III. HOPF Bifurcation in R2.

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 ) .