s not constant i n a neighborhood of

0

can be a t t r a c t i v e , and only when t h e b i f u r c a t i o n occurs f o r (ii) I n t h e other c a s e p(po)€

p(p)

Q t h e r e a r e closed t r a j e c t o r i e s on t h e t o r u s

When

p(po)t

the t o r u s

.

, some a r e a t t r a c t i v e ,

7 we

32$ .

Now, i f

p(po)+ Q

but does not s a t i s f y t h e

do not know how i s t h e s t r u c t u r e of t h e flow on

9P .

I n f a c t , a n important r e s u l t of M. Herman [g 1 says t h a t t h e set of such t h a t

s a t i s f i e s t h e conditions of theorem

p(p)

p o s i t i v e Lebesgue measure.

We have t o o

{p;p(p)

E

$3

both p o s s i b i l i t i e s a r e observable i n experiments when

7.

Exercise.

7 has

and expressed i n

C

F

CI

generically a

p

varies.

%

*

i s a s r e g u l a r a s we wish f o r t h e computation,

by

Fp(z) = X(p)z + A2(prz) +

a s i n paragraph 1. Assume t h a t closed curve i s given by

p

of p o s i t i v e measure.

P r e c i s e computation of t h e i n v a r i a n t c i r c l e

Let us assume t h a t

(1)

When

Q and s a t i s f i e s t h e conditions of theorem 7 , t h e n t h e

flow i s quasi-periodic on conditions of theorem

>

So t h e flow s p i r a l s between two such closed

the o t h e r r e p e l l i n g on t h e t o r u s . curves.

3CL

p

0

#

... 0

.

Then show t h a t t h e i n v a r i a n t

So,

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

100

( ~ 1 ~ needs ’ ~

Show t h a t t h e term i n

-(aA3 0 , z ) z(cp,p)

,

u 2 -(O,z)

b

, Aj(O,z) ,

d2h ~ ( 0 ) For t h e computation of d!J it i s p o s s i b l e t o use t h e way of t h e f i r s t p a r t of the proof of

, Aq(O,z)

aiL

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

, A5(0,z)

,

.

and

theorem 5 .

8. Domain of a t t r a c t i v i t y and uniqueness of t h e i n v a r i a n t c i r c l e . Let u s consider t h e map F

u

( 1 ) (F,(z)(

= IzI(1

+u

i n i t s canonical form ( l . 5 ) y then ( i f A 5 0

Rehl- alzl 2 + O(1zI4 +

L e t u s assume t h a t a> 0 ( i n t h e case a

1zI2 +

#

1)

Iu12)).

0 i t is easy t o modify t h e proof

t o g e t similar r e s u l t $ . We consider

I

E

> 0 and K

i f z€E1 ;

1

> 0 ( l a r g e enoughland we

d e f i n e t h e annuli E

1

IFv(z

(2)

We may choose E small enough [ s a y

E

6

) and 11-11 small enough t o show t h a t

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

if zEE1

2

101

U E2, t h e r e e x i s t s n such tkiat

This completes t h e proof of theorem 1 , t o show t h a t t h e domain of a t t r a c t i v i t y of t h e i n v a r i a n t c i r c l e c o n t a i n s t h e a n n u l u s

I n t h i s proof we u s e t h e assumption X 5 # 1 , t o b e a b l e t o choose

E

=

t o e n t e r t h e frame of theorem 1 .

It i s n o t necessary i n f a c t . I n t h e case X5 = 1 , we may do t h e same proof a s at theorem 1 b e f o r e t h e 2nd-step i n posing t h e change of v a r i a b l e :

r = ro(u) (

+ p 1 l 4 x ) , ( p 1 i 4 i n s t e a d of p 1/ 2 )

.

It i s c l e a r i n ( 2 ) t h a t i f me r e p l a c e K l p l 2 by K[vi3/2 and the result

3) s t i l l h o l d s . . .

E

by p

1/4

Bifurcation of Maps and Applications

102

Comments on Chapter 111.

( s e e a l s o t h e comments a t t h e end of $111.5 and §111.6).

The type of b i f u r c a t i o n i n t h i s c h a p t e r i s c a l l e d Hopf b i f u r c a t i o k a u e t o the f a c t that

E. Hopf (1942) has f i r s t given an a n a l y t i c way t o compute

t h e closed o r b i t b i f u r c a t i n g from a f i x e d point f o r a f i n i t e dimensional vector f i e l d (see [ 251 f o r a t r a n s l a t i o n of h i s p a p e r ) . The proofs of Lemma 1 and theorem 1 come from 1221, [ 2 5 ] except f o r t h e regularity i n

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

further bifurcations.

.

yp

This one i s u s e f u l f o r

The use of normal forms i s due t o J. MCtjER.

They a r e

extensively used i n a l l t h i s chapter. Theorem 2 g i v e s a p a r t i a l answer t o an open q u e s t i o n which i s what happens if

CY =

O ? (a being defined by (l2), (15) of $1). See remark 1. This q u e s t i o n

i s r e l a t e d t o t h e following open problem:

assume

f i n d a change of v a r i a b l e s such t h a t t h e map

where

@;)L

F

LL

Xn

0

#

1 Vn€N\{O)

, then

t a k e s t h e form

i s smooth.

Theorem 4 ( t h e weak resonance case) 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 study of chapter IV ( s t r o n g resonance).

The terminology i s due t o

V.I. ARNOLD [l]. Theorems 3,4,5 give a n i d e a o f t h e way t o compute t h e asymptotic expansion of t h e r o t a t i o n number of the r e s t r i c t i o n of

F

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

yP The only problem f o r t h e reader who wishes t o do t h a t i s t o f i n d enough t i m e

P

before h i s retirement. Nevertheless, the knowledge of t h e r o t a t i o n number a s a f u n c t i o n of

i s fundamental i f we wish t o understand t h e asymptotic behavior of t h e

I*] see

f o o t n o t e on n e x t page.

p

*

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

diffeomorphism on I n paragraph

y

P

4 we

2

( s e e the comments of '$111.5 and

103

§111.6).

compute a l l t h e c h a r a c t e r i s t i c s of t h e p e r i o d i c s o l u t i o n

which b i f u r c a t e s , even t h e period, i n t h e aim t o show t h a t nothing i s impossible t o t h e one who i s awaked e a r l y morning. I n paragraph

5 and 6 we p r e c i s e t h e behavior of t h e t r a j e c t o r i e s and of

t h e s o l u t i o n s on t h e b i f u r c a t e d t o r i . from

Theorem

6 comes from [7] and theorem 7

[31].

(*) T h i s type

o f b i f u r c a t i o n f o r v e c t o r f i e l d s was first observed by Poincare (18921, l a t e r i n a s p e c i a l c a s e by Andronov (193o)j E.Hopf (1942) gave t h e f i r s t proof i n a more g e n e r a l case f o r t h e f i e l d s . F o r t h e maps Neimark (Ook1.129 (1959)) gave i n d i c a t i o n s on t h e r e s u l t , Sacker (1964) was t h e f i r s t t o g i v e t h e f u l l proof a n d t h e non resonant conditions ( s e e r e f e r e n c e s i n r251 a n d [ X i ) .

This Page Intentionally Left Blank

IV.

FIXED POINTS

SUBHARMONIC BIFURCATIONS OF

I N iR2

(STRONG

RESONANCE).

1. The g e n e r a l study.

Let us consider a map F

P

%F

P

(0) =

iif_’.

A.

A

P

, a s i n chapter

i n XI2

s a t i s f i e s t h e following assumption.

has two conjugated eigenvalues

=

If

a. dP I k( p) I F

h(p)

, w i t h lhol = 1 , k o # + l of

A

P

, such t h a t

, satisfies

ho

IH.2.

xo

and

ho

A s i n chapter 111, we assume t h a t t h e eigenvalue

A(o)

111, such t h a t

I

~

=

>

0

(Hopf c o n d i t i o n ) .

c2 near o , we can w r i t e

is

By doing e x a c t l y the same t h i n g s a s i n c h a p t e r I11 (good choice of a b a s i s i n R2

, and i d e n t i f i c a t i o n of

lR2

and

) we a r r i v e a t a map F

C

P

in

c

of t h e form (2 1

FP(z) = X ( d z + R ( z , z , p )

where

[ R ( z , Z , p ) / = O(lz12)

if

F

is

C2

I n t h i s c h a p t e r we study t h e case when we have chosen t h e s m a l l e s t

n

. Xn = 1 , n

2 3 , assuming

that

.

We saw during t h e proof o f Lemma 1 of c h a p t e r 111, t h a t (i) if Ck-2

h.2 = 1

, and i f F i s of c l a s s Ck , k 2

p-dependent change of coordinates i n

put i n t o t h e form

105

C

2

, there exists a

, such t h a t

F

P

can b e

106

Bifurcation of Maps and Applications

4

Lo = 1

(ii) if

Ck-3

,

i s of c l a s s

and if F

Ck , k

p-dependent change of coordinates i n

23 ,

there exists a

, such t h a t

C

F

can be

p

put i n t o t h e form

I n f a c t , l e t us d e r i v e a more g e n e r a l r e s u l t , which c o n s i s t s i n f a c t of a g l o b a l change of v a r i a b l e which leaves only t h e unremovable terms. Let us assume t h a t H..2,

:L

=

1, n

23

and make t h e assumptions

and l e t us remark t h a t we cannot expect any f i x e d point f o r

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

0

if

k i s not a multiple of

n

.

H.l. and Fk

P

This comes from a

simple use of t h e i m p l i c i t function theorem a s i n chapter 11. Now, i n s t e a d of solving t h e equation n FP(z) = z

(5) in

C

, we

s h a l l write t h e equivalent system:

I

FP(xk) = x

...,n-1

~ f+o r ~ k = 1,

,

L e t us make t h e following change of v a r i a b l e s :

(7)

{ x ==

P

107

Subharmonic bifurcations

The system (6) becomes:

where

i s expressed i n terms of

xk

i n FP(z) , which i s th p equation i n ( 8 ) :

Hence, t h e first respect t o

yl,

k

(n-1)

...,'n-1

z

y

, thanks t o ( 7 ) . The p r i n c i p a l p a r t

9

, g i v e s the following c o e f f i c i e n t of y

equations of (8)

ho

#

...,n-1)

are s o l v a b l e w i t h

, by t h e simple use o f t h e

(p,y,)

a s functions of

i m p l i c i t f u n c t i o n theorem, because

(p=l,

i n the

q

1E-l f o r

..

p = 1,. , n - 1

.

Then we

have

-

0 E Ck i f F i s Ck , k 1 2 , and where we consider y , Y, as P independent v a r i a b l e s . Let us n o t e t h e following property of t h e f u n c t i o n s

where

0

P

.

.

To s e e t h i s , l e t us change xptl

,

for

satisfied.

(10)*

, p

= x:-'ep(p,Yn)

(10)

,..., n-1

p = 1

y

P and

into

xn

.

=

1,. . , n - 1

y Xp-l PO

i n (7).

into

x1

By t h e uniqueness of t h e s o l u t i o n

. This changes

x

P

into

, and t h e system (8) i s s t i l l (9),we then o b t a i n t h e property

Bifurcation of Maps and Applications

108

t o solve

NOW,

x

where

Writing

=

z

(8),it remains t o s o l v e one equation i n c

- n-1 C

1 k-1 +n o Y

r(k-1)(q-1)8q(p,yn) q=l. O

i n s t e a d of

iyn

, we

.

k = 1,...,n

n

o b t a i n a n equation (11) of t h e form

2 I f ( p , z , z ) l = O ( l p l IzI + 1 z I 2 )

where

:

f

is

k

C

, and because of

(10) we

have :

We remark t h a t we.then o b t a i n an equation where a l l the terms which can b e cancelled by changing v a r i a b l e s a s i n c h a p t e r I11 have disappeared! z

i n (12)i s not t h e x1 = z +

(14)

z n-1

;; C

q=l

in (5). Oq(~,nz)

i s a s o l u t i o n of ( 1 2 ) .

where

1.1, z

(p,z)

i s a s o l u t i o n of ( 1 2 ) , then

x1 for

i n (14) becomes

x2

...,n-1

(15) and define w r i t t e n as

w i l l be

Fn CL

, Now, because of t h e property (l3), i f (k,Xoz)

More g e n e r a l l y

i s a l s o a s o l u t i o n of (E), and (po,X:z)

, which g i v e s n f i x e d points

p = 0,1,

by (7) a s i n ( 1 4 ) .

.

I n f a c t t h e f i x e d p o i n t of

The point

i s a s o l u t i o n of (12) X1YX2’

* *

,x

n

defined

Moreover, l e t us put

2 incp z = r e fl(p,r,cp)

= e

-2irCQ

f ( p , r e2inQ

, r e-2incp)

,

then (12) can be

S u b h a r m o n ic b i f u r c a t i o n s

109

and

where

, p being prime t o n

a r g A = 2rr*p/n 0

l/n

- periodic

in

k2

,>

<

0

or

(by t h e Bezout-theorem,

cp

% p/n

, such t h a t

0

Now, i f the fixed point s o l u t i o n of

x1

+

.

Hence, i n f a c t

is

t h e r e e x i s t s two i n t e g e r s

k2 = l / n

of order

fl

n

kl '

).

i s expressed by (14) i n terms

(16), the family of n fixed points of F" , from

of

(p,r,cq)

x1

i s obtained by replacing

cp

by

q+k/n

, k

=

0,1,.. .,n-1

P

, i n (14).

This i s t h e generalization o f what we found i n chapter 11. Study of the case

A:

= 1

.

Let us assume (without loss of generality) t h a t that

F

i s of c l a s s

The c o e f f i c i e n t s the original

(19)

Fu,

Ck

ai(p)

,

k

-> 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

Bifurcation of Maps and Applications

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

,

Bifurcation of Maps and Applications

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

<

0

.

Hence t h e eigenvalues of t h e d e r i v a t i v e 1+36

with

ul+O(s

2

)

a p

'3 ( 0 ) FIL

are

, 1+3s02 + O ( e 2 )

o1 and u2 r e a l of opposite s i g n s .

Therefore f o r

E f 0

one of

t h e s e eigenvaluesis e x t e r i o r t o t h e u n i t d i s c , and t h e theorem i s proved, according t o chapter I.

+ 0(c2),

114

Bifurcation of Maps and Applications

Let us now show what happens when

I

.

So2(0) = 0

Theorem 2. Assume t h a t

F

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

is written i n

C

Ck

,

k

24

h: = 1 , ho

#

1

.

i s of c l a s s

near

,

0

Assume t h a t

with a good b a s i s f o r the l i n e a r p a r t .

t h a t t h e assumptions

to,

=

0

, when F

w

Then, i n general,

t h e r e does not e x i s t any b i f u r c a t e d f i x e d point of o r d e r 3 .

If

k

-> 6

, and

t h e assumption H.3 o r H I . 3 on t h e non-linear term of theorem 1 of chapter 111, holds, then t h e r e s u l t of t h i s theorem holds, i . e . t h e r e e x i s t s a n i n v a r i a n t circle, attractive i f its bifurcates f o r

I for

1

p

<

0

,

p

>

0

,

r e p u l s i v e if it b i f u r c a t e s

t h e r e g u l a r i t y being t h e same a s i n chapter 111.

Proof: A f t e r t h e change o f . v a r i a b l e s of t h e type of chapter 111, our map i s now:

I n f a c t , here we can suppress t h e second order term

So2(p) = O(p)

, because

t h e equation (11)of chapter I11 i s of t h e type Vo2(cl) =

go* h(I.4

(PI

-w2

which i s now r e g u l a r f o r

p

?L2(0)

-

-

near

Ao(h,0

+

O(l.4

2X1) +O(d

, because of Re I , #

change of v a r i a b l e s l e a d s t o a map, without t h e term i n

.

0 -2 z

.

Now, t h i s We put now

t h e map i n p o l a r form and change coordinates a s i n chapter I11 t o o b t a i n (17). Here we obtain

Subharrnoni c bi f u r e a t i o n s

f(cp)

with

1/3 , and B

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

C*

0

=

115

1/3

.

The method used

i n t h e proof of theorem 1 of c h a p t e r 111, t o e l i m i n a t e t h e term

p f(cp)

in

( 3 0 ) , a p p l i e s here i n t h e same way, hence t h e theorem too.

Now l e t us look f o r t h e f i x e d p o i n t s of o r d e r 3 of

F

I-L

.

After a suitable

change of v a r i a b l e s , our map takes t h e form:

hence

where

.

a = a(o)

Now i t i s easy t o s e e t h a t i t i s impossible t o o b t a i n a

b i f u r c a t i o n of a f i x e d p o i n t near (32)

#

a r g A1

arg(aio)mod

I n f a c t t h e p r i n c i p a l p a r t of

of

0

I I

z

#

0

,

p

if

.

f”z) - z

3 z ( ~ l + a i o i z ~ 2+ ) O ( I ~ I i 2 i 3 + where we wish

$

w i l l be

iCLi2izi+ i Z i4)

= 0

real.

Hence t h e theorem i s proved. Remark.

If (32) does not hold, we have i n f a c t

the map

f : cpn Z P

the invariant circle.

.

A study of

a s i n chapter 111.3 l e a d s g e n e r i c a l l y t o two f a m i l i e s of

f i x e d p o i n t s o f order 3 ( s e e theorem

(on t h e c i r c l e )

Q2 = 0 i n ( 3 0 ) .

One

4 in

chapter

I I I . 3 ) , which a l t e r n a t e on

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

Bifurcation of Maps and Applications

116

Lo4

Study of t h e case

= 1

.

Without l o s s of g e n e r a l i t y , we can assume t h a t class

.

Ck , k _> 3

with

fE Ck ,

if k

>7

. .

If we w r i t e

0

=i

, and t h a t F i s of

The property (13) l e a d s t o an equation (11) of t h e form

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

(33) i n function of Aj(p,z))

),

5

(k)

P9 Here we f i n d :

al =a,(o)

,

(cgefficients i n

a2 =a,(o)

, and

of t h e homogeneous parts

F (z) P

z = r e2incp

, (12) t a k e s here t h e

form :

where

g f Ck

, g(p,r,tp+l/4)

2

= g(P,r,tp) = C f d l ) L / +

IIllr2 + r4 ) I -

Theorem 3. Assume t h a t F i s of c l a s s Ck , k 2 4 near 0 , t h a t t h e r assumptions H . l , H.2 hold, and t h a t A. = + i . Define a1 and a2 by

(34) part.

- (35) when

F

P

is written i n

C

, with t h e good b a s i s f o r t h e l i n e a r

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

a

13m

1

a

>

1

117

there does not e x i s t any fixed p o i n t of order

1

4

.

b i f u r c a t i n g from 0 a a there e x i s t s two one-parameter f a m i l i e s of f i x e d (ii) If 13m < 1 1

151

$1

points of order

4 bifurcating

from

0

:

t h e f i x e d points being

.

(37)

(

a

lPl

+-Ikll

4cp0

=

a 2

1

plhl+

1 -2rr arf3[-(

Moreover, i f near

=

o .

F

If

a

a

2

i s analytic, lall

> 1a21

b i f u r c a t e on the same s i d e of

i s unstable.

‘11

.

(mod 1) p(j),cp(’)

and

x(j)

>

, i . e . the two f a m i l i e s

0

a r e analytic i n

e

k1(’)*

p1(2)

9=0

, and a t l e a s t one of the f a m i l i e s

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

118

i

[all

If

<

b21

'" ('). 1

1

PI (2) < 0 , i . e . the two f a m i l i e s b i f u r c a t e on

, and both a r e unstable.

opposite s i d e s of p = 0

Remark 1. The ' s t a b i l i t y of

0

for c~ < 0

,

and i t s i n s t a b i l i t y f o r

r e s u l t from chapter I and from the condition Remark 2 .

If FE Ck

term

,k>6

and i f

a2 = 0

Rehl>

0

.

p

same reason as i n theorem 2 .

, i t i s possible t o cancel the

Hence it i s possible t o f i n d an i n v a r i a n t

c i r c l e a s i n theorem 2 , i f the assumption H.3 of chapter I11 holds

.

T h i s invariant c i r c l e s t i l l e x i s t s when

a

1

< 1x11 1 3 m ~ 1 1 and paragraph 4 for t h e idea of p r o o f ) .

la21 (see [33]

J

Proof of theorem 3 .

In order t o look f o r t h e p r i n c i p a l p a r t of the equation (36), we put

We obtain Alp1 + a1 + a e

(38) where

-ainq0

2

p1

= w1(o)

is real i f f

0

by the change of variables (11) of chapter 111, f o r t h e

a,(p)z3

(Re a l f O )

>

,

cpo=cp(o)

+ o(1)

.

=

Hence

0

%

, i s a s o l u t i o n of ( 3 7 ) , which

,

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

Let us assume t h a t (39) i s s a t i s f i e d , and t h a t

Hence ( i ) i s proved. (pl,cpo)

with

119

s o l v e s ( 3 7 ) . The equation (36) takes t h e form:

, and

glE Ck-3

gl(o,plJcp) = 0

g1(sJ~,cp+1/4) =gl(6,pl,[p), as i n d i c a t e d by (17).

, and we can solve (40) with r e s p e c t t o

h(O,Pl(0),(PO) = 0

s

-8i~tae

=

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

(%,cp)

Writing (40) i n t h e form h ( 6 , h J c p ) = O i n

by t h e i m p l i c i t f u n c t i o n theorem. we have

Moreover

ah

ah

- (O,%(O)J’P~)

c ,

(OJP~(O),V~= )

-8i.n~

0

2

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

- 8iaa2e

Ilkl

i.e.

-8incp0 cpl =aEC

a 1 p1 +8iX(p1+ x;)cp,

=

a hl

J

, t h i s is ivertible

which i s r e a l i z e d i f t h e i n e q u a l i t y (39) i s s t r i c t . of the d e f i n i t i o n of

,

gl

gl

them

(l).

p1

p:’),

j = 1,2

2)>

0

.

if

la2)

> lal]

,

0

,

p

1

c

i n (37): l e t us c a l l

We have e a s i l y t h e following property:

, whereas

#

Now‘because of (13) and

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

Now, i f (39) i s s t r i c t , we have two s o l u t i o n s f o r

a 1 Re(p + -) 1 A,I

iff

kl(1).b p )

<

0

.

if

la1] >la21

The evenness

and t h i s l a s t property then g i v e s a l l t h e r e s u l t s on t h e e x i s t e n c e of t h e

t w o branches of f i x e d p o i n t s of o r d e r 4.

The a n a l y t i c i t y of the branches

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

120

when

F

i s a n a l y t i c , r e s u l t s e a s i l y from t h e i m p l i c i t f u n c t i o n theorem.

Let

us now study t h e s t a b i l i t y of t h e b i f i r c a t e d branches of f i x e d p o i n t s of order 4 .

We use f o r t h i s the form

(4) of t h e map F

P

,

obtained a f t e r a

change of v a r i a b l e s of t h e type used i n chapter 111:

It i s easy t o s e e t h a t by c o n s t r u c t i o n we have

defined by (34), (35).

a(.)

= hoal

, p(o)

= Aoa2

Now

leads 'to :

Now, l e t us consider

where map i n

(44)

x(c) z'

i s one of t h e

4 fixed points # 0

of one family.

The new

can be w r i t t e n a s :

F~' 4 ( z l ) = [ 1 + 4 d 1 + 8 a l l x ( c ) ( 2 + o(e 3 ) 1 z ' +(4alx 2 ( E ) +12a2G(s)2 +

,

Subharmonic bifurcations

The d e r i v a t i v e of

(%A1

+

a t the o r i g i n can be represented by a matrix

F14

U

Id + 4 c2A + O(e3)

121

, where a 1e

2al

The eigenvalues of

4incpo

+ 3a2e

-4irrcp0

, u1 and u2 a r e such t h a t :

A

u + u = 2( plReAl+ 2 Re al) 1 2

But we know t h a t

< O< assume p1

u 1 2

and *

(72

<

<

(p;l)+

pp)

Re

a

51

1)(p(2)+

Re

a

2) < A1

0

, and

if

(1)- (2) < 0

'4

, hence

0 i n t h i s case f o r both branches.

If

p1(l).

'"7.(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

.

.

Bifurcation of Maps and Applications

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 .

Subharmonic b i f u r c a 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)

,

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

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

Bifurcation of Maps and Applications

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

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

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

Bifurcation of Maps and Applications

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

<

121

I b ( a , x l )I

+lpl

1zI2 + 1z15)

and f o r i n s t a n c e

< 0 ( t o have a s u p e r c r i t i c a l b i f u r c a t i o n ) .

The i d e a of

[fj

r

c o n s i s t s t o s t u d y t h e behavior of t h e i t e r a t e s of t h e map F

v i a t h e study of a d i f f e r e n t i a l e q u a t i o n i n 6: :

such t h a t t h e time 1 map

( 3 ) F,,(z) = i ~l,(z) This l e a d s t o

+

$

P

satisfies :

o(lzl 51.

%

o(p) = pi1 + o(p

2

%

%

a l ( 0 ) = a l , a 2 ( 0 ) = a2'

Now, u s i n g t h e f u n c t i o n E ( z ) d e f i n e d by ( i d e a of [33]

E =

1 ~ 1 ~ z22 Z2 - Re(al %

%

-4 ,

a2 z )

) :

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

129

dE and which s a t i s f i e s - (z) < 0 i f I z I i s l a r g e and which i s e q u i v a l e n t t o dt

1zI4 when (z/ i s large, it can be shown v i a t h e Poincar6 - Bendixon theorem p l u s some p r o p e r t i e s of

if

p

5

t h e p e r i o d i c s o l u t i o n s o f (21,

that

0 , 0 i s a s y m p t o t i c a l l y s t a b l e , i t s domain of a t t r a c t i o n b e i n g a l l

a,

while i f p > 0 t h e r e i s a unique p e r i o d i c o r b i t , i n v a r i a n t under t h e map

z -+I

i z and such t h a t i t s domain of a t t r a c t i o n i s CC

\

(0).

The r e s u l t s on t h e d i f f e r e n t i a l equation ( 2 ) l e a d t o t h e e x i s t e n c e of

an i n v a r i a n t c i r c l e f o r a map which approximates w e l l F

P

. Then,

u s i n g t h e method

of c h a p t e r 111, we may " p e r t u r b t h i s c i r c l e " t o prove t h e Hopf b i f u r c a t i o n i n t h i s resonant case t o o .

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

130

Comments on Chapter I V . The form of t h e theorems

1,3,4seems new.

For d i f f e r e n t i a l equations i n

an i n f i n i t e .dimensional space they were proved i n [ 151 ( s e e $ 2 ) .

An open

problem i s t o know whether t h e r e e x i s t s a smooth change of v a r i a b l e s l e a d i n g t o a map

where

F

!J

of t h e form

g ( A z,xoz) = A g P

O

O P

(z,i)

.

What we d i d i s d i f f e r e n t , b u t s u f f i c i e n t

f o r looking a t p e r i o d i c p o i n t s . Theorem 2 i s here j u s t t o prove t h a t i t may e x i s t a n i n v a r i a n t c i r c l e i f the condition f o r s t r o n g resonance i s not r e a l i z e d . Note t h a t i n t h i s chapter we show how t o compute a resonant s o l u t i o n using t h e map F

1!

. In practical

cases, for a non-autonomous differential equation as in

paragraph 2, t h e b e s t way would be t o proceed d i r e c t l y on t h e d i f f e r e n t i a l equation, looking f o r nT-periodic n o n - t r i v i a l s o l u t i o n s . Computations a r e much s h o r t e r and may be done i n a Banach space t o o ( s e e [15]

) , This may be done

even i n t h e weak-resonant c a s e s , i f t h e s u i t a b l e a l g e b r a i c conditions a r e f u l l f i l l e d ( s e e paragraph 111.3 f o r t h e c o n d i t i o n s ) . Let us conclude t h a t i n t h e case X o = i

(A:=,)

, an

open problem i s t o know

whether t h e r e e x i s t s a b i f u r c a t e d i n v a r i a n t c i r c l e when t h e r e are f i x e d p o i n t s of order

4 bifurcating

from 0. The conjectures about t h i s case a r e m a i n l y i n

b] .

V.

I N V A R I A N T MANIFOLOS AN0 A P P L I C A T I O N S .

The aim of t h i s chapter i s t o give a powerful t o o l t o enable one t o reduce an i n f i n i t e dimensional problem t o a f i n i t e dimensional one o r a f i n i t e dimensional one t o one of a lower dimension.

The c e n t e r manifold

theorem w i l l give us t h i s t o o l and we s h a l l use t h i s theorem t o reduce some problems i n t o t h e problems t h a t we have considered i n chapters 11, 111, I V . We s h a l l a l s o g i v e i n t h i s chapter p r e c i s e r e s u l t s on t h e behavior of t h e i t e r a t e s of a point f o r a, map, near a f i x e d p o i n t , i n t h e hyperbolic situation. This type of r e s u l t and more g e n e r a l s i t u a t i o n s a r e developed i n a general way i n t h e l e c t u r e notes of HIRSCH, PUGH, SH[IB [ l o ]

.

Here, we g i v e

elementary proofs, which a r e mainly i n t h e papers by O.E. LANFCBD[22] and by O.A. LADYZHENSKAYA and V.A. Banach space

If

E

on R

i s of c l a s s

F

C1

.

L e t us d e f i n e a map

SOLB\TNIKOV [21].

We assume t h a t

F

in a

F(0) = 0 and w r i t e

i n a neighborhood of

0 in

E

, we have LE a ( E )

DR(0) = 0 and

= s~~(llxll,Il~ll)

for

llxl\ and

RE CkJ1

and

IlylI

3y >

t h e spectrum a(L)

e(r)

small enough.

-t

r + O

If

,

o

F

i s of c l a s s

CkJ1

, k 1 1 , then

e ( r ) 5 yr

.

We assume some conditions on

of t h e l i n e a r operator

L

, o r i n f a c t , on i t s n a t u r a l

0 such t h a t

complexified o p e r a t o r on t h e complexified Banach space 131

EC

.

W e assume

,

132

Bifurcation of Maps and Applications

u(L) =

u no2 = j4 1

with

Ul U a

2

'

,

Let us define, a s i n chapter I, the projectors L

and which a r e associated with the p a r t s

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

L2

is

and

a1

t o the invariant subspaces

L

( r e a l ) a r e denoted by

P1 , P2 which commutes with

and

L1

L2

.

=

and

El = P E 1

The spectrum of

, and we assume t h a t the norm i n E

u2

of the spectrum

a2

is

L1

El@ E2

o(L)

E2 = P2E

, t h a t of

u1

s a t i s f i e s the proB

.

k _> 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

<

two unique manifolds

5

( s t a b l e and unstable manifold), which a r e graphs of the maps 'pl: El 'p2:

(k=O if

E + El 2

F

is

, respectively. C

1

if k l l if k = 0

The rnaps

'pi

a r e of c l a s s

) , have the properties

, W1(0)= W2(0)= 0

,

3-

, 1 6 :R + R + such t h a t Q ( r ).-t

l l ~ j ( x ) l5 / @(llxll).llxll , j =1,2 f o r a small

r+O XE

0 and

E

j '

Ck2l

.

1< a

+

, E2

.

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

3)

The manifolds

vl

Moreover, i f

xE

%

and

m,

R1

and

%

t h a t , if x4

a r e l o c a l l y i n v a r i a n t under t h e map

( r e s p . $)

defined and tends t o

0 when

p

i n a neighborhood of

m,

133

, then 3

2

,

.

F

and

p

-t m

.

These p r o p e r t i e s c h a r a c t e r i z e

0

.

In particular,

with

0

II#(x)ll

Fp(x) ( r e s p . F-’(x))

-> 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

<1

,

such t h a t

j = 1,2

.

Let us define

9 t o be the l i p s c h i t z constant of

R

134

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

near

0

.

When the neighborhood of

0 i s small enough,

T!

i s a s small a s

we want. i ) Existence and c h a r a c t e r i z a t i o n of t h e unstable manifold

Let us define A

in E

7

put the

YT

{v : E 2 ( 7 )

=

, where E2(r)

yI/x2- x;11] radius

0

Co

.

2

-+

; hp(x,)II

El

5

7

,

m2-

IIv(x,) -cp(x;)ll

i s t h e closed b a l l centered a t

The space A'

- topolo@;yon i t :

v

Lemma 1.

ip

r

, the map @v:E,(T)

(x ) = ~~x~+ p 2 ~ [ x 2 + cp(x2)1

cp2

,

i s a b i l i p s c h i t z homeomorphism, such t h a t

-1 @v EE2(7)1

.

c E2(7)

Proof. We have

where

1= o(1) g2

as

= -2 ( X )

cp2

T

-+

0

.

Writing

, we have

,

of

i s a complete metric space, i f we

L e t us f i r s t show the lemma:

For a small enough

0

5

for

4

E2

c p A' ~

v

defined by

'

Invariant manifolds and a p p l ica t i o n s

and the second member i s a s t r i c t contraction i n Hence t h e r e i s a unique

enough.

f o r a small enough

,

7

Let us now define

where

x2

y

contraction i n

A

0

Y

,

.

small

r

By ( h ) , we a l s o have

by

.

*v-1(z2) f o r g2E E 2 ( r )

=

-

mCp-1(x,)

for

and the lemma i s proved.

0

If we can show t h a t

A>

x2 =

x2E E 2 ( r )

135

5 maps Ayr

i n t o i t s e l f and t h a t i t i s a s t r i c t

then we s h a l l obtain a unique fixed point

in

'p2

:

This w i l l mean t h a t i f we s t a r t with a 'point image of

will be

x

Z2E E 2 ( r )

.

graph i s

'p2:

%

=

22 +

(Scp2)(Z2)

=

x = x2+ v2(x2)

Z2+ cp2(Z2)

in

E

-+ El

.

Now, f o r

tpE 'A

vr

the

, provided t h a t

This w i l l prove the l o c a l invariance of the manifold

E2

,

%

whose

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

136

f o r a small enough

x2 = erp-1(g2)

where

llx*- $11

5

b2

.

7

,

This proves t h a t 9rpE A'

xi

Ildx,)

-1

= Crp

- tp'

(Z2)

(xi)/I

v7

.

Moreover

.

But (4) l e a d s t o

T

small enough, i n A

*

Because of

we have

and

9 i s a s t r i c t contraction, f o r

has a unique f i x e d p o i n t manif old

m,.

tp2E A

v , which

0

0

YT

.

The map

gives t h e l o c a l l y i n v a r i a n t

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

Let us remark t h a t t h e closed subspace of 'A those

such t h a t

cp

, is

cp(o) = 0

a s t r i c t contraction i n A

for

F

0

YT

where

cp2(o) = 0

3

.

.

c o n s i s t s of

Hence, by t h e

The f a c t t h a t 3

l e a d s t o a l o c a l l y a t t r a c t i n g manifold

is

?$

:

l e t us consider a point

F(x)

r , which

i n v a r i a n t under

uniqueness of t h e f i x e d p o i n t , we have

137

= f;,+

0.

<

,

x2E E(T)

x = x1 +x2

cpE 'A

Y7

such t h a t

E(r/2)

then we have

1 i s given by ( 1 1 ) .

t o show t h a t

in

,

To s e e t h i s i t s u f f i c e s t o d e f i n e

and use (11)with

cp

and

'p2

, making t h e remark

t h a t , with our choice:

To show t h a t

cpE A'

y7

, it

s u f f i c e s t o show t h a t

we have

where

1=

M

o(1)

when

7 -I0

, and

\lcp(x;)\l

5

T

.

But

Bifurcation of Maps and Applications

138

where t h e s e r i e s converges i n norm i n El

:

Now, (13) and (16) give

which proves (l2) and t h e l o c a l a t t r a c t i v i t y .

result 2) of theorem 1 f o r

??$

.

We a l s o deduce from (12) t h a t : i f VpEW

Z€%

, 3 x = x 1 + x 2 E E(r(2)

F[X2+

I

such t h a t

2 = Zl+ S2'2E E(7/2) $(x 1+ x2) =g1+Z2

satisfies

,

then

. Conversely, i f

with

I n f a c t t h i s a l s o proves t h e

x2

= 4

Z2+ 'p2(z2)E % , we

-1 ((si2)E E2(7)

'p2 Q2(x2)] =

.

This can be w r i t t e n

Hence t h e manifold

c h a r a c t e r i z e d as follows: it i s t h e set of

2 = P1+ P2E E(r/2) with

w e l l determined.

s2 +rp2(Z2)

r K T E ( r / 2 )i s

have

FP(xl+ x2) =

such t h a t

xl+

Z2

.

VpEN

,

there exists

x = x l + x2E E(7/2)

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

By

139

( 5 ) , we a l s o have

which shows t h a t

z2)

F-'(S1+

+

p++m

0

-

ii) Existence and characterization of the s t a b l e manifold

9-

We proceed i n t h e same way, defining a complete metric space A ; ~ a s previously except with the indexes 1 and 2 permuted. The map

21

(18)

0

S i n Av

= L

i s defined by

,

x + PIR[xl+ cp(xl)]

11

s o t h a t a fixed point

tpl

of

9

satisfies

which insures the invariance of the graph fact that

3 i s a s t r i c t contraction i n A'

obtain a unique fixed point

Now, we have

'pl

of

9

3 YT

i n A'

of

Q1

.

The proof of the

i s straightforward.

yr

I

Then we

140

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

with

This l e a d s t o a n e s t i m a t e on

which shows t h a t

'pl

of the type of

F(x) = j ;1+cp1(21) E

.

(7)

+E2)

if

x=

, and for a small enough

-i 0 To end the c h a r a c t e r i z a t i o n of ?$, , l e t us show P+* t h e following equivalent property: 6 > 0 such that, i f xf , then T

, Fp(x)

TR1"(E 1

(16). Moreover,

3

3 p E M with

ll@(x)l\ >6

.

To show t h i s we make a change of v a r i a b l e s i n a neighborhood of

0

in

E :

This change i s of t h e form I d e n t i t y + small p e r t u r b a t i o n , hence it i s a

b i l i p s c h i t z homeomorphism between two neighborhoods of

37 >

0 with

I n t h e new v a r i a b l e

(23 1

UH

u

G(u) = @

,

t h e map F

is written

O F o f -1(u) = Lu +

Z(U)

,

0 in

E

, such t h a t

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

-

g ( o ) = 0 , i\%(u)-R(u’)ll

where

u’E E ( T )

5 f\//u-u’I/, ll= o(1)

for

141

7 -t

0

,u ,

.

Now, by construction, i f Then F(x)E

%

and

u1 = 0 , “ E m 2

PIG(u) = 0

.

.

Assume t h a t

F(x)E E ( T )

.

Hence

and i n t h e same way

This gives

Now, f o r and

G

u

9

i n a s u f f i c i e n t l y small b a l l ,

extends t h e p r o j e c t i o n on

E2

.

i s such t h a t

bil-7

>1 ,

This obviously proves t h e end of

p a r t 3 ) of theorem 1. iii) Regularity of t h e manifolds.

Let us assume t h a t

for

i s analogous.

F i s Ck’l , k

2

1

, and consider

% .

The idea c o n s i s t s o f showing t h a t t h e map

The proof S

acts

from a new metric space i n t o i t s e l f a s a contraction, and t o choose a s u i t a b l e complete metric space

Y,M,T

Bifurcation of Maps and Applications

142

Let us define

we note

.

M = (Mo,...,hQ

It i s known t h a t t h i s space, with t h e

i s complete ( s e e

[22]

or [24, p. 41-43).

Co metric:

Note t h a t , i n t h e f i n i t e dimensional

case, t h i s Gesult follows from t h e Ascoli theorem. Take

-1 ..

eCp (x,)

where

x2 =

where

o ( 1 ) + 0 when

4, =

0,

. . . , j -1

(4) l e a d s t o

a s previously, then

Mo=7

.

.

Then we have

T

+ 0

, and

C

j'

j = 2,

...,k+l

o d y depend on

Md

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

Now the d e f i n i t i o n (6) of the map

3

143

, leads t o

This gives, because of (28):

5

where

by

only depends on

This leads t o t h e choice of

lb2(4)(Z2)ll 5 Iv$

*

M2

.

,

l a r g e enough t o obtain

For higher order d e r i v a t i v e s , we have the same phenomenon, which allows

us t o chosse successively

%

3

5

, . ' * , Mk

y w i l l depend on the choice of 3 i s a map from A

So

in the Ak v,M,T

_>

2 ).

For k

2

1 ,

Y,M,

7

r

, MI,. . .,%

into itself.

9

. i s a s t r i c t contraction,

topology, a s seen i n (ll), hence, because of the completeness of

Co

t h e r e g u l a r i t y r e s u l t s of theorem 1 hold.

'

Example 1.

For

k

Y l a r g e enough t o have

we choose

here

(for

E =B2

, let

us consider t h e map

F ( x , y ) H (X,Y)

:

144

Bifurcation of Maps and Applications

k1 < 1 < k2

where

.

The s t a b l e manifold w i l l be

the unstable manifold will be

%$=

%

= {(x,y);y=cp1(x)]

.

{(x,y);x =cp2(y) ]

Let us w r i t e

We obtain

a2 =

bl - h2

, a =

b (b - 2 a )i ) 1 2 11

>

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

’

Bifurcation of Maps and Applications

146

!I

Theorem 2 .

(Center manif old theorem)

Assume F and

b
,j

t o be =

C1

or

for

Ck’’

o,l, ...,k + l

has t h e property

$ , non-unique

e x i s t s a manifold

, defined i n a neighborhood of

Ckyl

(i)

( ~ ~ ( =0 0 )

(ii)

$

i s l o c a l l y i n v a r i a n t under

(iii)

$

i s l o c a l l y a t t r a c t i n g under

neighborhood of Remark.

I n the l i n e a r case

Fn(x)

, then

0

(F = L )

,

0

Then t h e r e

,

cp2 : E2-’E1

0 and such t h a t

if k _ > 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: <

1 , j =o,l,.

The problem i s t o pass round the impossibility t o define

$1, because when

Z2‘2E E2(‘r)

3 l i k e i n (6) of

, x2 does not necessarily belong t o E2 ( T )

To do t h i s , l e t us begin b y doing the change of variables Then we modify t h e map by multiplying R theorem 2 .

..,k+l .

This gives us t h e new map

by

XHX/T

=

2

P :E -+B+ defined i n t h e 2

.

.

9

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

a(?)

where

=

1

7

R(T?)

= ?l +G2

,

,

and

.

(unit b a l l i n El) and f o r any ? E E2 2

147

$ i s defined f o r ? E E1(l) 1

Moreover, we have

which shows t h a t , i n a small neighborhood of t h e o r i g i n t h e map i s unchanged, and t h a t our new map i s now defined i n Let us now suppress the roofs on

.

R(x)

where

x

=

h

.

E1(l) @E2 F

and

,

x

and w r i t e

p(G2)6(?)

We have the following e s t i m a t e s :

i s a number such t h a t

x1+ x2E E1(l)

E2

h

+ 0

as

T-)

0

,

and where

.

The 'proof i s now analogous as t h a t of theorem 1, with t h e f u n c t i o n a l space

With the

Co

- topology,

t h e metric space

i s complete.

as

.-

Bifurcation of Maps and Applications

148

A.

We can choose

small enough such t h a t , i f

M

5 1 , for

any

cpE

$

t h e map

gV: E2

*

E2

defined by

= L2x2 + P2R[x2+cp(x2)1 9 (x i s a b i l i p s 9c h i2t z homeomorphism.

---

.

Remark.

_-

We can choose

Ir

as small as we wish provided t h a t

r

is also

small enough. Proof of Lemma 2 . The proof i s n e a r l y t h e same a s t h e lemma 1, except t h a t , h e r e

and

(8)

/Icp(x2)\l 5 1 i f

5

lIx2-x;ll

M

51 .

We o b t a i n t h e estimates

1-2b21\ b2 ll~2-y 9

which gives t h e l i p s c h i t z constant of Let us now d e f i n e

where

-

-I x2 = gCp (x,)

,

cp

3 as previously, by

i 2 E E2

.

.

x2E E2

,

149

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

If we can show t h a t

maps

gr

%0

i n t o i t s e l f and t h a t it is a s t r i c t con-

t r a c t i o n , then we s h a l l o b t a i n a unique f i x e d p o i n t

cp2

of

9

in

% -

This w i l l prove, a s previously, t h e e x i s t e n c e o f an i n v a r i a n t manifold whose graph i s

cp2 : E2

.

El

-I

This manifold i s defined everywhere i n

b u t i t i s only i n v a r i a n t f o r t h e “modified“ map

F

.

This l e a d s t o t h e

e x i s t e n c e of a l o c a l l y i n v a r i a n t manifold f o r t h e t r u e map t r u n c a t i n g by

F (before

p ) , which coincides with the modified one i n a l i t t l e

neighborhood of t h e o r i g i n . i) Existence of a g l o b a l l y i n v a r i a n t manifold f o r t h e modified map.

Now, f o r

since

b

and f o r

cpE

1 < 1 and blb2 < 1

This proves t h a t

where

-x ,.

x2 = IT (x,)

But, (6) l e a d s t o

and then

=

and

h

.

@€A;

, xi

%

@-Q’’

h/M

.

($)

.

small enough, we have

E2

’

150

Bifurcation of Maps and Applications

This estimate and (12) l e a d t o

II(%')

(14) where for cp2

Ci

-

(z2) I/ 5

(%I

$

, and A/M

T)

, which

F

of the cut-off f u n c t i o n

p

cp(0) = 0

have

( ~ ~ ( =0 0)

\lcp2(x)ll

.

IIV-V'

/Io

~ I I v - v ' IIo

=

2

4 '

Hence t h e r e i s a unique f i x e d p o i n t

i s non-unique i n g e n e r a l because of t h e choice

.

Now, the closed subspace

Assuming only

= o(llxl1)

l-2b2i

This g i v e s a l o c a l l y i n v a r i a n t manifold under

, i s i n v a r i a n t under d

that

(bl+ 2h)b2h

3 i s a s t r i c t contraction i n

small enough.

f o r t h e modified map.

t h e "true" map

know i f

(bl+ 1 +

< 1 f o r h small enough. Then

A (i.e. in

(f,)

when

F

.

t o be

$

, of

cp

such

Hence, by t h e uniqueness, we

i t i s an open problem t o

C1

x-1 0 i n E2

.

So, we cannot use t h i s

m2.

property, a s f o r theorem 1, t o prove t h e l o c a l a t t r a c t i v i t y of

ii) Global a t t r a c t i v i t y of the i n v a r i a n t manifold f o r t h e modified map.

Let us show t h a t our g l o b a l manifold f o r the modified map i s a t t r a c t i n g ( n o t o n l y l o c a l f o r t h e modified map!).

This w i l l l e a d t o t h e l o c a l

a t t r a c t i v i t y f o r the "true" map.

It s u f f i c e s t o show t h a t

VxE E

such t h a t

l\P1x\l

51 ,

t h e n i f we

note F(x) = we have

//:,/1

5

1 and

f1+ 2 2

'

ll~l-cp2(f2)//

w i l l lead t o

dist(Fn(x),%)

-t

0

n-1m

5 aIIx1-'p2(x2)I/

f o r an

a <

1

.

This

I n v a r i a n t manifolds and a p p l i c a t i o n s I n f a c t , we have

hence

i

112 1,1

Zl = L1x 1 + P1R(xl+ x2) g2

_< b l + h 5

Let us consider

z2

,

= L x + P2R(x + x2) 2 2 I

and

which l e a d s t o

x* = @

*

= L2 x2

hence

1 for

7

v2

*

+

small enough.

(z2) , then

-1

P2R[x2

+

*

v2(x2)1

,

151

152

Bifurcation of Maps and Applications

Pose

CI

=b + 1

)I

+ (bl+

2X) (l-2b2X)-Ib2h

<1

for

T

small enough, hence t h e

g l o b a l a t t r a c t i v i t y i s proved f o r t h e modified map, and t h e l o c a l a t t r a c t i v i t y of t h e manifold f o r t h e “ t r u e ” map holds.

FE C

I

(k=o)

near

0

Hence theorem 2 i s proved f o r

.

Regularity of the manifold.

iii)

Let us assume t h a t

F

is

,

Ckyl

k

2

1

, we

s h a l l proceed a s f o r

theorem 1, with t h e f u n c t i o n a l space

Th’is space i s complete with t h e

metrix (see [ZZ] o r [25] p .

Co

We can compute as i n theorem 1

where

hl,.

..,‘k

-+

0 when

T

0

-t

.

This leads, a s f o r theorem 1, t o the estimates

...,k

j = 1,

,

41-43).

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

153

and

with

+

0

r+o

So,

. Ak i n t o i t s e l f , because we e a s i l y v e r i f y t h e

3 i s a map from

l a s t property:

D(%p)(O)

is a s t r i c t c o n t r a c t i o n i n t h e

C

.

cpE Ak

for

= 0

0

We have proved i n i) t h a t

topology, then t h e theorem 2 i s proved.

E x p l i c i t computation of the q u a d r a t i c term of Let us observe t h a t our

(p2

of i t s Taylor series a r e unique!

where

R (2)

s h a l l obtain

cp

if

+

.

L e t us w r i t e

FE C 2 ' l

cp2(x) = cpL2)(x,x)

2

i s not unique i n g e n e r a l , b u t t h e c o e f f i c i e n t s

i s b i l i n e a r , symmetric, bounded i n cp2E C 2 ' l

9

O(llxf)

E

, and b y theorem 2 we

, then we a l s o pose

, xE

E2

By i d e n t i f i c a t i o n , w e o b t a i n

hence

We observe t h a t , because of

2 1 2

b b

< 1 , t h i s s e r i e s converges i n norm and

determines t h e unique s o l u t i o n of ( 2 0 ) . of

v2

A l l t h e terms of t h e Taylor s e r i e s

may be obtained through t h i s way, provided enough r e g u l a r i t y of

F

154

Bifurcation of Maps and Applications

and the conditions

b bJ

1 2

< 1 realized.

E2 i s f i n i t e dimensional, we can compute d i r e c t l y p(2) by i d e n t i f i c a -

If

2

t i o n using a s u i t a b l e b a s i s o f Example 3.

E2

, a s i n example 2 of paragraph 1.

Non-uniqueness, non-analyticity of

L e t us consider t h e map F

% .

i n B3 : (xo,yo,zo 1-

(X,YtZ)

h(Yo- zos) ds

0

z = z where easily

h

is

ti

F(O) =

Let us note

X

C

m

function on R

, with

o ,

=

, then

(x,y,z)

1 L x = (,x, 1

0,o)

L2X = (0,y-z,z)

The Taylor expansion of

h

can be w r i t t e n

h(0) = h’(0) = 0

.

We have

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

even if it does not converge i n general. 2

i s the graph of

cp :R

A center manifold f o r the map (22)

- + R such that

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

F

.

The formula (20) gives 2 2 cp(2)(Y,z) = a y + B y 2 + y z (1-e

-1

)a

=

h2e 0

s-1

ds

/

(1-e -1)p - 2 e - l a = 12$(1-s)eS-lds 0

, so we obtain

f o r any regular c e n t e r manifold f o r (22). L e t us look f o r a global expression of some center manifolds f o r ( 2 2 ) !

If we modify

h

into

E E Cm such t h a t

( t h i s modification i s always possible).

Now w e pretend t h a t the graph of

155

156

Bifurcation of Maps and Applications

Hence it i s non-unique because of t h e p o s s i b l e

i s a c e n t e r manifold f o r (22)! choices on

which modify t h e manifold (27) even i n a neighborhood of t h e

o r i g i n (without modifying t h e Taylor expansion a t

(0,O)

of

cp !)

.

I n f a c t , i t can be proved e a s i l y t h a t

Lj0

eo h(yo-ozo)do+J 1e S - l c(yo-szo)ds =

-m

0 -m

0

.

and t h i s shows t h e g l o b a l invariance of t h e graph of (27) under l e a d s t o the

local invariance

The Taylor expansion of

,

es G(yo-zo-szo)ds

of t h i s graph under cp

i n (27) a t

a s i t can be e a s i l y shown using ( 2 3 ) i n

-h

for

.

F

(0,O)

This

can b e w r i t t e n

y-oz

i n a neighborhood of

0

The formula (28) shows t h a t even i f t h e map (22) is a n a l y t i c i n a neighborhood of

0 in

pi3

, f o r instance

i f we assume

a n a l y t i c , t h e n we cannot f i n d

h

i n g e n e r a l a c e n t e r manifold which i s a n a l y t i c i n a neighborhood of

0

.

We

s e e on (28) t h a t a necessary condition t o g e t an a n a l y t i c c e n t e r manifold (which w i l l be unique) i s t o assume t h a t

hk

decrease f a s t enough as

C q! h z9 9 92 2

k

+ m

converges f o r

h

a n a l y t i c on a l l R

so t h a t a

z

#

0

an4 t h a t t h e

.

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

Remark.

157

The map (22) i s t h e time 1 map of t h e d i f f e r e n t i a l equation

introduced i n [ 2 2 ] t o g i v e one example of non-uniqueness and n o n - a n a l y t i c i t y of the c e n t e r manifold.

The proof i n [ 2 2 ] i s d i r e c t l y done on t h e d i f f e r e n t i a l

% .

system (29) which i s not s o f a i r because of t h e non-uniqueness of

The

way t o o b t a i n t h e i n v a r i a n t manifolds f o r d i f f e r e n t i a l equations, from t h e theory on maps i s t o put t h e cut-off f u n c t i o n d i r e c t l y on t h e system; s o t h e time 1 map s a t i s f j t s t h e s u i t a b l e property f o r t h e use of t h e method of t h i s paragraph.

3.

Application t o b i f u r c a t i o n problems. Let

b e a r e a l Banach space and l e t

E

defined i n a neighborhood of neighborhood of

is

Ck , k

2

1

0

.

.

O

F

be a family of maps:

P-

, such t h a t F (a) = O , and P-

E

4

E

i n a real

p

We assume t h a t t h e map R x E -+ E

We d e f i n e the continuous family of bounded l i n e a r o p e r a t o r s

and assume I.._____

For

p = 0

, t h e spectrum

unit c i r c l e and a part

0")

1

of

A

0

i s t h e union of a p a r t

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

(O)

O2

on t h e

158

Bifurcation of Maps and Applications

We c a l l

El( O )

, EAo)

t h e subspaces of

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

, Pio’

oio’ U

t h e p r o j e c t o r s on E(O)

1

i n v a r i a n t under

E

, which

of t h e spectrum, and we note

0:”’

and

A.

which commute w i t h

ELo’

A

0

.

The idea i s t o reduce t h e study of t h e asymptotic behavior of t h e i t e r a t e s under

F

P

of a p o i n t i n

which has a lower dimension t h a n E t h e r e s t r i c t i o n of t h e map

M

F

c1

,

E

.

a study on a new map i n

to

Eg( 0 )

For t h a t , we p r o j e c t on t h i s subspace

on a l o c a l l y i n v a r i a n t and a t t r a c t i n g manifold

.

P

To obtain a family of l o c a l l y a t t r a c t i v e and l o c a l l y i n v a r i a n t manifolds

under

F

P

,

li. i n a neighborhood of

for

0

,

we introduce t h e map

H : R X E +IRXE defined by

The map

H i s Ck by c o n s t r u c t i o n and

.

DH(O,Oj[p,xl = [P,A~XI= L[P,x]

(5)

The spectrum of

L

c o n s i s t s of

o (’)

1

a s s o c i a t e d w i t h t h i s s e p a r a t i o n are

(O)

u02

u El] .

{ o ] x E~( O ) and

theorem 2 we need t o assume

I

F-

H.2:

W X E 2( 0 ) has t h e property

(P),

.

The i n v a r i a n t subspaces R X E 2(o)

.

To use t h e

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

Note t h a t t h i s c o n d i t i o n i s r e a l i z e d i f

159

E (O)

i s a H i l b e r t space o r i s 2 f i n i t e dimensional, which i s our main i n t e r e s t i n what follows. The

assumptions of theorem 2 a r e s a t i s f i e d f o r t h e map c e n t e r manifold

of c l a s s

Ck-l,1

m

, so there e x i s t s a

H

which i s t h e graph of

i n a neighborhood of

(0,o)

.

Moreover we note

and we have

(7)

'P(0,O) = 0

, W(0,o)

=

6 if

k >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

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

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 :

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

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

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

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)

i n a neighborhood of

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

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

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

Bifurcation of Maps and Applications

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

i n a neighborhood of

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

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

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

<

Re X1 > 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.

i n a neighborhood of

0

More-

,

.

Bifurcation of Maps and Applications

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

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

& 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<.-b.--

u") 2

1

value of t h e r e a l operator A.

= {Xo,xo)

, lhol

condition), where

h,

=

1

,

ho#+l

.

, and ho i s a simple eigen-

Moreover

i s defined by ( 2 8 ) .

The formulas (11) and (l2) l e a d t o

ReA1> 0

(Hopf

,

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

168

which defines t h e operators

i n (15); the map G

other c o e f f i c i e n t s

‘pll

and

becomes a map i n

P

5,

and

cp

02

C

on E2(01

.

Let us put

:

$(l) can e a s i l y be computed too, using (15).

P9 We make now t h e change of variable in

C

:

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

169

then t h e new map takes t h e form G ' ( z ' ) =Ao(l+$.l)z' CL

(40)

+

'q+ o( (

z'

2 2
I 2+1

PI

I z ' I + 1 z ' I 3) I z I ) I

,

which i s good t o compute t h e i n v a r i a n t closed curve o r the periodic points of chapters I11 and I V .

5'

I f we need t o eliminate

i n the l i n e a r p a r t of

G'(z') P

we can make a change of coordinates of the type

2

z " = z ' + p ( a ( p ) z ' +P(P)Z')

(41)

and f i n d a(p)

, @ ( p ) by the i m p l i c i t function theorem.

r e s u l t s of chapters I11 and IV on the map E

,

f o r the map F

CL

X = PRe([z'

(42)

a c e used the

(40), t h e corresponding p o i n t s i n

, w i l l be: LA!

' 1 -zl]co)

1-X2

+ p

(pll(z'(~o+~5,)

+

0

4. Applications t o d i f f e r e n t i a l equations. 4.1.

The non-autonomous case.

Let us consider the d i f f e r e n t i a l equation i n a r e a l Banach space

where

r9

H

L ( t ) i s a family of unbounded l i n e a r operators with a fixed domain P

, and N ( t , . )

contains

P

&

.

a family of unbounded nonlinear operators whose domain

We denote

E

the space

fi

with a s t r u c t u r e of Banach space

( i n most cases t h i s w i l l be the graph-norm of

Lp(t))

, and we assume t h a t

,

Bifurcation of Maps and Applications

170

( 31

Np(t,X)

+O(t!XtIE)4

+NF)(t;X,X,X)

= NF)(t;X,X)

in

H

4 22

.

a r e q - l i n e a r symmetric bounded: E -t H L e t us P9 assume t h a t t h e Cauchy problem (1)with X ( 0 ) = XoE E , i s s o l v a b l e s o t h a t :

where

and

N(')

P

N

t h e r e e x i s t s a unique s o l u t i o n of (1):

for is PE I

tE[O,T]

,

Ck

k

,

open i n t e r v a l

pc

I

,

*

and such t h a t t h e map X :EXRXR+ -+ E

l a r g e enough*, w i t h X (XO,p,t)

.

-t

t-to'

Xo

in

E

These assumptions a r e r e a l i z e d i s t h e case when H = R n function

n2x R"

-+

4 ~ "x )d '

, uniformly i n , and t h e

:

i s r e g u l a r enough, and (4) i s defined on

(-T,T)

.

I n t h e i n f i n i t e dimensional c a s e , l e t us give two examples Example 1.

*It

i s s u f f i c i e n t t o have h e r e

Co

in

t

,

Ck

i n other variables.

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 2

u = u ( t , x ) E R , xE h2 bounded regular domain of R

where

a(p)

, b ( p , t ) a r e s c a l a r functions and B(p,t)

regular i n t h e i r arguments.

The nonlinear term

, we

0 i n RXW2

i n a neighborhood of

171

.

a vector function,

M (t,* , * ) P

i s such t h a t ,

have

For instance the quadratic term can be w r i t t e n 2

vu * C ( p, t ) *Vu+ u D( p, t ) . V u + e( p, t ) u where

i s a symmetric matrix i n R

C(p,t)

e(p,t)

2

, D(p,t) a vector function and

a s c a l a r function, which a l l a r e r e g u l a r i n t h e i r arguments.

assume t h a t

a(p)

>

0 for

an

boundary condition on

.

pE I

H = L2(n)

I n t h i s example

We

and 8 =

$(n) n<(n)

i f we have the D i r i c h l e t

(we use the usual notations f o r Sobolev spaces,

see 1241). The assumptions necessary t o g e t t o t h e f a c t t h a t a l l terms, except

(4), a r e

a(p)Au

r e a l i z e d i n t h i s example due

, i n t h e second member of ( 5 )

play the r o l e of a perturbation f o r the h e a t equation i n fi

.

The proof of

the existence of a regular s o l u t i o n of the form (4) uses the a n a l y t i c i t y of the semi-group generated by the heat equation [ 1 9 ] ; a proof f o r a more general problem may be found i n [ 1 4 1

.

Example 2.

at =

p Av-Vp- ( v . V ) w ( t , p ) - ( w ( t , ~ ) . V ) V -

V-V =

o ,

~ l a n= o

in 3

,

(V-V)V

,

Bifurcation of Maps and Applications

172

i s a bounded r e g u l a r domain i n R3

where

system, where

(6),we

This i s t h e Navier-Stokes

i s t h e v e l o c i t y f i e l d of t h e f l u i d and

V(t,x)

i n a non-dimensional form, and fact, t o get

.

p

t h e pressure,

i s t h e inverse of t h e Reynolds number.

p

In

s u b s t r a c t e d t o the t r u e v e l o c i t y f i e l d a s o l u t i o n

s a t i s f y i n g t h e t r u e boundary conditions i n presence of a n eventual e x t e r n a l force field. This system i s not t r i v i a l l y i n t h e form (1). To g e t it we p r o j e c t t h e system (6) on a divergence f r e e vector f u n c t i o n space which i s orthogonal t o

So,

the gradients.

Vp

i s eliminated and we have

automatically i n our space 4

.

0.V = 0 and

VIan = 0

The proof of t h e e x i s t e n c e of a r e g u l a r

s o l u t i o n of t h e form (4) and on more general p a r t i a l d i f f e r e n t i a l weakly non-linear systems i s given i n [25], [ 141.

For a system with non-linear terms

of t h e same order of d e r i v a t i o n a s t h e l i n e a r ones, see t h e important work of G. RA F'RATO and P. GRISVARD [5]

f o r t h e type of proof i n t h e simplest f u n c t i o n

space. Let us now define t h e l i n e a r operator

(7)

Dx X(O,p,t) = S p ( t ) E ;e(E) 0

, t 2

0

This operator solves t h e l i n e a r problem

with

X(t) = S (t)Xo P

.

.

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

173

We a l s o need t h e operator s o l v i n g

X(t) = S (t,s)Xo

which defines SP(t,s)

and

P-

, t 2s

.

The r e g u l a r i t y p r o p e r t i e s of

S ( t ) a r e deduced from t h o s e of P

X

i n (4) and we have i n

4E) :

( SJt'

0) =

s P ( t ) , s P( t , t ) =

1

,

I n the f i n i t e dimensional case, we have

b u t i n g e n e r a l , i n t h e i n f i n i t e dimensional c a s e t h i s i s n o t t r u e . We s h a l l e x t e n s i v e l y use t h e Taylor expansion of neighborhodd of

where

0 i n RxE

.

We have i n

E

X

for

(p,X ) 0

in a

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

174

even i n the i n f i n i t e dimensional case, provided t h a t some r e g u l a r i t y condition on

and some r e s t r i c t i o n on t h e space of values of

So

(see

L (7) a r e r e a l i z e d 1

[ 131 f o r i n s t a n c e ) .

I n t h e same order of i d e a s we s h a l l need t o compute some s c a l a r products, hence it i s u s e f u l t o know what i s t h e a d j o i n t of L e t us assume t h a t

*

I-L

*

.

s u i t a b l e operation, as f o r

g

c1

L (t) i s t h e unbounded a d j o i n t of

d e f i n e d ) , with a f i x e d domain 69

-

S (t,s)

,

= L p Y

becoming a Banach space

in

aE*) .

L (t) (densely

P * E after

a

Let us consider t h e problem

t <s

,

so t h a t i t s s o l u t i o n may be w r i t t e n

s"

and l e t us assume t h a t [see and

[13]

* , 69

for

I

s a t i s f i e s t h e same r e g u l a r i t y p r o p e r t i e s as

f o r an example) and t h a t

tfs

.

f i n i t e dimensional case. Lemma 5

P

S

b

and

s I-1

a c t s from H

S

These conditions a r e , of course, r e a l i z e d i n t h e So, we have t h e

P

i n t o 69

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

175

0

Remark.

This lemma reduces t o t h e Lemma 5 of c h a p t e r I11 i n t h e f i n i t e

dimensional case. Proof.

xE E , yE E

Let us consider f o r any

It i s e a s y t o check t h a t

f'(7) = 0

.

Y

, t h e d u a l i t y product ( i n H)

So, making

T

=

, and

t

= s

we

obtain

hence t h e lemma 5 , by d e n s i t y of

E

and

E

*

in

H

and

*

H

We assume now t h a t t h e c o e f f i c i e n t s i n (1) a r e p e r i o d i c i n

.

t

of p e r i o d

T

.

So, by c o n s t r u c t i o n we have now

(20)

S ( t + T , s + T ) = S (t,s) I.L P

Hence, if we define t h e map

F

P

The p o s i t i o n of t h e spectrum of C

.

i n a neighborhood of

S (T) P

0 in E

, by

with r e s p e c t t o t h e u n i t c i r c l e i n

, will g i v e t h e r e s u l t s on t h e s t a b i l i t y o r not of t h e 0 s o l u t i o n of (l),

176

Bifurcation of Maps and Applications

according t o theorems 1 and 2 of chapter I. Let us assume t h a t

A(p)

i s a non-zero eigenvalue of

s~(T)

, it i s

c a l l e d a Floquet multiplier f o r the l i n e a r system ( 8 ) . We have

where

C(p) E E

\{O]

.

Let us define

u(p) F C

such that

then the vector function

T periodic and regular i n E

is

.

Moreover

is a

T-periodic solution

of

Let us assume that E

S (T)E P

and t h a t it is a compact operator i n

$H;E)

( t h i s i s of course r e a l i z e d i n the f i n i t e dimensional case, and a l s o t r u e

i n t h e cases of examples 1 and 2 f o r i n s t a n c e ) . for (28)

*

and we can define

S (T) P

S;(T)

S*(d

=

*

5

(p)c E

~ C * ( P )

*

(S(cl),

i s a n eigenvalue

So,

such t h a t

S*(d)

= 1

.

Moreover, we define

(29) which i s a

*

S,(t)

= e'(k)t

E,(t,a)C*(p)

, t 5

T - p e r i o d i c vector function i n E

0

*

, and s a t i s f i e s

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

177

Note t h a t , by lemma 5 we have

Let us c a l l A

P

the operator

, and l e t us make the assumptions

Sp(T)

and H . 2 o f paragraph 3 on the operator A. invariant and a t t r a c t i v e manifold the map F

P

M

P

in

in

E

E

.

We have a l o c a l l y

such t h a t the r e s t r i c t i o n of

i n a neighborhood of

may be w r i t t e n for xE E)':

H.l

0

:

where

and

To get higher order terms of the map G

c1

i n ELo'

, we may use the formulas

(14) and (15) of paragraph 3 , with (l2), (13), (14), (15) f o r t h e expansion of

X(Xo,w,T)

.

More p r e c i s e l y , l e t us assume t h a t

of So(T) ,

\Xol

=

1

.

ho

i s an i s o l a t e d simple eigenvalue

Then, the formulas ( 2 7 ) , (28), (29), ( 3 0 ) of

178

Bifurcation of Maps and Applications

paragraph 3 , give here :

with

thanks t o (26) and (31). I n t h e case of assumptions H . l and H.2.a o r H.2.b of paragraph 3 we can compute the c o e f f i c i e n t s of ( 3 3 ) o r ( 3 8 ) of paragraph 3 :

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

179

(39)

These formulas were used i n [ 3 ] t o compute e x p l i c i t e l y 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 t o r u s f o r a

4 dimensional d i f f e r e n t i a l system of p h y s i c a l

i n t e r e s t ( s o (39) i s more than j u s t a joke!). Let us now assume t h a t we o b t a i n i s o l a t e d p e r i o d i c p o i n t s of period b i f u r c a t i n g from the f i x e d p o i n t

G

P

,

P

.

Because of t h e l o c a l a t t r a c t i v i t y

M

t o t h e b i f u r c a t i o n of s o l u t i o n , and f o r closed curve

F

,

, they a r e on M , s o we o b t a i n n - p e r i o d i c p o i n t s P P Thanks t o t h e p r o p e r t i e s (19) and ( 2 3 ) , t h i s s i t u a t i o n corresponds

of t h e manifold for

for

0

n

yP

p

for

nT - p e r i o d i c s o l u t i o n s of (1) from t h e t r i v i a l close t o F

P

in

0 E

.

0

If we o b t a i n a b i f u r c a t e d i n v a r i a n t

, we r e f e r t o t h e d i s c u s s i o n of c h a p t e r 111.5,

s p e c i a l l y t o the Lemma 6, t o prove t h a t we have an i n v a r i a n t two dimensional t o r u s f o r t h e d i f f e r e n t i a l equation (l), and t o know t h e n a t u r e of t h e s o l u t i o n s

180

Bifurcation of Maps and Applications

on t h i s t o r u s . The a t t r a c t i v i t y o r t h e i n s t a b i l i t y of

nT

periodic solutions or

i n v a r i a n t two dimensional t o r u s i s known by using t h e theorems of chapters 11, 111, I V on 4.2

G

.

L

The autonomous case. Let us first consider a d i f f e r e n t i a l equation i n a r e a l Banach space a s

(l), except t h a t

L

P

and

N

a r e independent of

P

t

.

A l l t h e r e s u l t s of

l t . 1 apply of course here, i n a simpler way because we have i n f a c t t h e

semi-group property

which s i m p l i f i e s some formulas.

f o r any f i x e d

to > 0

.

For a b i f u r c a t i o n s t u d y we can u s e t h e map

I n f a c t , t h i s way i s not i n t e r e s t i n g i f we look

f o r the b i f u r c a t i o n of f i x e d p o i n t s of t h e s t a t i o n a r y f i e l d , because when we obtain a f i x e d p o i n t of of period.

(42)

to

.

F

P

, t h i s may correspond t o a closed o r b i t f o r

For t h i s t y p e of study, it i s b e t t e r t o w r i t e

L X + N ( X ) = O in

F

CL

H

, [ 121 1.

and t o use t h e standard Lyapunov-Schmidt decomposition ( s e e [29], Nevertheless, l e t

X(p,t )

be a n i s o l a t e d f i x e d p o i n t of

F

CL

:

(1)

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

181

then

i s a family of f i x e d p o i n t s of

F

1J.

too, because

Hence, because of t h e i s o l a t e d n e s s of t h e f i x e d p o i n t , we n e c e s s a r i l y have %(p,s)

=

X(p,to)

independent of

, to

s

> o and

X(p,to)

i s a f i x e d p o i n t of t h e f i e l d (42),

.

If w e a r e looking f o r t h e b i f u r c a t i o n of closed o r b i t s f o r t h e equation

t h e most i n t e r e s t i n g way, t o g e t an e x p l i c i t computation, i s t o use a standard Lyapunov-Schmidt method (see [29] , [ 121 1. t o o t h e maps.

Let us assume that

curve i n v a r i a n t under

F

P

.

Y (t ) P

O

Nevertheless, we can use

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

Then, by t h e same type of proof a s previously,

i s i n f a c t independent of

to and i s a t r a j e c t o r y of t h e f i e l d (43)

(see e x e r c i s e 2 of chapter I ) . Let us make t h e following assumption: The spectrum of

Go ,

Lo

contains 2 simple i s o l a t e d eigenvalues

the remaining of t h e spectrum s a t i s f y i n g

Rea<- 5

<

0

.

oo=iLoo and

Then, i f

some s u i t a b l e r e g u l a r i t y conditions are r e a l i z e d ( t r u e on examples of t h e type

Bifurcation of Maps and Applications

182

4.1) it can be proved t h a t S o ( t o ) has t h e two simple eigen-

described i n values

k

= e

lo on the u n i t c i r c l e , t h e remaining of i t s

and

spectrum being s t r i c t l y i n t h e open u n i t d i s c . h

n 0

#

1 for .n=1,2,3,4

map on t h e manifold

, and t h e i s o l a t e d b i f u r c a t e d closed curve yP(to)

O

$4.1,w i t h a s i m p l i c a t i o n i n doing T = t o ,

The c o e f f i c i e n t s are t h e same a s i n u ( 0 ) =iv)

0

co(t)

'

l i n e a r operators

=

c(0)

s0(s)

to such t h a t

and compute a s i n d i c a t e d i n t h e paragraph 3 t h e

M (t ) P

Let us choose

= c*(o)

, c:(t)

,

, pi( O )

L(O)

i

,

and i n using t h e f a c t t h a t t h e

commute.

The computation of t h e

period of t h i s 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 may be achieved i n a simple way a s i n 111.4.

The o r b i t a l s t a b i l i t y and t h e l i m i t phase a r e obtained i n t h e

same way too. The most i n t e r e s t i n g use of the method developed here i s on t h e b i f u r c a t i o n from a closed o r b i t t o a n i n v a r i a n t two dimensional t o r u s . the d i f f e r e n t i a l equation i n

and l e t us assume t h a t

(45)

x

H

tH Xo(t,p)

sufficiently regular i n = X0(t,P) + Y

Let us consider

(t,p)

,

is a

T(b) - p e r i o d i c s o l u t i o n of

taking values i n t h e domain of

Gb

(44),

.

fitting

,

we g e t

which i s a non-autonomous d i f f e r e n t i a l equation f o r in

9 4.1.

Y

, of t h e t y p e discussed

We assume that t h e s u i t a b l e r e g u l a r i t y on s o l u t i o n s i s obtained

a s f o r (1), and we use t h e n o t a t i o n s of

54.1.

The p a r t i c u l a r i t y i s t h e

.

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

Prn

6.

1 i s an eigenvalue of

S&T(p)]

in

E

183

.

This i s due t o t h e invariance of the equation (44) with respect t o t r a n s l a t i o n s i n time

t

, and the proof i s i d e n t i c a l t o the proof

given i n 11.3. To study an eventual b i f u r c a t i o n from t h e i n v a r i a n t c i r c l e , we construct t h e Poincard map F

a s i n 11.3, i n

P

E

.

We use on t h i s map the reduction

t o a map i n a one o r two-dimensional space using the manifold

M

P

, and then

we use t h e r e s u l t s of chapters 11, I11 o r I V . If t h e eigenvalue 1 of

S p [ T ( ~ ) ] i s i s o l a t e d and semi-simple, we can

use exactly the same way a s i n 11.3 t o f i n d e x p l i c i t e l y of f i r s t r e t u r n of the t r a j e c t o r i e s

PY

= 0

P O

.

axO

*

satisfies

ax

(-$(O,k),X*)

The hyperplane

#

0 for

p

close t o

* *

X E E

such t h a t

0

.

2 t h a t we may use c o n s i s t s of a l l Yo i n E such t h a t

The r e a l equation which defines

which i s of the form

Let us c6nsider

,

(~(O,O),X) = 1

hence

, where Yo

7(Yo,p)

and the time

P

I n f a c t t h i s assumption i s not necessary, b u t it gives simplest

expressions f o r f u r t h e r computations.

(47)

F

7(Y0,p)

f ( Y O , p , r ) = 0 with

i s now

Bifurcation of Maps and Applications

184

Hence we can solve (49) and

But

Dy

T(

0, b)

#

7

= T(Y ,p) 0

i n g e n e r a l here:

0

0

i t i s easy t o s e e t h a t we have i n E

*

verifies

:

and we can compute a l l terms of the Taylor expansion:

where we note

a s i n 4 . 1 with t h e same expressions (l3), Poincarg map F

u

in

i

i s now, f o r

The advantage of (55) i s t h a t b y a complicated form of

F

P

T(Yo,p)

(14) f o r

the coefficients.

t?

a c t s i n a f i x e d space

.

mr

f?

; we pay t h i s

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

185

Let us p r e c i s e l y w r i t e

then, we have

Moreover, we can compute

(59)

which reduces t o the formulas of 11.3 i f

.

$4.1 f o r );2

~(l =) 0 b

, using (13) and (14) ' o f

Let us make the assumptions H . l and H.2.b of paragraph V.3 on t h e spectrum of

A.

.

This i s e q u i v a l e n t t o t h e following:

The spectrum of open u n i t d i s c and a p a r t circle.

So[T(Q)] 0")

2

The eigenvalues 1, A.

=

i s t h e union of a p a r t

{13 U {Ao] U

Exo]

, Xo #?l

0'')

1

, on

i n the the unit

, io a r e simple and we have t h e Hopf-condition

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

186

il

dplk(~)l

lp=o

>

, 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

<

0 t h e closed t r a j e c t o r y

Xo(-,p)

i s exponentially s t a b l e ( t h e r e i s a l i m i t phase, a s i n d i c a t e d i n §II.3),

>

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),

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

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

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

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

Bifurcation of Maps and Applications

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

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

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.

*

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

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 <*(t) a r e defined w i t h P P 1 u(p) = m ( L o g l k ( P ) + k ) i n t h e case ( i ) .

case ( i ) . Note t h a t

The simplest choice f o r 7

P

and

T ( ~ )

k

X* i s X *= 5o (*0 ) a s i n t h e preceding case, so

a r e s t i l l expressed by (66) and (68),a s w e l l a s

p, *

by

(70). Now, we can compute t h e map

G

P

in R

, given by t h e formula ( 3 2 )

paragraph 3, which represents t h e r e s t r i c t i o n of manifold

M

!J

.

We obtain

F

P

of

on t h e one dimensional

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

with

A

1

and w i t h

(64), ao2 given by (35) w i t h To i n s t e a d of

s t i l l given by

u(o)

a

coefficient

0 i n t h e case (ii)and

=

computed i n

03

193

in/To

T

i n t h e case ( i ) . The

(36), with To i n s t e a d of

T

, has t o b e

added t o t h e term -ho(Lo(0)C(o),5*(o))

(80)

,S,(s)l , C0o (*s ) ) e2U(O)Sds

TO (No2[s;Co(s) 0

(due t o

a

T ( ~ ) ( Y ~ , Y ~ t)o) g e t t h e c o e f f i c i e n t 0

of (79).

03

So we a r e a b l e t o compute f i x e d p o i n t s o r p e r i o d i c p o i n t s of p e r i o d 2

for

G

w

.

in W

These b i f u r c a t e d s o l u t i o n s correspond t o t h e b i f u r c a t i o n

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

(44), d i f f e r e n t

to

Xo(

*,

i n t h e case of a f i x e d p o i n t , and a period

.(Y0,p)

i n t h e case of a p e r i o d 2 point f o r close t o

G

P

.

2To and may be computed e a s i l y .

p)

, which has a period

7(Yo,LL) + T [ F ~ ( Y ~ ) , ~ ]

I n t h i s l a s t case t h e p e r i o d i s The s t a b i l i t y of t h e s 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 s i s t h e same a s 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 of t h e map

G

p

in R

The s o l u t i o n

where

&'

X(t)

. of

(44) can be obtained from xER

i s given by (54) and (13),

(14), ( l ? ) , and

, by

Bifurcation of Maps and Applications

194

1 i s not a semi-simple

The l a s t case we want t o consider i s when eigenvalue f o r

, which i s the general case when 1 i s double. We

So[To]

don't need the perturbation theory, because t h e eigenvalue of a r e not s o c l o s e l y r e l a t e d t o the eigenvalues of SOITo] does not commute i n t h i s case.

I

I

S [T(p)]

P

5PS P[ T(p)]

, because ho

and

L e t us make t h e assumption:

Hl.2.c. The spectrum of

So[To]

i s the union of a p a r t

1

i n the open u n i t

d i s c and of the double non-semi-simple eigenvalue 1. Moreover, we have

I

the Hopf condition

A1 > 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

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

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

Bifurcation of Maps and Applications

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 .

Bifurcation of Maps and Applications

198

The s o l u t i o n

X(t)

of (44) i s obtained from

xER

by t h e formulas

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

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 ) .

This Page Intentionally Left Blank

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

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

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

. Assume t h a t ,

, unstable

when

for

T

small enough,

p

T1 x 0

p > 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

.

Bifurcation of Maps and Applications

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 <0

d e p e n d i n g on t h e s i g n of

a n o t h e r q u a n t i t y which we a s s u m e t o b e non z e r o , t h e r e b i f u r c a t e s a

- torus *

2

i n v a r i a n t under

FcL a n d d e p e n d i n g c o n t i n u o u s l y on

1

.

Its s t a b i l i t y is t h e o p p o s i t e o f t h a t o f t h e p e r s i s t i n g circle close t o 1

T x 0

.

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

FcL t o t h i s

2

-

is i s o t o p i c t o iden-

torus

tity. Moreover t h e d i s t a n c e t o der

P

, whereas

i t is of

T

1

x 0

order

o f t h e i n v a r i a n t c i r c l e i s of or-

1~1%

for t h e invariant

2

- torus.

COMMENTS ON THE ASSUWTIONS

(i) From

I/ f o l l o w s t h a t t h e r e s t r i c t i o n of t h e o r i g i n a l flow t o

t h e i n v a r i a n t t o r u s , f o r which

Fo

is t h e P o i n c a r g map, d e f i n e s t h e same

f o l i a t i o n (up t o d i f f e o m o r p h i s m ] as a q u a s i - p e r i o d i c f l o w [same p r o o f as Lemma 6 o f 111.5) c o n d i t i o n on

o

.

N o t i c e however t h a t i n

3/

t h e r e i s no di ophant i ne

i t s e l f , s o t h e f l o w need n o t be e q u i v a l e n t t o a q u a s i

p e r i d o c f l o w (we c a n n o t u s e t h e Kolmogorov t h eo r em o f i n view o f Arnold-Hermann

sis I/

*

th e o r e m (see

[9 1

1 the

111.6)

.

Anyway,

s i t u a t i o n o f hypothe-

o c c u r s w i t h non z e r o p r o b a b i l i t y and s o i s w o r t h s t u d y i n g .

There i s a l o s s of r e g u l a r i t y w i t h r e s p e c t t o t h e i n i t i a l which i s p r e c i s e d i n [ 4 ]

F

,

H i g h e r order b i f u r c a t i o n s

[ii) The non-resonance

assumptions

207

are f u n d a m e n t a l , b u t i t i s

3/

u n c l e a r w h e t h e r or n o t i t i s p o s s i b l e t o g e t r i d c o m p l e t e l y o f t h e diop h a n t i n e a p p r o x i m a t i o n a s s u m p t i o n s . A s o n l y t r u n c a t e d n o r m al f o r m s are i m p o r t a n t f o r t h e p r o o f , t h i s c o u l d be r e a s o n a b l y e x p e c t e d . The s t r o n g l y

3 r, 0 5 r 5 4

r e s o n a n t cases when

r

no +

q

E

u)

0

, are

Z

(iii) Assumption

and

q

E

studied i n d e t a i l s i n

such t h a t

Z

is t h e most s t r i n g e n t and a l s o t h e most i n t e -

2/

r e s t i n g b e c a u s e o f i t s g e o m e t r i c a l meaning : i t i m p l i e s t h a t invariant a l l the

5

subbundle 2

-

tori

,

-

2

t o r i of a

T 1 x J? is

Go

CR

.

[4 ]

IV of

-

of

1

T

1

-

parameter

x E

Go l e a v e s

family f o l i a t i n g t h e

a n d t h a t , on e a c h of t h e s e

c o n j u g a t e t o a map o f t h e form

L e t u s r e c a l l t h a t i n t h e case o f Hopf b i f u r c a t i o n f o r maps (see 111)

1

t h e e x i s t e n c e of a Dx Fo[O) trum o f

-

f a m i l y of i n v a r i a n t c i r c l e s f o r

parameter

i s a u t o m a t i c f r o m t h e s p e c t r a l h y p o t h e s i s , Here i f t h e sp ecf f o is as i n a s s u m p t i o n

2/

except t h a t

is n o t g e n e r a t e d

o2

by a n e i g e n v a l u e [which i s t h e g e n e r a l case f o r s u c h a s p e c t r u m ) t h e rest r i c t i o n of

G

5,

to

0

may well h a v e a d e n s e o r b i t !

Some g e m e t r i c a l h y p o t h e s i s o f t h i s k i n d is d e f i n i t e l y needed if o n e l o o k s f o r a n i c e b i f u r c a t i o n t h e o r e m : namely, i f one starts w i t h a

1

- parameter

f a m i l y of maps

F

v.

:

-

1

T1xT , T x 0

T

1

x E , T

f o r which t h e c o n c l u s i o n s o f t h e th e o r e m h o l d , t h e i n v a r i a n t which a p p e a r s b e i n g t h e image o f a map f r o m

T2

to

T

1

x E

1 2

x 0

-

torus

o f t h e form

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

208

Go

then

(0, (iv)

2

leaves i n v a r i a n t each

The word

$1 'I

-

torus

image of a map

,

( 0 , K X ~ [ ~$ ,) )

D

i n general

"

K

constant.

i n t h e theorem means t h a t t h e c o n c l u s i o n

h o l d s provided two q u a n t i t i e s a r e non zero. These ones a r e b e s t understood a f t e r some changes o f v a r i a b l e s , so l e t us r e f e r t o t h e p r o o f o f t h e theo-

I n f a c t , t h e y correspond t o t h e R u e l l e -

rem f o r an e x p l i c i t statement;.

Takens assumptions t h a t eigenvalues o f t r a n s v e r s e l y and t h a t

0

i s

Dx Fo(0)

cross the u n i t c i r c l e

vaguely a t t r a c t i v e

"

"

(or repelling] f o r F

0 .

A warning i s necessary a t t h i s p o i n t : thanks t o t h e f i r s t c o n c l u s i o n a f t h e theorem,

one may assume

F ( T I x 0) I.I

1

c T x 0

.

The assumptions do n o t i m p l y t h a t t h e spectrum o f

eigenvalue f o r

I n fact, the

j4

o

.

a2 p a r t o f t h e spectrum o f

i n t o an annulus a t values o f

I.I f o r which

f(0,

s m a l l (af-

a,,, may

t e r a change o r coordinates) so t h a t an o p e r a t o r Qo

for

a.

b

be d e f i n e d as contains

an

is expected t o explode

0, p )

has r a t i o n a l r o t a -

t i o n number, b u t t h e assumptions may be shown t o i m p l y t h a t i t s t a y s away f r o m the u n i t c i r c l e as soon as

.

3

p

i s non zero, and small,

CENTER MANIFOLD THEOREM The f i r s t i d e a i s t o reduce t h e i n i t i a l problem t o a problem i n

T

1

x F?

by c e n t e r m a n i f o l d technique as i n

V.3

.

For t h a t l e t u s use

H i g h e r order b i f u r c a t i o n s

209

t h e following. C e n t e r manifold theorem Let

consider a

US

F : T1 x Y, T 1 x 0

(i)

u

1

U o2

c--t

Ck

,k

T1

x E, T 1 x 0

5 3

L e t u s assume t h a t t h e

, map

k

such t h a t

-3

F

is a n u n i o n

with b

=

sup h E Ul

<

(hi

inf

h E o2

Ihl

and assume t h a t t h e i n v a r i a n t s u b s p a c e s with

s p e c t o g r a p h of

(T

i

=

Let us note T(g)

of

a

9

k 3

C

1 (T ; E)

associated

h a v e t h e form

(i = 1, 2 )

&i

‘i

=

Ck3 1 (T ; Eil

the

o3 :

(then

E

=

El @ E2)

k 3 s p e c t o g r a p h of t h e t a n g e n t map t o

T1 x R

*

T1xR

d e f i n e d by

, and d e f i n e

We f u r t h e r m o r e a s s u m e t h a t

(so i n p a r t i c u l a r

b < 1)

g

:

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

21 0

(ii) L e t u s d e n o t e by

Ei

,

.

i = 1 , 2

We assume t h a t

(iii)

-

p : E2

and

As i n

A;?(9) =

%

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

is i n d e p e n d a n t of

, which

Ck

outside of t h e u n i t b a l l o f

0

Then, t h e r e e x i s t s a s u b m a n i f o l d i n g e n e r a l , g r a p h o f a map

'p2 : T 1 x

d e f i n e d i n a neigbourhood of

T

is

( c e n t e r m a n i f o l d ) , non u n i q u e

E2-

if

k 24

3/

%

is 1 o c a l l y . i n v a r i a n t under

4/

X$

is l o c a l l y a t t r a c t i n g under

1 x 0

p--"

, then

.

El

, such

T1 x 0

0

?%

2/

E

i s a neigbourhood of

, of

class

,

Ck3 1

that

0

=

,

1

.

'p2[8, 0)

t T1 x

.

9

E2

I/

(9, x )

D 'p,(9,

0)

=

0

% is t a n g e n t F , F , i.e. if a (

and a l l i t s p o s i t i v e i t e r a t e s u n d e r t h e distance of

Fp(e, x )

to

F

%$ g o e s

to

T

1

Let u s n o t e t h a t t h e dependance i n

.

The a s s u m p t i o n

A2

0 € T'

leads

are close t o to

0

when

V.2.

t o an assumption v i a

c o n s t a n t i s a v e r y s t r o n g one and it seems h a r d

t o g e t r i d o f i t i n g e n e r a l . The i n t e r e s t e d r e a d e r w i l l p r o o f of t h i s theorem i n

x E2]

point

The proof of s u c h a theorem, looks v e r y much l i k e t h e p r o o f of

B

in

Dx I(0, 0 )

we assume t h a t t h e r e e x i s t s a f u n c t i o n

V.2

of class

R+

Ai(9)

11.1

of

find

a detailed

H i g h e r order b i f u r c a t i o n s

REMARK

2 11

To o b t a i n e x p l i c i t e l y t o t h e T a y l o r e x p a n s i o n of

:

is s u f f i c i e n t t o i d e n t i f y t h e homogeneous terms i n

x

rp2

, it

of t h e same orders

;.n t h e f o l l o w i n g i d e n t i t y :

where

R

is t h e p r o j e c t i o n on

i

m(9,

X)

=

[i = 1, 2 )

Ei

A(0) x

+

of

R

d e f i n e d by

R(9, x )

and

F o r i n s t a n c e , if we n o t e (6

I

(p2(f3,

x)

=

E ( e ; x, x ]

o b t a i n , by i d e n t i f i c a t i o n i n

4

. PROOF

(4)

+ O(1xl 3 1

; x, x )

R ( 0 , x ) = R("(0

+ O ( I x /3 )

for

x

in

E2

and

, then

we

:

OF THE WIN THEOREM. STEP 1 : REDUCTION TO THE DINENSION

~~

To r e d u c e t h e problem t o t h e dimension

a l i n e a r operator

2

.

~~~

~

A2( 9)

i n d e p e n d e n t of

0

2

we h a v e m a i n l y t o o b t a i n

a f t e r s u i t a b l e c h a n g e s of

v a r i a b l e s . To do t h i s we p r o c e e d i n two s t e p s , t h e f i r s t s t e p w i l l i g n o r e t h e r o l e of

.

Let u s define the vector spaces

Bifurcation of Maps and Applications

21 2

Ei(e)

=

Thanks t o t h e assumptions

111.1

of

[43

I/ and

f o an easy proof)

(1 1

=

E

E1(B)

and the associated p r o j e c t o r s Moreover, f o r any

(4

1

c (e)

X

u

E

=

a

z

x € E

@

x

(pile) 3,

2/

of the main theorem we have (see

1

E2(0)

p Ei cblJ

:

we have i n a

=

such t h a t

u(e)

[TI; E ( E ) ]

u = (Re z , I m z )

where

: E-R2

x

,

unique way

,

2( e )

9 E T1

I

satisfy

xo(e] + Z Y,(e)

? (0)

i= 1, 2,

t h e d i r e c t sum

pi(@)

C (T ; E)

Then, t h e r e e x i s t s a map

and f o r any

ei,

{ x t E ; 3 cpi €

,

.

,

21 3

H i g h e r order bifurcations

From

Let

T

1

(5)

LIS

we can compute f o r i n s t a n c e

-

now make t h e f o l l o w i n g change o f v a r i a b l e s i n

x E x

n2

:

(71

1

(9, x, u ] XI

=

x2

=

c

F

x2)

+ c

P1(9) x

with

(9) u

5

(9) x

L e t us note that the component The map

(9, x,,

may be considered as

-

w i l l n o t p l a y any r o l e i n t h e p r o o f .

u

:

T1 x E x

8

T

1

8

x E x

f i n e d by (8

1

F (9,

P

x,

Ll9

=

and the change o f v a r i a b l e s

Z ( 9 , xI9

(9)

x21

=

with

It

is easy t o check t h a t

(Fig,

(f(0, (7)

x,

PI, @(9,

xs h1Y 01

9

l e a d s t o t h e new form :

x13 x2,

@ , ( e , XI,

x29 IL),

@ 2 ( 9 ,x l s

x2#

8

de-

21 4

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

where i n

s(s)

we have t h e m a t r i x r e p r e s e n t a t i o n

The s e c o n d s t e p i n t h e way t o o b t a i n a c o n s t a n t o p e r a t o r c o n s i d e r t h e map

i :

-

T1xExF?xR

A2

consiststo

T 1 x E x R2 X I ?

d e f i n e d by

8 %XI'

(131

Is

XZ'

u s i n g t h e same i d e a as i n

PI V.3

=

.

XI' x,l,

(F,l9,

This

n

F

t h e c e n t e r m a n i f o l d theorem. The o p e r a t o r

PI

I

w i l l play t h e r o l e of associated t o

F^

F

in

has the

form (141 so

E(e,

in g(F?)

xl,

x2,

, the

n i f o l d theorem is

d

=

(deII

A ~ ( ~ +) bx 1~ b h

operator corresponding t o t h e :

R % X ~

A2

+ b2bh,

IL),

o f t h e c e n t e r ma-

Higher order b i f u r c a t i o n s

21 5

which i s n o t y e t c o n s t a n t . It i s s u f f i c i e n t i n f a c t t o d o a t r a n s l a t i o n for e l i m i n a t i n g t h e

bi(e)

E ,

in

i = 1

t i o n s we first ch a n g e of v a r i a b l e i n

,2

.

To s i m p l i f y t h e computa-

9 w i n g t h e assumption

main t h eo r em , i n s u c h a way t o r e p l a c e

by

g(0)

8

+ wo

1

of the

(mod I)

.

Now w e s e e k a t r a n s l a t i o n o f t h e f o r m

t h e n t h e r e w i l l be n o more

bi(9)p

main t h eo r em on t h e p a r t o f t h e l i t y t o solve 0,

, To s o l v e

3/

with

r = 1

(17) (17)

.

L.

-

with r e s p e c t t o in

y2

, we

.

mi

in

The a s s u m p t i o n

spectrograph y1

,

2/

of t h e

o,, g i v e s t h e p o s s i b i -

j u s t u s i n g t h e contraction

have t o u s e t h e d i o p h a n t i n e c o n d i t i o n

I n t h e f o l l o w i n g , we w i l l e x t e n s i v e l y use t h e

Bifurcation of Maps and Applications

21 6

.

1

LEW

2/

I/,

Under the assumptions

and

of the main theo-

3/

rem, t h e e q u a t i o n

(If31

y(e

where

E

cy

Cm(T1; C)

, if

E' > 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

Bifurcation of Maps and Applications

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

Bifurcation of Maps and Applications

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-

Higher order b i f u r c a t i o n s

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

Higher order b i f u r c a t i o n s

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

<0

i s analogue)

.

The new map

may b e w r i t t e n (suppress t h e primes and make a l i t t l e abuse o f n o t a t i o n s ) :

F

223

Higher order b i f u r c a t i o n s

uses t h e assumption 3

The f i r s t change o f v a r i a b l e s

(r = I and

r = 3)

o f t h e theorem :

where

(6)

j

8,

and

yPs

f,(d

+ A,

5,,(e)

+

satisfy

sl(e + wol

A:(X,I~+'

-

s&el

Yps(e + w,)

= 0

- yPq(e) = 0

, p+q

= 2

.

We l o o s e a l i t t l e r e g u l a r i t y f o r t h e new c o e f f i c i e n t s o f t h e map which becomes

1

(suppress t h e p r i m e s ) :

(7)

with

El3

,

z

, 71

=

d e ,

z

7)

=

f(e

e+

w0 +

$Cf,(e)

+ h,fe,z)l

2

+ 7 rA,(elz +

w r i t t e n i n t h e same

form as

A3

+

3

oh

x 2 ( e E + a,(e , 4 1 1 + G ( V3 1 i n

(2)

.

The second change o f v a r i a b l e s uses no d i o p h a n t i n e c o n d i t i o n :

I

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

224

(9)

For i n s t a n c e ,

and we use lemma 2

with

1.

= 0

,2 ,4 ,

t o show t h a t t h e new map

F

m y

be w r i t t e n (suppress t h e primes) :

where

~ ( 7 )i s

a continuous f u n c t i o n

of

11

going t o

0 when

7

goes t o

0

,

H i g h e r order b i f u r c a t i o n s

Re XI

Note t h a t

is t h e i n t e g r a l (5.5)

( s t r i c t crossing condition) t h e main theorem.

225

and i s assumed t o be non zero

a t t h e Lemma 4 which l e a d s t o t h e f i r s t p a r t o f

To f i x the ideas, we assume now

which j u s t s . y s t h a t t h e p e r s i s t i n g c i r c l e is s t a b l e f o r

c~

.

>0

L e t us w r i t e t h e rap (10) i n p o l a r coordinates, as f o r t h e Hopf b i f u r c a t i o n : we pose

The map (10) becomes :

L e t us assume

( i n the case

K

now t h a t

Re

>

, we

0

i n s t e a d o f t h e p r e s e n t case

should have t o c o n s i d e r t h e case

>

0)

.

The l a s t change o f v a r i a b l e s i s n a t u r a l l y

*

In t h e case when

Re

~y

= 0

, we

p

:

might proceed as i n 111.2

.

<0

Bifurcation of Maps and Applications

226

where

>

0

and l e a d s t o t h e new rmp ( l i t t l e abuse o f n o t a t i o n s ) : 2 2 JI I Y 1 1 ) = 0 + Wo + 7 (wl + + $

1

Y

1

v

2

$+no+*(Imhq+ro

=

We now use t h e "usual" p r o o f

(as i n

111.1

and

t h e e x i s t e n c e o f a b i f u r c a t e d t o r u s f o r t h e map *

2 : T R -

graph of

y

proof o f

111.1

.

5)

F

I-L

t o show

,

obtained f r o m t h e

which is invariant under t h e map (16)

, The

a p p l i e s a l s o for t h e r e s u l t on t h e s t a b i l i t y o f t h i s t o r u s .

Hence t h e - w i n theorem i s proved

7

3

2

.

AN EXAMPLE

E = R

The f o l l o w i n g example, w i t h

expects generic examples o f t h i s k i n d f o r

F

Let

where

I-L

a(e)

: T

>0

1

1

x R -T

0

Moreover

,

R

i s h i g h l y non g e n e r i c

dim

E > 1

.

,

b u t one

be d e f i n e d by

i s v e r y smooth and

I t i s easy t o show t h a t f o r any o f center

x

,

and r a d i u s

i f max a ( @ )

-

1

+

~1

min a l e )

4

.

,

the

,&spectrograph

of

Fp

i s the c i r c l e

i s not too large, i t i s possible t o f i n d

227

H i g h e r order b i f u r c a t i o n s

C

>0

such t h a t , f o r

~1

Fp(T

small

1

x

11

CO ,C

c

,

T ’ x [ O , C[

hence t h e i n t e r s e c t i o n

contains then, f o r

p

no i n t e r s e c t i o n w i t h 1

(KL1-.,T

x 0

as

> 0 small a compact set T

1

x 0

, and

- 0 )

L e t u s t r y t o see w h e t h e r

then

If

Log x,(e)

wo

choose

. K

K

P

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

I-1

i n v a r i a n t under T

1

with

F P

x 0

is t h e graph of a f u n c t i o n

is e a s i l y seen t o be s o l u t i o n o f t h e d i f f e r e n c e equation

is too well a p p r o x i m a t e d by r a t i o n a l numbers, i t i s p o s s i b l e t o a(Q)

a n a l y t i c s u c h t h a t t h e a b o v e e q u a t i o n h a s e v e n no m e a s u r a b l e

s o l u t i o n ! So, t h e q u e s t i o n is, what c a n Note t h a t i f we are a b l e t o f i n d

one s a y on

xo

K

LL

i n t h i s case

?

solution o f t h i s equation, it

c o r r e s p o n d s t o a c h a n g e of v a r i a b l e s s u c h t h a t

a ( @ )Is r e p l a c e d by 1 f o r t h e

new map. Then by t h e u s u a l way we are a b l e t o compute t h e b i f u r c a t e d circle.

This Page Intentionally Left Blank

B I B L I O G R A P H Y

V.I.

ARNOLD

Loss o f s t a b i l i t y o f s e l f - o s c i l l a t i o n s c l o s e t o resonance and v e r s a 1 d e f o r m a t i o n s o f e q u i v a r i a n t v e c t o r 2, 1-10 (19771. f i e l d s . Funk. Anal. Ego. P r i l o z .

11,

R.

BDNIC and

J . FRAMPTON

Smooth f u n c t i o n s on Banach m a n i f o l d s . J . Math. 15, 877-898 ( 1 9 6 6 ) . Mech. -

R. BOUC, M. OEFILIPPI, G. I O O S S

On a p r o b l e m o f f o r c e d n o n l i n e a r o s c i l l a t i o n s . N u m e r i c a l example o f b i f u r c a t i o n i n t o an i n v a r i a n t t o r u s . J. N o n l i n e a r A n a l . Theory, Methods and A p p l i c a t i o n s . 2, 2, 211-224 ( 1 9 7 8 1 .

A. G.

B i f u r c a t i o n s de t o r e s i n v a r i a n t s . Arch. Mech. Anal. [ t o a p p e a r ) .

CHENCINER

IOOSS

[4 b i s ] A. CHENCINER G.

IOOSS

G. DA-PRATO e t P. GRISVARD

Rational

P e r s i s t a n c e e t b i f u r c a t i o n de t o r e s i n v a r i a n t s . Arch. R a t . Mech. Anal. [ t o a p p e a r ) .

E q u a t i o n s d ' Q v o l u t i o n a b s t r a i t e s n o n - 1 i n Q a i r e s de t y p e p a r a b o l i q u e ( t o appear].

J . GUCKENHEIMER On t h e b i f u r c a t i o n o f maps o f t h e i n t e r v a l . I n v e n t i o n e s mathematicae, 39, 1 6 5 - 1 7 8 ( 1 8 7 7 1 . J.K.

HALE

Ordinary d i f f e r e n t i a l equations.

New-York,

Wiley 1969.

M. HERMAN

Sur l a c o n j u g a i s o n d i f f e r e n t i a b l e des d i f f e o m o r p h i s m e s du c e r c l e des r o t a t i o n s . P u b l . I.H.E.S. Paris [ t o appear).

M.

HERMAN

Mesure de Lebesgue e t nombre de r o t a t i o n . L e c t u r e Notes i n Maths. No 597. B e r l i n - H e i d e l b e r g New-York. S p r i n g e r 1 9 7 7 .

M.

W.

I n v a r i a n t m a n i f o l d s . L e c t u r e Notes i n Maths. No 583. B e r l i n - H e i d e l b e r g - New-York. Springer 1977.

C.C.

M.

HIRSCH, PUGH SHUB

229

230

Bib1 i o g r a p h y

[Ill

G . IOOSS

[I21

G.

IOOSS

B i f u r c a t i o n e t s t a b i l i t g . Pub. Math. d ’ 0 r s a y no 31 ( 1 9 7 4 ) .

[I31

G.

IOOSS

B i f u r c a t i o n o f a p e r i o d i c s o l u t i o n o f t h e Navier-Stokes e q u a t i o n s i n t o an i n v a r i a n t t o r u s . Arch. R a t i o n a l Mech. Anal. 2, 1, 35-56 (19751.

1141 G.

IOOSS

S u r l a deuxieme b i f u r c a t i o n d ’ u n e s o l u t i o n s t a t i o n n a i r e d e s y s t h e s du t y p e N a v i e r - S t o k e s . Arch. R a t i o n a l Mech. Anal. 64, 4, 339-369 ( 1 9 7 7 ) .

E x i s t e n c e e t s t a b i l i t i ! de l a s o l u t i o n p g r i o d i q u e second a i r e i n t e r v e n a n t dans les problsmes d ’ 6 v o l u t i o n du t y p e N a v i e r - S t o k e s . Arch. R a t i o n a l Mech. Anal. 47. 301-329 119721.

[li]G. IOOSS and D.D. JOSEPH

B i f u r c a t i o n a n d s t a b i l i t y o f nT-periodic s o l u t i o n s branching f r o m T - p e r i o d i c s o l u t i o n s a t p o i n t o f resonance. Arch. R a t i o n a l Mech. Anal. 66, 2, 135-172 ( 1 9 7 7 ) .

[I61 V.I.

IUOOVICH

On t h e s t a b i l i t y o f s e l f - o s c i l l a t i o n s o f a l i q u i d . D o k l . 3, 574-576 (19701. Akad. Nauk. S.S.S.R..

[I71 V.I.

IUOUVICH

Appearance o f s e l f o s c i l l a t i o n s i n a f l c l i d . Mek. 35, 638-655 (19711.

1181 O.D. JOSEPH

[I91

T.

I201

K. KIRCHGHSSNER and J . SCHEURLE

[211

O.A. and V.A.

KATO

195,

P r i k l . Mat.

S t a b i l i t y o f f l u i d m o t i o n s I, 11. S p r i n g e r T r a c t s i n N a t u r a l P h i l o s o p h y . V o l . 27 and 28 B e r l i n - H e i d e l b e r g New York. S p r i n g e r 1976. Perturbation theory f o r l i n e a r operators. Berlin-Heidelberg New York. S p r i n g e r 1966.

B i f u r c a t i o n . Symp. on Dynamical Systems. V o l . 1 ed. by C e s a r i , Hale, LaSalle. Acad. Press (1976).

LAOYZHENSKAYA On t h e l i n e a r i z a t i o n p r i n c i p l e and on i n v a r i a n t m a n i f o l d s f o r problems o f rnagnetohydrodynamics. Zap. SOLONNIKOV Nauch. Sem. Lorni, 38, 46-93, L e n i n g r a d (19731.

Bib1 i o y r a p h y

231

[22]

O.E.

[23]

F. LEMAIRE

[ 241

J L. LIONS e t E. MAGENES

Problemes aux l i m i t e s non homoggnes, Dunod, P a r i s 1968.

[25]

J.E. MARSDEN and M. MCCRACKEN

The Hopf b i f u r c a t i o n and i t s a p p l i c a t i o n s . A p p l i e d Math. Sciences B e r l i n - H e i d e l b e r g - New-York. S p r i n g e r 1976.

[ 261

G.H.

E i g e n f u n c t i o n branches o f n o n l i n e a r o p e r a t o r s , b i f u r c a t i o n s . L e c t u r e Notes i n Maths n o 104. B e r l i n - H e i d e l b e r g - New-York. S p r i n g e r 1969.

[27]

0. RUELLE and F. TAKENS

[28]

R.J.

SACKER

On i n v a r i a n t s u r f a c e s and b i f u r c a t i o n o f p e r i o d i c s o l u t i o n s o f ordinary d i f f e r e n t i a l equations. New-York U n i v . INN-NYU, 333 (19641.

[29]

D.H.

SATTINGER

T o p i c s i n s t a b i l i t y and b i f u r c a t i o n t h e o r y . L e c t u r e Notes i n Math. No 309. B e r l i n - H e i d e l b e r g - New-York. S p r i n g e r (1972).

1301

J . SCHEURLE

[31]

S.

[32]

M.M. and V.A.

VAINBERG

Y.H.

WAN

[33]

LANFORO I11

.

PIMBLEY J r .

STERNBERG

Bifurcation of periodic solutions i n t o invariant t o r i . L e c t u r e Notes i n Maths No 322, 159-192. B e r l i n - H e i d e l b e r g - New-York. S p r i n g e r 1973. B i f u r c a t i o n de Hopf pour' les a p p l i c a t i o n s dans un cas sonant. C.R. Acad. S c i . P a r i s , A(19781 ( t o a p p e a r ] .

Vol.

re-

1,

19,

On t h e n a t u r e o f t u r b u l e n c e . 167-192 (19711.

and t h e i r

Comm. Maths. Phys.

20, -

S e l e c t i v e I t e r a t i o n and A p p l i c a t i o n s . J . Math. Anal. Appl. E. 3, 596-616 (19771.

C e l e s t i a l mechanics,

11. New-York.

Benjamin 1969.

The t h e o r y o f b r a n c h i n g o f s o l u t i o n s o f n o n - l i n e a r e q u a t i o n s . N o o r d h o f f I n s t . P u b l . Leyden 1974.

TRENOGIN

B i f u r c a t i o n i n t o i n v a r i a n t t o r i a t p o i n t s o f resonance. Arch. R a t . Mech. Anal. ( t o a p p e a r ) .

Bibliography

232

[34]

V.I.

ARNOLD,

Lectures on bifurcation in versa1 families. Russ. Math. Surveys 2 7 . 2 (19721, p. 54-123.

[35]

V.I.

ARNOLD,

Complements on the theory o f ordinary differential equations. Nauka 1978 ( I n Russian).