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On the Tk→Tk+1 Bifurcation Problem
Singularities €2 Dynamical Systems S.N. Pnevmatikos (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1985
ON
k
T H E T 'T
k+l
35
BIFURCATION PROBLEM
Dietrich Flockerzi Universitat Wurzburg West Germany
I. INTRODUCTION
In 1979 G.R.Sell[l3] and A.Chenciner and G.Iooss[l] published results describing a Hopf-type bifurcation of a k-dimensional into a (k+l)-dimensional invariant torus within a one-parameter family of differential equations or maps,respectively.The present contribution gives an extension of these results since it will
not
beassumedthat
i) the normal spectrum of the k-dimensional torus crosses from IR+ to E7 in a transversal fashion (or that the corresponding transversality condition in Theorem 111.2.11 of [ l ] holds), ii)the quadratic and cubic terms are the only significant nonlinearities. The proof of our main result in Section IV i s rather involved
(cf.
[ 5 ] ) . So we confine ourselves to an outline of the main ideas and put more emphasis on the derivation of the set up we use for this
kind of bifurcation problem. Just to give the flavor of the result we are aiming at we recall the essential features of the corresponding bifurcation problem in IR2.
To this end we consider an analytic one-parameter family of or-
dinary differential equations
;; =
(1.I ,)
f,(X), x t IR2
,
a
E I=(-cc0,ao),
with f (o)=oand we assume that the variational equation x=f' (o)x 0
0
along the trivial solutjon x=o of the unperturbed system (1 .l)o admits a family F of closed orbits C*,i.e. we assume P
Al)
f'(o) has eigenvalues i i w 0
0
with w o > o .
Then,by the implicit function theorem,the system (l.l)cc possesses a stationary solution x ( a ) which we may take to be the trivial solution x=o. Here -as always- we suppose that a
0
> o is sufficiently
small. We further assume that the closed orbits C * of F are of mul-
.
P
tiplicity m In terms of the spectrum of fi(o) this can be formulated as an m-th order contact condition:
D. Flockerzi
36 A2)
f i ( o ) h a s e i g e n v a l u e s am h m ( a ) i i w ( a ) f o r a E I w i t h m i - IN I Am(o)=/= o and w ( o ) = w o .
The a s s u m p t i o n s A1 and A2 imply t h a t x=o i s a f o c u s f o r i = f ' (o)x f o r a = # o and a c e n t e r f o r a = o . Now,the e f f e c t of t a k i n g t h e n o n l i ( 1 . 1 I u i n t o a c c o u n t s h o u l d b e t h a t some of t h e m c o p i e s of e a c h i n v a r i a n t c u r v e C* of F a r e b e n t t o t h e r i g h t ( s u p e r c r i t i c a l
n e a r i t i e s of
P
bifurcation f o r a>o),some t o the l e f t ( s u b c r i t i c a l bifurcation f o r a < o ) and some a r e l e f t i n p l a c e ( v e r t i c a l b i f u r c a t i o n f o r a = o ) . A
vague a t t r a c t o r o r r e p e l l o r c o n d i t i o n l i k e A3)
x=o i s a f o c u s o f f i n i t e m u l t i p l i c i t y f o r ( 1 . 1 )
0
e x c l u d e s t h e v e r t i c a l b i f u r c a t i o n and i m p l i e s t h e e x i s t e n c e of l e a s t one t r a n s c r i t i c a l b r a n c h of b i f u r c a t i n g p e r i o d i c
at
solutions
p r o v i d e d m i s odd. T h i s c a n b e s e e n by p u r e l y t o p o l o g i c a l a r g u m e n t s (PoincarG-Bendixson t h e o r e m ) . The number of b i f u r c a t i n g b r a n c h e s
I
t h e i r d i r e c t i o n of b i f u r c a t i o n and t h e s t a b i l i t y p r o p e r t i e s o f t h e b i f u r c a t i n g p e r i o d i c s o l u t i o n s c a n b e d e t e r m i n e d b y a n a l y t i c a l means f o r any m ( c f . [ 3 ] ) . I n t h e p r e s e n t p a p e r w e want t o i n v e s t i g a t e what t h e c o r r e s p o n d i n g h y p o t h e s e s f o r t h e above i n d i c a t e d Tk+Tk"
bifurcation are that
a l l o w t o answer q u e s t i o n s i n v o l v i n g t h e number of b r a n c h e s o f d i m e n s i o n a l t o r i b i f u r c a t i n g from a g i v e n k - d i m e n s i o n a l
(k+l)-
torus,their
d i r e c t i o n o f b i f u r c a t i o n and t h e i r s t a b i l i t y p r o p e r t i e s .
Thus w e
w i l l c o n s i d e r a Cm-smooth o n e - p a r a m e t e r f a m i l y of d i f f e r e n t i a l equations (1-2)a
5 = Fa(5
/
5 E IRn ,
a E I=(-ao/ao)
p o s s e s s i n g a smooth f a m i l y o f k - d i m e n s i o n a l k+2
/
invariant t o r i
Ma
with
one may t h i n k o f Ma a s b e i n g g e n e r a t e d by k weakly coup-
l e d van d e r P o l o s c l l a t o r s . T h i s f a m i l y Ma t a k e s o v e r t h e r o l e of t h e f a m i l y x ( a ) of s t a t i o n a r y s o l u t i o n s o f not-too-restrictive
(1.1)
.We w i l l establish
c r i t e r i a which a l l o w t h e e x i s t e n c e of
(k+l)-di-
m e n s i o n a l i n v a r i a n t t o r i Ma t e n d i n g as a whole t o Mo f o r a - t o . I n S e c t i o n I1 w e g i v e an example where j u s t p a r t o f a 2 - t o r u s u n d e r g o e s a Hopf-type b i f u r c a t i o n t o a p i n c h e d t o r u s . By a s e c o n d example w e i l l u s t r a t e t h e p o s s i b i l i t y t h a t one p a r t o f a 2 - t o r u s c a n u n d e r g o s u c h a b i f u r c a t i o n t o a p i n c h e d t o r u s b e f o r e t h e rest of it d o e s s o t h a t i n t h e end a p r o p e r 3 - t o r u s h a s b i f u r c a t e d . The c a s e o f b i f u r c a t i o n from a p e r i o d i c s o l u t i o n ( k = 1 ) i s much e a s i e r t o d e a l w i t h t h a n t h e c a s e k 2 2 . T h e r e a r e e s s e n t i a l l y two ways
On the
7k Tk" Bifurcation Problem
37
--f
of p r o v i n g t h e T1+T2 b i f u r c a t i o n : f i r s t , b y c o n s i d e r i n g t h e f i r s t re-
t u r n map on a t r a n s v e r s a l h y p e r s u r f a c e one i s l e d t o t h e Hopf b i f u r c a t i o n problem f o r maps ( c f . [ l l ] ) , s e c o n d l y , b y u s i n g F l o q u e t t h e o r y one c a n r e d u c e t h i s b i f u r c a t i o n problem t o t h e Hopf b i f u r c a t i o n p r o b -
l e m from a s t a t i o n a r y s o l u t i o n . F o r k22 t h e r e i s n o F l o q u e t t h e o r y a v a i l a b l e , and i t i s known t h a t t h e r e d u c t i o n of a q u a s i - p e r i o d i c l i n e a r system t o a l i n e a r system w i t h c o n s t a n t c o e f f i c i e n t s i s n o t always p o s s i b l e . Thus t h e r e i s a problem i n d e t e r m i n i n g p r e c i s e l y a c r i t i c a l v a l u e of t h e p a r a m e t e r a t which t h e s t a b i l i t y o f t h e i n v a r i a n t t o r u s Ma changes i n a way which a l l o w s t h e b i f u r c a t i o n of a ( k + l ) - d i m e n s i o n a l t o r u s . I n t h e s e q u e l w e mainly have t h i s case k>2 i n mind. I n t h e n e x t s e c t i o n w e d e s c r i b e how t h e t h e o r y o f t h e s p e c trum o f an i n v a r i a n t m a n i f o l d ( c f . [ 1 2 ] ) c a n be u s e d t o d e f i n e s u c h a critical value. 11. THE
BIFURCATION SET UP
W e c o n s i d e r system ( 1 . 2 )
and assume t h a t t h e f l o w on Ma i s gene-
r a t e d by
where Tk d e n o t e s t h e s t a n d a r d t o r u s of dimension k . The components of y a r e t h u s d e f i n e d modulo 2.rr
. Here
it i s g e n e r a l l y n o t s u f f i c i e n t
t o require t h e existence of the k-torus
just for
a=o.
I f one h a s t h e
above b i f u r c a t i o n i n mind Mo c a n n o t b e n o r m a l l y h y p e r b o l i c and t h u s t h e c o n t i n u a t i o n o f Mo t o Ma f o r a E I i s n o t always p o s s i b l e (compare t h e remark f o l l o w i n g H l i i b e l o w ) . A d a p t i n g t h e h y p o t h e s e s A1 and A2 t o t h e p r e s e n t s i t u a t i o n amounts - i n somewhat l o o s e t e r m s -
t o the
f o l l o w i n g assumption: The " v a r i a t i o n a l e q u a t i o n " o f t h e u n p e r t u r b e d s y s t e m ( 1 . 2 )
0
a l o n g t h e t o r u s Mo p o s s e s s e s a f a m i l y F o f i n v a r i a n t (k+l)dimensional t o r i
M* of " m u l t i p l i c i t y m" P
.
I n what f o l l o w s w e w i l l make p r e c i s e what i s meant by " v a r i a t i o n a l e q u a t i o n " and by " m u l t i p l i c i t y m " . The f i r s t o b j e c t i v e i s t o i n t r o d u c e a l o c a l c o o r d i n a t e s y s t e m near M
v i a a d e c o m p o s i t i o n o f t h e t a n g e n t s p a c e o f lRn a t y i n t o t h e sum of t h e k-dimensional t a n g e n t s p a c e Yci(y) t o Ma a t y , t h e "non-
c r i t i c a l " p a r t Za(y) of t h e normal s p a c e t o Ma a t y and t h e " c r i t i c a l " p a r t Xci(y) o f t h e normal s p a c e which r e f l e c t s t h e f a c t of M a n o t bei n g normally h y p e r b o l i c f o r a l l
ci
E1,i.e.
v i a a decomposition
38
D. Flockerzi
(2.2)
IRn
for all ( y , a ) i ~ ~ x ~ .
+ Y,(y) + Z a ( y )
= Xa(y)
F o r t h e H o p f - t y p e b i f u r c a t i o n w e a r e a i m i n g a t X,(y) s h o u l d b e a n k 2 - d i m e n s i o n a 1 , s m o o t h l y v a r y i n g s u b s p a c e o f IR f o r a l l (y,a)CT X I . I n o r d e r t o d e r i v e such a decomposition ( 2 . 2 ) w e consider the l i n e a r skew-product
flow
11 (
to'Y0,a,t)
= (E
(Yo,U,
.
k
on IRn X T X I i n d u c e d by ( 1 . 2 ) with i n i t i a l value y
t )S 0 , Y ( Y o , a , t ), a )
Here, Y d e n o t e s t h e s o l u t i o n o f
(2.1
a t t = o and 6 d e n o t e s t h e f u n d a m e n t a l m a t r i x
0
L = F i ( Y ( y o , u , t ) ) C w i t h 6 ( y 0 , a , o ) = I . A s i n [ 1 3 , V ] w e de-
s o l u t i o n of
n o t e t h e i n d u c e d t a n g e n t i a l a n d n o r m a l l i n e a r s k e w - p r o d u c t f l o w s by iIT a n d I T N , r e s p e c t i v e l y . F o r t h e n o t i o n o f
t h e s p e c t r a of such flows
w e r e f e r t o [ 1 2 1 o r [ 1 3 ] . Then w e h a v e a d e c o m p o s i t i o n a s i n ( 2 . 2 )
i f the following s p e c t r a l conditions hold: S 1 ) TIT
and TIN h a v e d i s j o i n t s p e c t r a f o r
af
o . (Note t h a t
o
belongs
t o b o t h s p e c t r a f o r a = o i f k;;.l.) 5 2 ) ITN h a s a " c r i t i c a l " s p e c t r a l s e t C c(-6,6)f o r a E I w i t h a t w o dimensional a s s o c i a t e d s p e c t r a l subspace Xa(y) such t h a t
c0 =io},
C~;IR+
=/=
fi *
C ~ ~ I R - =
f o r a+o.
The r e m a i n i n g s p e c t r a l i n t e r v a l s a r e bounded away f r o m o by 6 5 3 ) Xli(y) v a r i e s s m o o t h l y i n k a l l ytT
.
.
a n d X o ( y ) i s d i s j o i n t f r o m Yo(y) f o r
Under t h e s e c i r c u m s t a n c e s X ( y ) and Z ( y ) c a n b e p a r a m e t r i z e d i n a 0 0 2 smooth way by xEIR a n d , r e s p e c t i v e l y . The c o r r e s p o n d i n g d i - f f e r e n t i a l equation w i l l be 2n-periodic
i n e a c h component of y and
w i l l have t h e f o l l o w i n g form:
G (2.31,
= n(y)x+asl
-
1
(y,a)x+uR
(y,a)z
2
,
$
=
5
= B(y)z+cxB ( y , a ) x + a B 2 ( y , a ) z
w
+ Y(x,y,z,a) 1
2
w h e r e X and Z a r e o f o r d e r O ( / x l + 1 z I g,(y). (2.4)
iX ( x , y , z , a ) ,
2
+
)
Moreover t h e m a t r i x B ( y ) i n ( 2 . 3 )
Z(x,y,z,a)
,
X E I R
2
,
YETk I mn-k-2
I
and w h e r e Y ( o , y , o , a ) e q u a l s s a t i e s f i e s t h e following: k
= B ( o t + y ) z a d m i t s a n e x p o n e n t i a l d i c h o t o m y f o r a l l y ET 0
0
( c f . [ 5 1 ) .The c o n s i d e r a t i o n s t o f o l l o w a l w a y s r e f e r t o s y s t e m ( 2 . 3 ) a
with the property
(2.4).
The f o l l o w i n g e x a m p l e s r e v e a l t h e i m p o r t a n t r o l e t h e n a t u r e o f the rotation vector
p l a y s and m o t i v a t e o u r h y p o t h e s i s H I b e l o w .
On the
r' -+ Tk"
39
Bifurcation Problem
ExurnpLc 1 ( ~ 6 . 1 6 1 ) W e c o n s i d e r t h e f o l l o w i n g 4-dimensional
w i t h a r a t i o n a l r o t a t i o n v e c t o r =;
s y s t e m o f t h e form ( 2 . 3 ) ,
(1/2 1)
$Q
and wo
1 w i t h ( x , y l ,y2)EIR2 x T x T 1 . With p o l a r c o o r d i n a t e s
:
( r , e ) f o r x and t h e
change of c o o r d i n a t e s (2.6)
$ =
2y 1 -y 2
I
+
= -Yl+Yl
w e w r i t e ( 2 . 5 ) a i n t h e form (?.7)a
3
r = a(l+2cos$)r-r
e =
0
,
,
&
= asin$ =
,
(1-asin$)/2.
The ( q r @ ) - s y s t e md e s c r i b e s a f l o w on a 2 - t o r u s h a v i n g two p e r i o d i c o r b i t s f o r Iaif r u s of of
0.
The s e t { r = o } c o r r e s p o n d s t o t h e i n v a r i a n t 2-to-
( 2 . 7 ) a which i s a t t r a c t i v e f o r a=o. The l i n e a r i z e d e q u a t i o n
(2.710 admits a family F of i n v a r i a n t 3 - t o r i .
i n d e p e n d e n t of
($,$)
The ( r , e ) - s y s t e m i s
and c a n b e a n a l y z e d by p h a s e p l a n e t e c h n i q u e s .
F o r t h e c h o i c e o f c o o r d i n a t e s as i n F i g u r e 1
Fig. 1 t h e two s k e t c h e s i n F i g u r e 2 show t h e f l o w o f t h e ( r , O ) - s y s t e m f o r
a < o and a > o :
Fig. 2
40
D. Flockerzi
The c o r r e s p o n d i n g i n v a r i a n t s e t s o f t h e p e r i o d map on t h e s u r f a c e { y 2 = o m o d 2 n l -one may t h i n k of y 2 p l a y i n g t h e r o l e of t h e t i m e -
are
t h u s p i n c h e d t o r i a s shown i n F i g u r e 3 .
T h u s , i n s t e a d o f a o n e - s i d e d b i f u r c a t i o n t o a 3 - t o r u s o n e h a s a twos i d e d b i f u r c a t i o n t o i n v a r i a n t s e t s t h a t can b e v i s u a l i z e d a s t h r e e d i m e n s i o n a l t o r i which a r e p i n c h e d a l o n g a c i r c l e . F o r a more e x t e n s i v e s t u d y of t h i s r e s o n a n c e phenomenon w e r e f e r t o 1 6 1 .
Exarnp-Ye 2 i e d . [ l o l l The e s s e n t i a l f e a t u r e s of t h e above example c a n b e u s e d t o show t h a t t h e T2+T3 b i f u r c a t i o n i s v e r y s e n s i t i v e t o p e r t u r b a t i o n s . C o n s i d e r (2.8)€ where
E
r3
= u r * 1 y = 1 2
+
+
E
2
rcos(2y,-y2)
1 2 s i n ( 2 y -y ) 2 1 2
--E
B
I
,
= w
0
+
2
o(E
y2= 1 ,
i s j u s t an a u x i l i a r y p a r a m e t e r m e a s u r i n g t h e s i z e of t h e p e r -
t u r b a t i o n . The u n p e r t u r b e d s y s t e m ( 2 . 8 ) 0 u n d e r g o e s a s u p e r c r i t i c a l
{ r = o }t o t h e i n v a r i a n t
Hopf b i f u r c a t i o n from t h e i n v a r i a n t 2 - t o r u s
3 - t o r u s {r=Jcil. By means of t h e p e r t u r b e d s y s t e m ( 2 . 8 ) € w e i l l u s t r a t e t h a t p a r t o f an i n v a r i a n t 2 - t o r u s c a n u n d e r g o a Hopf b i f u r c a t i o n b e f o r e a p r o p e r 3 - t o r u s a p p e a r s . By i n t r o d u c i n g t h e new b i f u r c a t i o n parameter B v i a system ( 2 . 8 )
G
=
2
B,the s c a l i n g r + E
r
and t h e t r a n s f o r m a t i o n ( 2 . 6 )
can be w r i t t e n a s E
8 = 0
Fig. 4
O=E
2
r ( ~ + c o s + -2r 2
0
+ O ( E )
,
I
+
=
E
2
sin+
I
; = l + O ( E 2) .
2
On the Tk -+
Tk" Bifurcation Problem
41
Thus,for i n c r e a s i n g @ , t h e i n v a r i a n t 2-torus undergoes a p a r t i a l H o p f b i f u r c a t i o n t o a p i n c h e d t o r u s a t B=-1 and a s e c o n d s u c h b i f u r c a t i o n a t ~ = 1 .The i n v a r i a n t s e t s o f
-now t o f u l l 3 - t o r u s -
(2.8)
for the
p e r i o d map on { y =omod2n} a r e shown i n F i g u r e 4 . 2 I n view o f t h e s e examples w e a s k f o r a n o n r e s o n a n t r o t a t i o n vec-
tor
whose components a r e i n d e p e n d e n t o v e r Q . Our g e o m e t r i c i d e a o f
the family F requires a l l solutions of
2
(2.9)
= n(wt+y)x
yETkf
(c(;y)-cy)
t o b e bounded. ( T h i s i s n o t i m p l i e d by o u r s p e c t r a l a s s u m p t i o n s f s e e [9].)
We t h e r e f o r e c a n t a k e n ( y ) t o b e o f t h e form
[a]).
There i s n o l o s s o f g e n e r a l i t y i n t a k i n g c ( y ) e q u a l t o i t s
(cf.
mean v a l u e p r o v i d e d ( 2 . 9 ) a d m i t s a n o n t r i v i a l almost p e r i o d i c s o l u t i o n f o r some y t T k f s i n c e t h e n any s o l u t i o n o f ( 2 . 9 ) w i l l b e a l m o s t p e r i o d i c ( f o r any y E T k f c f . [ 8 1 ) . W e r e s t r i c t o u r a t t e n t i o n t o t h i s c a s e and assume
Now,the e x i s t e n c e o f t h e f a m i l y F of
(k+l)-dimensional i n v a r i a n t to-
r i M Y f o r t h e l i n e a r i z e d s y s t e m o f ( 2 . 3 ) 0 i s obvious.We would l i k e P t o add a remark: R.A.Johnson h a s shown i n [ 7 , 8 ] t h a t a l r e a d y f o r
dimy=2 t h e r e need n o t e x i s t a n o n t r i v i a l a l m o s t p e r i o d i c s o l u t i o n of ( 2 . 9 ) even i f a l l s o l u t i o n s o f
( 2 . 9 ) a r e bounded.The c r u c i a l p o i n t
i n proving t h e nonexistence of such a s o l u t i o n i s t h e c o n s t r u c t i o n of an a n a l y t i c f u n c t i o n c and a s q u a r e - i n t e g r a b l e , b u t n o t c o n t i n u o u s antiderivative i)
y
on T2 w i t h t h e f o l l o w i n g p r o p e r t i e s :
t 2 c h a s mean v a l u e o a n d J c ( w s + y ) d s i s unbounded f o r a l l YET t
i i ) y (wt+y) - y ( y ) = J
where
E,, i n
- W=
(w,
c ( o s + y ) d s f o r a l l yET2 and a l l t E l R
l ) T i s a s u i t a b l e i r r a t i o n a l number.
A s it t u r n s o u t t h e above assumption o f n o n r e s o n a n c e on
0 alone
w i l l n o t s u f f i c e f o r o u r p u r p o s e s . I n 1 1 1 A.Chenciner and G.Iooss have c o n s i d e r e d some c a s e s o f s t r o n g r e s o n a n c e where w depends r a 0 t i o n a l l y on 0. H e r e w e w i l l a s k f o r t h e s t r o n g v e r s i o n o f n o n r e s o nance i n form of a d i o p h a n t i n e c o n d i t i o n f o r Hlii)
There e x i s t p o s i t i v e c o n s t a n t s (11
satisfies
[v.w(
>K~(VI
-K
K
0
-
W=(W
and
0 K
, u ) ~ . W e assume
such t h a t
1+k for a l l o = f = v t ~
42
D. Flockerzi
A c a r e f u l a n a l y s i s o f o u r a r g u m e n t a t i o n b e l o w shows t h a t H l i i i s onAs l y n e e d e d f o r a f i n i t e number of v A s ( i f u d e n o t e s ( v v ) F Z x Z k ) . 0'
( w i t h vo=o,vo=l)
A . C h e n c i n e r a n d G.Iooss h a v e n o t e d h y p o t h e s i s H l i i allows the continuation of the torus M
0
t o t h e t o r i Ma f o r a G I ( [ 2 ] ) .
W e now p r o c e e d t o make t h e s t a t e m e n t a b o u t t h e " m u l t i p l i c i t y o f
M :
"
precise. In cylindrical coordinates ( x , y , z ) = ( r , o l y I z ) = ( r , $ , z ) E IR+ ~ T ~ + ~ ~ I R ~ - ~ - ~
syste
( 2 . 3 ) n i s now o f t h e f o r m
i
6
(2.10
=
nA ( 6 , c r ) r
= w
+
aA ( @ , u ) z
2 + Q(r,$,z,a), 1
= B($)z
+
R(r,@,z,u),
+ aBl(tJfa)r + aB2(tJru)z + Q ( r , $ I z I u ) ,
w h e r e R and Q are o f s e c o n d o r d e r i n ( r , z ) . By a smooth c h a n g e of variables
r
(2.11)
+
+
r + uU($,cc)r
aV($,cc)z
one i s l e d t o s o l v e 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 o f t h e form
(2.12) where lues
u$ ( $ )
x
- O
+ x($)
=
7=
i s a known 2 n - p e r i o d i c
7.v a n i s h
f u n c t i o n o n Tk+'.
when p e r f o r m i n g ( 2 . 1 1 )
s i v e l y because of H l i i -
x($)
mean v a l u e o f -the
i . can
U n l e s s a l l mean vab e computed r e c u r -
t h e r - e q u a t i o n t a k e s t h e form
= amK m r + a m + l [ o ( r ) + O ( / z l ) + ] O(r2+1z/*),
~ ~ = + o
( c f . [ 5 ] ) . W e assume t h a t t h e f o l l o w i n g s i t u a t i o n p r e v a i l s :
H2)
T h e r e e x i s t s a n mEN t h e mean v a l u e s
s u c h t h a t K :? =k o w h e r e a s m m are a l l e q u a l t o 0 .
7l f . . . , x m - l
I n c a s e of t h e Hopf b i f u r c a t i o n c o n s i d e r e d i n S e c t i o n I t o t h e c o n d i t i o n A2.
H 2 reduces
The e x i s t e n c e o f s u c h a n i n t e g e r m c a n be t a k e n
f o r t h e d e f i n i t i o n o f t h e m u l t i p l i c i t y o f M*. P
H 2 i s t h e e x a c t formu-
l a t i o n o f t h e p r e v i o u s g e o m e t r i c i d e a and s t a t e s i n a n a l y t i c a l t e r m s t h a t t h e n o r m a l s p e c t r u m o f Ma , i . e .
t h e s p e c t r a l s u b s e t C,,has
m-th o r d e r c o n t a c t w i t h t h e t a n g e n t i a l s p e c t r u m a t
U=O.
an
Finally we
s t a t e t h e a s s u m p t i o n c o r r e s p o n d i n g t o A3 : H3)
Mo i s v a g u e l y h y p e r b o l i c f o r ( 2 . 3 ) 0 ( c f . ( 3 . 3 ) b e l o w ) .
Having e s t a b l i s h e d t h e s e t up f o r o u r b i f u r c a t i o n p r o b l e m w e formul a t e o u r goad i n p r e c i s e t e r m s : Show t h e e x i s t e n c e o f i n v a r i a n t ( k + l ) - t o r i Ma o f t h e f o r m (2.13)
M,
= { z = z ( r , $ , a ) , r=;($'cc), $ t T k + ' l
f o r IC. > o
On the
Tk Tkt +
’
43
Bifurcation Problem
f
w i t h a p p r o p r i a t e f u n c t i o n s 2 and
such t h a t M a + M
(cf .Section 1V.B).
0
as a - t o
111. NORMAL FORMS FOR ( 2 . 3 )
W e c o n f i n e o u r s e l v e s t o o u t l i n e t h e d e r i v a t i o n o f t h e normal
forms. For t h e d e t a i l s w e r e f e r t o [ 5 ] . A f t e r t h e t r a n s f o r m a t i o n ( 2 . 1 1 ) w e make t h e s u b s t i t u t i o n s
(3.1)
r
+ E
introducing
m
r
,
z
E
-t
m
z
,
= EB
01
a s t h e new b i f u r c a t i o n p a r a m e t e r . A s w e g o a l o n g w e
E>O
w i l l be l e d t o v a r i o u s choices of
B ( i n terms o f
E)
and t h u s t o t h e
e x a c t s c a l i n g s w i t h € = € ( a ) . I n a f i r s t s t e p w e compute a n approximat i o n t o t h e c e n t e r manifold of
( 2 . 3 ) a i n t h e new v a r i a b l e s
(3.1) v i a
a transformation (3.2)
z
-t
z +
E W ( r , @ , ~ , f i ) .
Because o f t h e e x p o n e n t i a l dichotomy i n ( 2 . 4 ) t h e a r i s i n g 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 which a r e o f t h e form - B ( @ ) w ( $ ) + w ( @ ) * w + Q($) = $
h a v e u n i q u e smooth 2 n - p e r i o d i c
0
s o l u t i o n s . By means o f
(3.2) w e a r r i v e
a t a s y s t e m o f t h e form
2
=
5:
+
ER*(r,@,E,B)
= w
+
EO(IZ\),
O(E),
= B($)z
+
N
E O ( ~ Z I )+ O ( E ) ,
where N i s a n a r b i r a r i l y l a r g e f i n i t e p o s i t i v e i n t e g e r . I n v a r i a n t manifold theory t e l l s us t h a t t h e z - t e r m
i n t h e r a d i a l e q u a t i o n can
a h i g h e r o r d e r term (see S e c t i o n V). I n t h e
be c o n s i d e r e d a s b e i n g
s e c o n d s t e p w e r e p l a c e t h e $ - d e p e n d e n t c o e f f i c i e n t s of t h e l e a d i n g powers o f r i n R* by t h e i r c o n s t a n t mean v a l u e s . by a n a v e r a g i n g t r a n s f o r m a t i o n
This canbeachieved
r + r + c U ( r , @ , ~ , B )w h e r e t h e c o r r e -
s p o n d i n g 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 a r e o f t h e form ( 2 . 1 2 ) . B e c a u s e o f t h e d i o p h a n t i n e c o n d i t i o n H l i i t h e y h a v e u n i q u e smooth 21rp e r i o d i c s o l u t i o n s w i t h mean v a l u e
0.
The r e s u l t i n g r a d i a l e q u a t i o n
w i l l t h e n be o f t h e form = P(r,E,B)
+
P ~ ( ~ , $ , E , o +) E O C ~ Z = ~ ) O(E)
where - b e c a u s e o f H3- t h e r e e x i s t s a n il tN s u c h t h a t o n e h a s 0
(3.3)
P ( r , c , o ) = E2L,mKor2~om+l, KO+
0.
D. Flockerzi
44
p R c a n t h u s be assumed t o be o f h i g h e r order w i t h respect
to
By
E.
employing t h e Newton d i a g r a m f o r P ( r , E , @ ) = o ( w h e r e o n e marks t h e powers o f
E
on t h e a b s c i s s a and t h o s e o f 6 on t h e o r d i n a t e ) o n e no-
t i c e s t h a t t h i s a l g e b r a i c b i f u r c a t i o n e q u a t i o n c a n b e w r i t t e n as P ( r , E , @ )=
(3.4)
+
[email protected]+1
jcJ
1
h.0.t
The s e t J 1’ and f3-powers on t h e convex p o l y g o n a l
w i t h n o n z e r o c o n s t a n t s K . and n o n n e g a t i v e i n t e g e r s L 7
corresponds t o t h e p a i r s of
E-
l i n e i n t h e Newton d i a g r a m . The v a r i o u s s l o p e s -l/A ( h ) E Q - , h = l , .
. . ,H,
o f Newton’s p o l y g o n a l l i n e t h e n y i e l d t h e c o r r e c t r e l a t i o n b e t w e e n a and E by
(3.5)
a = €6,
6
=
+E
A ( h ) , h=l
,.. . , H ,
and l e a d t o H normal r e p r e s e n t a t i o n s o f
( 2 . 3 ) a i n t h e form
with appropriate positive rationals u (h)
.
The “Newton p o l y n o m i a l “
P t ( r ) i s t h e c o e f f i c i e n t o f t h e lowest E-power h t h u s n o monomial. IV.
i n P ( r , E , + E A ( h ) ) and
M A I N RESULT
A ) Under t h e h y p o t h e s e s Hl,H2 and H3 t h e above
H normal forms ( 3 . 6 I E
c a n b e d e r i v e d from ( 2 . 3 ) a . A n y b i f u r c a t i n g i n v a r i a n t ( k + l ) - t o r u s o f t h e form ( 2 . 1 3 ) w i l l b e o n e f o r o n e o f t h e s e B) I f
+ Ph
representations
(3.6)
.
o r P h h a s a s i m p l e p o s i t i v e zero p t h e n t h e r e e x i s t s a n i n -
v a r i a n t ( k + l ) - t o r u s Ma f o r ( 2 . 3 ) , o f t h e form
I (r,$,z,a): r=$(+,a),
z = & ( e ( $ , a ) ,$,a)
,
$ET~+’}
and 5 are g i v e n by m(h)) F ( $ , a ) = I a l m ( h )( P + O ( I ) ) , 2(P(+,a)t$,a) = o(lal w i t h m ( h ) = m / ( l + A ( h ) ) . The s t a b i l i t y p r o p e r t y o f Ma w i t h i n t h e c e n t e r f o r a > o o r a < o , r e s p e c t i v e l y . The f u n c t i o n s
s t a b l e m a n i f o l d f o l l o w s from t h e s i g n o f aP’ C)
(p)
.
ar h I f m i n H2 i s odd t h e n t h e r e e x i s t s a t l e a s t o n e h~~1,...,H]so
t h a t t h e “Newton s l o p e ” - l / A ( h )
i n ( 3 . 5 ) c o r r e s p o n d s t o an odd o r d i -
n a t e segment i n t h e Newton d i a g r a m of
(3.4). I f -l/A(h)
+
i s such a
s l o p e t h e n a t l e a s t o n e o f t h e p o l y n o m i a l s Ph o r Pi p o s s e s s e s an odd o r d e r p o s i t i v e z e r o . Such z e r o s c a n “ i n g e n e r a l ” b e r e d u c e d t o s i m p l e z e r o s ( c f . [ 5 , S e c t i o n V]:The o n l y e x c e p t i o n c a n be c a u s e d b y t h e n e e d o f i n f i n i t e l y many r e d u c t i o n s t e p s ) .
On the D)
Tk + 7k"Bifurcation Problem
45
I n t h e p a r t i c u l a r c a s e m=l t h e r e e x i s t s j u s t t h e Newton s l o p e
+
- l / A ( 1 ) = - 1 / ( 2 ~ ~ - 1 )I.n t h i s c a s e e i t h e r PI or P; h a s t h e s i m p l e p o s i t i v e z e r o P = / K ~ / K 1~ /\2 Q 0 , and one h a s a u n i q u e b i f u r c a t i o n o f an i n v a r i a n t (k+l)-torus e i t h e r f o r a>o o r f o r a
E) The number of p o s i t i v e z e r o s o f a l l t h e Newton p o l y n o m i a l s toget h e r d o e s n o t e x c e e d min(m,Eo). T h u s , " i n g e n e r a l " t h e r e are a t most m i n ( m , i ) b r a n c h e s of b i f u r c a t i n g ( k + l ) - t o r i . 0
V.
INVARIANT MANIFOLDS AND T
k+Tk+1
BIFURCATION
W e have a l r e a d y s k e t c h e d t h e p r o o f of p a r t A i n S e c t i o n 1 I I . P a r t s C , D and E f o l l o w from a t h o r o u g h d i s c u s s i o n of t h e Newton d i a g r a m
f o r ( 3 . 4 ) . H e r e , w e w i l l j u s t g i v e an o u t l i n e how t h e t h e o r y of i n -
+
v a r i a n t m a n i f o l d s c a n be u s e d t o p r o v e p a r t B. W e s u p p o s e t h a t Ph possesses a simple p o s i t i v e zero p,denote
+ Ph
s i m p l y by P and d r o p
t h e dependence on h i n t h e s e q u e l . I f A d e n o t e s
y/6
i n lowset terms
w e i n t r o d u c e t h e new p a r a m e t e r v by c = v 6 t o a v o i d f r a c t i o -
(y>0,6>0)
n a l powers of t h e p a r a m e t e r . With t h e t r a n s l a t i o n r+r+p s y s t e m (3.6)&
i s o f t h e form
c (5.1),,
6
v
60
( P ' ( p ) r + o ( l ) )+ v6~(/zj), 6 = w + O(v ), 6 = B ( $ ) z + li O ( \ Z l ) + O ( v N 6 ) .
=
I n o r d e r t o r e d u c e t h i s n-dimensional
system (5.1)
!J
t o a (k+2)-di-
m e n s i o n a l s y s t e m w e modify t h e r i g h t - h a n d s i d e s o it h a s t h e n e c e s s a r y g l o b a l p r o p e r t i e s . Because o f
4
= o
+
6
O(p ),
(2.4) t h e system
= B($)z
a l s o a d m i t s a n e x p o n e n t i a l dichotomy ( c f . [ 1 2 ] o r [ 4 1 ) . T h u s w e c a n d e f i n e t h e G r e e n ' s f u n c t i o n f o r t h e m o d i f i e d s y s t e m which e n a b l e s u s t o r e d u c e t h e n - d i m e n s i o n a l s y s t e m t o a ( k + 2 ) - d i m e n s i o n a l one on a c e n t e r manifold
A f u r t h e r averaging transformation
$+$+v
6
V(r,$,u)
-where f o r t h e
r e c u r s i v e computation of V t h e d i o p h a n t i n e c o n d i t i o n H l i i is used once more- g e n e r a t e s a weakly c o u p l e d s y s t e m (5.3)
r = v
6U
$ = w
+
-
(P'(p)r+G(l)), Oo(rlv)
+ O(v
6U+1
1
on t h e c e n t e r m a n i f o l d ( 5 . 2 ) . S i n c e
Q0
d o e s n o t depend on $ t h e n o r -
mal f l o w d o m i n a t e s t h e t a n g e n t i a l f l o w f o r ( 5 . 3 ) . Thus t h e r e e x i s t s
46
D. Flockerzi
an invariant manifold (5.4)
{r=:($,u)=o(p)
:
k+ 1 GET 1
on the manifold (5.2). All in al1,we thus have arrived at the desired (k+l)-torus Ma with the properties stated in Section 1V.B. We note that the various scalings (3.1) and (3.5) amount to
so that the functions
f and 5 of (2.13) and Section 1 V . B aregivenby
$ ( $ , a ) = am(h) ( p + > ( $ , p ) )
,
z(r,$,a) = am(h)2(rfG,u).
REFERENCES 1. A.Chenciner and G.1ooss:Bifurcations de tores invariants,Archive Rat .Mech .Anal.69 ( 1979)109-1 98. 2. A.Chenciner and G.Iooss:Persistance et bifurcations de tores invariants,Archive Rat.Mech,Anal.71 (1979)301-307. 3. D.Flockerzi:Existence of Small Periodic Solutions of ODE'S in IR2 , Archiv der Mathematik 33(1979)263-278. 4. D.Flockerzi:Weakly Nonlinear Systems and Bifurcation of Higher Dimensional Tori,in:H.W.Knobloch and K.Schmitt,Equadiff 82,Springer Lecture Note in Math. 101 7 ( 1983) 185-1 93. D.Flockerzi:Generalized Bifurcation of Higher Dimensional Tori, 5. University of Wurzburg preprint 97(1983),(to appear in the JDE). 6 . D.Flockerzi:Resonance and Bifurcation of Higher Dimensional Tori, University of Wiirzburg preprint 102 (1983). 7. R.A.Johnson:Measurable Subbundles in Linear Skew-Product Flows, Illinois J .Math. 23 ( 1 979 ) 183-1 98. 8. R.A. Johns0n:Analyticity of Spectral Subbundles ,JDE 35 (1980) 366-387. 9. R.A.Johnson:On a Floquet Theory for Almost-Periodic Two-Dimensional Linear Systems,JDE 37(1980)184-205. 1o.K.R.Meyer:Tori in Resonance,Univ.of Minnesota preprint 13(1983). 11.D.Ruelle and F.Takens:On the Nature of Turbulence,Comm.Math.Phys. 20( 1971)167-192. 12.R.J.Sacker and G.R.Sell:A Spectral Theory for Linear Differential Systems,JDE 27(1978)320-358. 13.G.R.Sell:Bifurcation of Higher Dimensional Tori,Archive Rat.Mech. Anal.69(1979)199-230. Dietrich Flockerzi Math.Institut der Universitat wiirzbwg Am HUbland,P8700 Wzburg,West Germany.
ON
k
T H E T 'T
k+l
35
BIFURCATION PROBLEM
Dietrich Flockerzi Universitat Wurzburg West Germany
I. INTRODUCTION
In 1979 G.R.Sell[l3] and A.Chenciner and G.Iooss[l] published results describing a Hopf-type bifurcation of a k-dimensional into a (k+l)-dimensional invariant torus within a one-parameter family of differential equations or maps,respectively.The present contribution gives an extension of these results since it will
not
beassumedthat
i) the normal spectrum of the k-dimensional torus crosses from IR+ to E7 in a transversal fashion (or that the corresponding transversality condition in Theorem 111.2.11 of [ l ] holds), ii)the quadratic and cubic terms are the only significant nonlinearities. The proof of our main result in Section IV i s rather involved
(cf.
[ 5 ] ) . So we confine ourselves to an outline of the main ideas and put more emphasis on the derivation of the set up we use for this
kind of bifurcation problem. Just to give the flavor of the result we are aiming at we recall the essential features of the corresponding bifurcation problem in IR2.
To this end we consider an analytic one-parameter family of or-
dinary differential equations
;; =
(1.I ,)
f,(X), x t IR2
,
a
E I=(-cc0,ao),
with f (o)=oand we assume that the variational equation x=f' (o)x 0
0
along the trivial solutjon x=o of the unperturbed system (1 .l)o admits a family F of closed orbits C*,i.e. we assume P
Al)
f'(o) has eigenvalues i i w 0
0
with w o > o .
Then,by the implicit function theorem,the system (l.l)cc possesses a stationary solution x ( a ) which we may take to be the trivial solution x=o. Here -as always- we suppose that a
0
> o is sufficiently
small. We further assume that the closed orbits C * of F are of mul-
.
P
tiplicity m In terms of the spectrum of fi(o) this can be formulated as an m-th order contact condition:
D. Flockerzi
36 A2)
f i ( o ) h a s e i g e n v a l u e s am h m ( a ) i i w ( a ) f o r a E I w i t h m i - IN I Am(o)=/= o and w ( o ) = w o .
The a s s u m p t i o n s A1 and A2 imply t h a t x=o i s a f o c u s f o r i = f ' (o)x f o r a = # o and a c e n t e r f o r a = o . Now,the e f f e c t of t a k i n g t h e n o n l i ( 1 . 1 I u i n t o a c c o u n t s h o u l d b e t h a t some of t h e m c o p i e s of e a c h i n v a r i a n t c u r v e C* of F a r e b e n t t o t h e r i g h t ( s u p e r c r i t i c a l
n e a r i t i e s of
P
bifurcation f o r a>o),some t o the l e f t ( s u b c r i t i c a l bifurcation f o r a < o ) and some a r e l e f t i n p l a c e ( v e r t i c a l b i f u r c a t i o n f o r a = o ) . A
vague a t t r a c t o r o r r e p e l l o r c o n d i t i o n l i k e A3)
x=o i s a f o c u s o f f i n i t e m u l t i p l i c i t y f o r ( 1 . 1 )
0
e x c l u d e s t h e v e r t i c a l b i f u r c a t i o n and i m p l i e s t h e e x i s t e n c e of l e a s t one t r a n s c r i t i c a l b r a n c h of b i f u r c a t i n g p e r i o d i c
at
solutions
p r o v i d e d m i s odd. T h i s c a n b e s e e n by p u r e l y t o p o l o g i c a l a r g u m e n t s (PoincarG-Bendixson t h e o r e m ) . The number of b i f u r c a t i n g b r a n c h e s
I
t h e i r d i r e c t i o n of b i f u r c a t i o n and t h e s t a b i l i t y p r o p e r t i e s o f t h e b i f u r c a t i n g p e r i o d i c s o l u t i o n s c a n b e d e t e r m i n e d b y a n a l y t i c a l means f o r any m ( c f . [ 3 ] ) . I n t h e p r e s e n t p a p e r w e want t o i n v e s t i g a t e what t h e c o r r e s p o n d i n g h y p o t h e s e s f o r t h e above i n d i c a t e d Tk+Tk"
bifurcation are that
a l l o w t o answer q u e s t i o n s i n v o l v i n g t h e number of b r a n c h e s o f d i m e n s i o n a l t o r i b i f u r c a t i n g from a g i v e n k - d i m e n s i o n a l
(k+l)-
torus,their
d i r e c t i o n o f b i f u r c a t i o n and t h e i r s t a b i l i t y p r o p e r t i e s .
Thus w e
w i l l c o n s i d e r a Cm-smooth o n e - p a r a m e t e r f a m i l y of d i f f e r e n t i a l equations (1-2)a
5 = Fa(5
/
5 E IRn ,
a E I=(-ao/ao)
p o s s e s s i n g a smooth f a m i l y o f k - d i m e n s i o n a l k+2
/
invariant t o r i
Ma
with
one may t h i n k o f Ma a s b e i n g g e n e r a t e d by k weakly coup-
l e d van d e r P o l o s c l l a t o r s . T h i s f a m i l y Ma t a k e s o v e r t h e r o l e of t h e f a m i l y x ( a ) of s t a t i o n a r y s o l u t i o n s o f not-too-restrictive
(1.1)
.We w i l l establish
c r i t e r i a which a l l o w t h e e x i s t e n c e of
(k+l)-di-
m e n s i o n a l i n v a r i a n t t o r i Ma t e n d i n g as a whole t o Mo f o r a - t o . I n S e c t i o n I1 w e g i v e an example where j u s t p a r t o f a 2 - t o r u s u n d e r g o e s a Hopf-type b i f u r c a t i o n t o a p i n c h e d t o r u s . By a s e c o n d example w e i l l u s t r a t e t h e p o s s i b i l i t y t h a t one p a r t o f a 2 - t o r u s c a n u n d e r g o s u c h a b i f u r c a t i o n t o a p i n c h e d t o r u s b e f o r e t h e rest of it d o e s s o t h a t i n t h e end a p r o p e r 3 - t o r u s h a s b i f u r c a t e d . The c a s e o f b i f u r c a t i o n from a p e r i o d i c s o l u t i o n ( k = 1 ) i s much e a s i e r t o d e a l w i t h t h a n t h e c a s e k 2 2 . T h e r e a r e e s s e n t i a l l y two ways
On the
7k Tk" Bifurcation Problem
37
--f
of p r o v i n g t h e T1+T2 b i f u r c a t i o n : f i r s t , b y c o n s i d e r i n g t h e f i r s t re-
t u r n map on a t r a n s v e r s a l h y p e r s u r f a c e one i s l e d t o t h e Hopf b i f u r c a t i o n problem f o r maps ( c f . [ l l ] ) , s e c o n d l y , b y u s i n g F l o q u e t t h e o r y one c a n r e d u c e t h i s b i f u r c a t i o n problem t o t h e Hopf b i f u r c a t i o n p r o b -
l e m from a s t a t i o n a r y s o l u t i o n . F o r k22 t h e r e i s n o F l o q u e t t h e o r y a v a i l a b l e , and i t i s known t h a t t h e r e d u c t i o n of a q u a s i - p e r i o d i c l i n e a r system t o a l i n e a r system w i t h c o n s t a n t c o e f f i c i e n t s i s n o t always p o s s i b l e . Thus t h e r e i s a problem i n d e t e r m i n i n g p r e c i s e l y a c r i t i c a l v a l u e of t h e p a r a m e t e r a t which t h e s t a b i l i t y o f t h e i n v a r i a n t t o r u s Ma changes i n a way which a l l o w s t h e b i f u r c a t i o n of a ( k + l ) - d i m e n s i o n a l t o r u s . I n t h e s e q u e l w e mainly have t h i s case k>2 i n mind. I n t h e n e x t s e c t i o n w e d e s c r i b e how t h e t h e o r y o f t h e s p e c trum o f an i n v a r i a n t m a n i f o l d ( c f . [ 1 2 ] ) c a n be u s e d t o d e f i n e s u c h a critical value. 11. THE
BIFURCATION SET UP
W e c o n s i d e r system ( 1 . 2 )
and assume t h a t t h e f l o w on Ma i s gene-
r a t e d by
where Tk d e n o t e s t h e s t a n d a r d t o r u s of dimension k . The components of y a r e t h u s d e f i n e d modulo 2.rr
. Here
it i s g e n e r a l l y n o t s u f f i c i e n t
t o require t h e existence of the k-torus
just for
a=o.
I f one h a s t h e
above b i f u r c a t i o n i n mind Mo c a n n o t b e n o r m a l l y h y p e r b o l i c and t h u s t h e c o n t i n u a t i o n o f Mo t o Ma f o r a E I i s n o t always p o s s i b l e (compare t h e remark f o l l o w i n g H l i i b e l o w ) . A d a p t i n g t h e h y p o t h e s e s A1 and A2 t o t h e p r e s e n t s i t u a t i o n amounts - i n somewhat l o o s e t e r m s -
t o the
f o l l o w i n g assumption: The " v a r i a t i o n a l e q u a t i o n " o f t h e u n p e r t u r b e d s y s t e m ( 1 . 2 )
0
a l o n g t h e t o r u s Mo p o s s e s s e s a f a m i l y F o f i n v a r i a n t (k+l)dimensional t o r i
M* of " m u l t i p l i c i t y m" P
.
I n what f o l l o w s w e w i l l make p r e c i s e what i s meant by " v a r i a t i o n a l e q u a t i o n " and by " m u l t i p l i c i t y m " . The f i r s t o b j e c t i v e i s t o i n t r o d u c e a l o c a l c o o r d i n a t e s y s t e m near M
v i a a d e c o m p o s i t i o n o f t h e t a n g e n t s p a c e o f lRn a t y i n t o t h e sum of t h e k-dimensional t a n g e n t s p a c e Yci(y) t o Ma a t y , t h e "non-
c r i t i c a l " p a r t Za(y) of t h e normal s p a c e t o Ma a t y and t h e " c r i t i c a l " p a r t Xci(y) o f t h e normal s p a c e which r e f l e c t s t h e f a c t of M a n o t bei n g normally h y p e r b o l i c f o r a l l
ci
E1,i.e.
v i a a decomposition
38
D. Flockerzi
(2.2)
IRn
for all ( y , a ) i ~ ~ x ~ .
+ Y,(y) + Z a ( y )
= Xa(y)
F o r t h e H o p f - t y p e b i f u r c a t i o n w e a r e a i m i n g a t X,(y) s h o u l d b e a n k 2 - d i m e n s i o n a 1 , s m o o t h l y v a r y i n g s u b s p a c e o f IR f o r a l l (y,a)CT X I . I n o r d e r t o d e r i v e such a decomposition ( 2 . 2 ) w e consider the l i n e a r skew-product
flow
11 (
to'Y0,a,t)
= (E
(Yo,U,
.
k
on IRn X T X I i n d u c e d by ( 1 . 2 ) with i n i t i a l value y
t )S 0 , Y ( Y o , a , t ), a )
Here, Y d e n o t e s t h e s o l u t i o n o f
(2.1
a t t = o and 6 d e n o t e s t h e f u n d a m e n t a l m a t r i x
0
L = F i ( Y ( y o , u , t ) ) C w i t h 6 ( y 0 , a , o ) = I . A s i n [ 1 3 , V ] w e de-
s o l u t i o n of
n o t e t h e i n d u c e d t a n g e n t i a l a n d n o r m a l l i n e a r s k e w - p r o d u c t f l o w s by iIT a n d I T N , r e s p e c t i v e l y . F o r t h e n o t i o n o f
t h e s p e c t r a of such flows
w e r e f e r t o [ 1 2 1 o r [ 1 3 ] . Then w e h a v e a d e c o m p o s i t i o n a s i n ( 2 . 2 )
i f the following s p e c t r a l conditions hold: S 1 ) TIT
and TIN h a v e d i s j o i n t s p e c t r a f o r
af
o . (Note t h a t
o
belongs
t o b o t h s p e c t r a f o r a = o i f k;;.l.) 5 2 ) ITN h a s a " c r i t i c a l " s p e c t r a l s e t C c(-6,6)f o r a E I w i t h a t w o dimensional a s s o c i a t e d s p e c t r a l subspace Xa(y) such t h a t
c0 =io},
C~;IR+
=/=
fi *
C ~ ~ I R - =
f o r a+o.
The r e m a i n i n g s p e c t r a l i n t e r v a l s a r e bounded away f r o m o by 6 5 3 ) Xli(y) v a r i e s s m o o t h l y i n k a l l ytT
.
.
a n d X o ( y ) i s d i s j o i n t f r o m Yo(y) f o r
Under t h e s e c i r c u m s t a n c e s X ( y ) and Z ( y ) c a n b e p a r a m e t r i z e d i n a 0 0 2 smooth way by xEIR a n d , r e s p e c t i v e l y . The c o r r e s p o n d i n g d i - f f e r e n t i a l equation w i l l be 2n-periodic
i n e a c h component of y and
w i l l have t h e f o l l o w i n g form:
G (2.31,
= n(y)x+asl
-
1
(y,a)x+uR
(y,a)z
2
,
$
=
5
= B(y)z+cxB ( y , a ) x + a B 2 ( y , a ) z
w
+ Y(x,y,z,a) 1
2
w h e r e X and Z a r e o f o r d e r O ( / x l + 1 z I g,(y). (2.4)
iX ( x , y , z , a ) ,
2
+
)
Moreover t h e m a t r i x B ( y ) i n ( 2 . 3 )
Z(x,y,z,a)
,
X E I R
2
,
YETk I mn-k-2
I
and w h e r e Y ( o , y , o , a ) e q u a l s s a t i e s f i e s t h e following: k
= B ( o t + y ) z a d m i t s a n e x p o n e n t i a l d i c h o t o m y f o r a l l y ET 0
0
( c f . [ 5 1 ) .The c o n s i d e r a t i o n s t o f o l l o w a l w a y s r e f e r t o s y s t e m ( 2 . 3 ) a
with the property
(2.4).
The f o l l o w i n g e x a m p l e s r e v e a l t h e i m p o r t a n t r o l e t h e n a t u r e o f the rotation vector
p l a y s and m o t i v a t e o u r h y p o t h e s i s H I b e l o w .
On the
r' -+ Tk"
39
Bifurcation Problem
ExurnpLc 1 ( ~ 6 . 1 6 1 ) W e c o n s i d e r t h e f o l l o w i n g 4-dimensional
w i t h a r a t i o n a l r o t a t i o n v e c t o r =;
s y s t e m o f t h e form ( 2 . 3 ) ,
(1/2 1)
$Q
and wo
1 w i t h ( x , y l ,y2)EIR2 x T x T 1 . With p o l a r c o o r d i n a t e s
:
( r , e ) f o r x and t h e
change of c o o r d i n a t e s (2.6)
$ =
2y 1 -y 2
I
+
= -Yl+Yl
w e w r i t e ( 2 . 5 ) a i n t h e form (?.7)a
3
r = a(l+2cos$)r-r
e =
0
,
,
&
= asin$ =
,
(1-asin$)/2.
The ( q r @ ) - s y s t e md e s c r i b e s a f l o w on a 2 - t o r u s h a v i n g two p e r i o d i c o r b i t s f o r Iaif r u s of of
0.
The s e t { r = o } c o r r e s p o n d s t o t h e i n v a r i a n t 2-to-
( 2 . 7 ) a which i s a t t r a c t i v e f o r a=o. The l i n e a r i z e d e q u a t i o n
(2.710 admits a family F of i n v a r i a n t 3 - t o r i .
i n d e p e n d e n t of
($,$)
The ( r , e ) - s y s t e m i s
and c a n b e a n a l y z e d by p h a s e p l a n e t e c h n i q u e s .
F o r t h e c h o i c e o f c o o r d i n a t e s as i n F i g u r e 1
Fig. 1 t h e two s k e t c h e s i n F i g u r e 2 show t h e f l o w o f t h e ( r , O ) - s y s t e m f o r
a < o and a > o :
Fig. 2
40
D. Flockerzi
The c o r r e s p o n d i n g i n v a r i a n t s e t s o f t h e p e r i o d map on t h e s u r f a c e { y 2 = o m o d 2 n l -one may t h i n k of y 2 p l a y i n g t h e r o l e of t h e t i m e -
are
t h u s p i n c h e d t o r i a s shown i n F i g u r e 3 .
T h u s , i n s t e a d o f a o n e - s i d e d b i f u r c a t i o n t o a 3 - t o r u s o n e h a s a twos i d e d b i f u r c a t i o n t o i n v a r i a n t s e t s t h a t can b e v i s u a l i z e d a s t h r e e d i m e n s i o n a l t o r i which a r e p i n c h e d a l o n g a c i r c l e . F o r a more e x t e n s i v e s t u d y of t h i s r e s o n a n c e phenomenon w e r e f e r t o 1 6 1 .
Exarnp-Ye 2 i e d . [ l o l l The e s s e n t i a l f e a t u r e s of t h e above example c a n b e u s e d t o show t h a t t h e T2+T3 b i f u r c a t i o n i s v e r y s e n s i t i v e t o p e r t u r b a t i o n s . C o n s i d e r (2.8)€ where
E
r3
= u r * 1 y = 1 2
+
+
E
2
rcos(2y,-y2)
1 2 s i n ( 2 y -y ) 2 1 2
--E
B
I
,
= w
0
+
2
o(E
y2= 1 ,
i s j u s t an a u x i l i a r y p a r a m e t e r m e a s u r i n g t h e s i z e of t h e p e r -
t u r b a t i o n . The u n p e r t u r b e d s y s t e m ( 2 . 8 ) 0 u n d e r g o e s a s u p e r c r i t i c a l
{ r = o }t o t h e i n v a r i a n t
Hopf b i f u r c a t i o n from t h e i n v a r i a n t 2 - t o r u s
3 - t o r u s {r=Jcil. By means of t h e p e r t u r b e d s y s t e m ( 2 . 8 ) € w e i l l u s t r a t e t h a t p a r t o f an i n v a r i a n t 2 - t o r u s c a n u n d e r g o a Hopf b i f u r c a t i o n b e f o r e a p r o p e r 3 - t o r u s a p p e a r s . By i n t r o d u c i n g t h e new b i f u r c a t i o n parameter B v i a system ( 2 . 8 )
G
=
2
B,the s c a l i n g r + E
r
and t h e t r a n s f o r m a t i o n ( 2 . 6 )
can be w r i t t e n a s E
8 = 0
Fig. 4
O=E
2
r ( ~ + c o s + -2r 2
0
+ O ( E )
,
I
+
=
E
2
sin+
I
; = l + O ( E 2) .
2
On the Tk -+
Tk" Bifurcation Problem
41
Thus,for i n c r e a s i n g @ , t h e i n v a r i a n t 2-torus undergoes a p a r t i a l H o p f b i f u r c a t i o n t o a p i n c h e d t o r u s a t B=-1 and a s e c o n d s u c h b i f u r c a t i o n a t ~ = 1 .The i n v a r i a n t s e t s o f
-now t o f u l l 3 - t o r u s -
(2.8)
for the
p e r i o d map on { y =omod2n} a r e shown i n F i g u r e 4 . 2 I n view o f t h e s e examples w e a s k f o r a n o n r e s o n a n t r o t a t i o n vec-
tor
whose components a r e i n d e p e n d e n t o v e r Q . Our g e o m e t r i c i d e a o f
the family F requires a l l solutions of
2
(2.9)
= n(wt+y)x
yETkf
(c(;y)-cy)
t o b e bounded. ( T h i s i s n o t i m p l i e d by o u r s p e c t r a l a s s u m p t i o n s f s e e [9].)
We t h e r e f o r e c a n t a k e n ( y ) t o b e o f t h e form
[a]).
There i s n o l o s s o f g e n e r a l i t y i n t a k i n g c ( y ) e q u a l t o i t s
(cf.
mean v a l u e p r o v i d e d ( 2 . 9 ) a d m i t s a n o n t r i v i a l almost p e r i o d i c s o l u t i o n f o r some y t T k f s i n c e t h e n any s o l u t i o n o f ( 2 . 9 ) w i l l b e a l m o s t p e r i o d i c ( f o r any y E T k f c f . [ 8 1 ) . W e r e s t r i c t o u r a t t e n t i o n t o t h i s c a s e and assume
Now,the e x i s t e n c e o f t h e f a m i l y F of
(k+l)-dimensional i n v a r i a n t to-
r i M Y f o r t h e l i n e a r i z e d s y s t e m o f ( 2 . 3 ) 0 i s obvious.We would l i k e P t o add a remark: R.A.Johnson h a s shown i n [ 7 , 8 ] t h a t a l r e a d y f o r
dimy=2 t h e r e need n o t e x i s t a n o n t r i v i a l a l m o s t p e r i o d i c s o l u t i o n of ( 2 . 9 ) even i f a l l s o l u t i o n s o f
( 2 . 9 ) a r e bounded.The c r u c i a l p o i n t
i n proving t h e nonexistence of such a s o l u t i o n i s t h e c o n s t r u c t i o n of an a n a l y t i c f u n c t i o n c and a s q u a r e - i n t e g r a b l e , b u t n o t c o n t i n u o u s antiderivative i)
y
on T2 w i t h t h e f o l l o w i n g p r o p e r t i e s :
t 2 c h a s mean v a l u e o a n d J c ( w s + y ) d s i s unbounded f o r a l l YET t
i i ) y (wt+y) - y ( y ) = J
where
E,, i n
- W=
(w,
c ( o s + y ) d s f o r a l l yET2 and a l l t E l R
l ) T i s a s u i t a b l e i r r a t i o n a l number.
A s it t u r n s o u t t h e above assumption o f n o n r e s o n a n c e on
0 alone
w i l l n o t s u f f i c e f o r o u r p u r p o s e s . I n 1 1 1 A.Chenciner and G.Iooss have c o n s i d e r e d some c a s e s o f s t r o n g r e s o n a n c e where w depends r a 0 t i o n a l l y on 0. H e r e w e w i l l a s k f o r t h e s t r o n g v e r s i o n o f n o n r e s o nance i n form of a d i o p h a n t i n e c o n d i t i o n f o r Hlii)
There e x i s t p o s i t i v e c o n s t a n t s (11
satisfies
[v.w(
>K~(VI
-K
K
0
-
W=(W
and
0 K
, u ) ~ . W e assume
such t h a t
1+k for a l l o = f = v t ~
42
D. Flockerzi
A c a r e f u l a n a l y s i s o f o u r a r g u m e n t a t i o n b e l o w shows t h a t H l i i i s onAs l y n e e d e d f o r a f i n i t e number of v A s ( i f u d e n o t e s ( v v ) F Z x Z k ) . 0'
( w i t h vo=o,vo=l)
A . C h e n c i n e r a n d G.Iooss h a v e n o t e d h y p o t h e s i s H l i i allows the continuation of the torus M
0
t o t h e t o r i Ma f o r a G I ( [ 2 ] ) .
W e now p r o c e e d t o make t h e s t a t e m e n t a b o u t t h e " m u l t i p l i c i t y o f
M :
"
precise. In cylindrical coordinates ( x , y , z ) = ( r , o l y I z ) = ( r , $ , z ) E IR+ ~ T ~ + ~ ~ I R ~ - ~ - ~
syste
( 2 . 3 ) n i s now o f t h e f o r m
i
6
(2.10
=
nA ( 6 , c r ) r
= w
+
aA ( @ , u ) z
2 + Q(r,$,z,a), 1
= B($)z
+
R(r,@,z,u),
+ aBl(tJfa)r + aB2(tJru)z + Q ( r , $ I z I u ) ,
w h e r e R and Q are o f s e c o n d o r d e r i n ( r , z ) . By a smooth c h a n g e of variables
r
(2.11)
+
+
r + uU($,cc)r
aV($,cc)z
one i s l e d t o s o l v e 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 o f t h e form
(2.12) where lues
u$ ( $ )
x
- O
+ x($)
=
7=
i s a known 2 n - p e r i o d i c
7.v a n i s h
f u n c t i o n o n Tk+'.
when p e r f o r m i n g ( 2 . 1 1 )
s i v e l y because of H l i i -
x($)
mean v a l u e o f -the
i . can
U n l e s s a l l mean vab e computed r e c u r -
t h e r - e q u a t i o n t a k e s t h e form
= amK m r + a m + l [ o ( r ) + O ( / z l ) + ] O(r2+1z/*),
~ ~ = + o
( c f . [ 5 ] ) . W e assume t h a t t h e f o l l o w i n g s i t u a t i o n p r e v a i l s :
H2)
T h e r e e x i s t s a n mEN t h e mean v a l u e s
s u c h t h a t K :? =k o w h e r e a s m m are a l l e q u a l t o 0 .
7l f . . . , x m - l
I n c a s e of t h e Hopf b i f u r c a t i o n c o n s i d e r e d i n S e c t i o n I t o t h e c o n d i t i o n A2.
H 2 reduces
The e x i s t e n c e o f s u c h a n i n t e g e r m c a n be t a k e n
f o r t h e d e f i n i t i o n o f t h e m u l t i p l i c i t y o f M*. P
H 2 i s t h e e x a c t formu-
l a t i o n o f t h e p r e v i o u s g e o m e t r i c i d e a and s t a t e s i n a n a l y t i c a l t e r m s t h a t t h e n o r m a l s p e c t r u m o f Ma , i . e .
t h e s p e c t r a l s u b s e t C,,has
m-th o r d e r c o n t a c t w i t h t h e t a n g e n t i a l s p e c t r u m a t
U=O.
an
Finally we
s t a t e t h e a s s u m p t i o n c o r r e s p o n d i n g t o A3 : H3)
Mo i s v a g u e l y h y p e r b o l i c f o r ( 2 . 3 ) 0 ( c f . ( 3 . 3 ) b e l o w ) .
Having e s t a b l i s h e d t h e s e t up f o r o u r b i f u r c a t i o n p r o b l e m w e formul a t e o u r goad i n p r e c i s e t e r m s : Show t h e e x i s t e n c e o f i n v a r i a n t ( k + l ) - t o r i Ma o f t h e f o r m (2.13)
M,
= { z = z ( r , $ , a ) , r=;($'cc), $ t T k + ' l
f o r IC. > o
On the
Tk Tkt +
’
43
Bifurcation Problem
f
w i t h a p p r o p r i a t e f u n c t i o n s 2 and
such t h a t M a + M
(cf .Section 1V.B).
0
as a - t o
111. NORMAL FORMS FOR ( 2 . 3 )
W e c o n f i n e o u r s e l v e s t o o u t l i n e t h e d e r i v a t i o n o f t h e normal
forms. For t h e d e t a i l s w e r e f e r t o [ 5 ] . A f t e r t h e t r a n s f o r m a t i o n ( 2 . 1 1 ) w e make t h e s u b s t i t u t i o n s
(3.1)
r
+ E
introducing
m
r
,
z
E
-t
m
z
,
= EB
01
a s t h e new b i f u r c a t i o n p a r a m e t e r . A s w e g o a l o n g w e
E>O
w i l l be l e d t o v a r i o u s choices of
B ( i n terms o f
E)
and t h u s t o t h e
e x a c t s c a l i n g s w i t h € = € ( a ) . I n a f i r s t s t e p w e compute a n approximat i o n t o t h e c e n t e r manifold of
( 2 . 3 ) a i n t h e new v a r i a b l e s
(3.1) v i a
a transformation (3.2)
z
-t
z +
E W ( r , @ , ~ , f i ) .
Because o f t h e e x p o n e n t i a l dichotomy i n ( 2 . 4 ) t h e a r i s i n g 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 which a r e o f t h e form - B ( @ ) w ( $ ) + w ( @ ) * w + Q($) = $
h a v e u n i q u e smooth 2 n - p e r i o d i c
0
s o l u t i o n s . By means o f
(3.2) w e a r r i v e
a t a s y s t e m o f t h e form
2
=
5:
+
ER*(r,@,E,B)
= w
+
EO(IZ\),
O(E),
= B($)z
+
N
E O ( ~ Z I )+ O ( E ) ,
where N i s a n a r b i r a r i l y l a r g e f i n i t e p o s i t i v e i n t e g e r . I n v a r i a n t manifold theory t e l l s us t h a t t h e z - t e r m
i n t h e r a d i a l e q u a t i o n can
a h i g h e r o r d e r term (see S e c t i o n V). I n t h e
be c o n s i d e r e d a s b e i n g
s e c o n d s t e p w e r e p l a c e t h e $ - d e p e n d e n t c o e f f i c i e n t s of t h e l e a d i n g powers o f r i n R* by t h e i r c o n s t a n t mean v a l u e s . by a n a v e r a g i n g t r a n s f o r m a t i o n
This canbeachieved
r + r + c U ( r , @ , ~ , B )w h e r e t h e c o r r e -
s p o n d i n g 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 a r e o f t h e form ( 2 . 1 2 ) . B e c a u s e o f t h e d i o p h a n t i n e c o n d i t i o n H l i i t h e y h a v e u n i q u e smooth 21rp e r i o d i c s o l u t i o n s w i t h mean v a l u e
0.
The r e s u l t i n g r a d i a l e q u a t i o n
w i l l t h e n be o f t h e form = P(r,E,B)
+
P ~ ( ~ , $ , E , o +) E O C ~ Z = ~ ) O(E)
where - b e c a u s e o f H3- t h e r e e x i s t s a n il tN s u c h t h a t o n e h a s 0
(3.3)
P ( r , c , o ) = E2L,mKor2~om+l, KO+
0.
D. Flockerzi
44
p R c a n t h u s be assumed t o be o f h i g h e r order w i t h respect
to
By
E.
employing t h e Newton d i a g r a m f o r P ( r , E , @ ) = o ( w h e r e o n e marks t h e powers o f
E
on t h e a b s c i s s a and t h o s e o f 6 on t h e o r d i n a t e ) o n e no-
t i c e s t h a t t h i s a l g e b r a i c b i f u r c a t i o n e q u a t i o n c a n b e w r i t t e n as P ( r , E , @ )=
(3.4)
+
[email protected]+1
jcJ
1
h.0.t
The s e t J 1’ and f3-powers on t h e convex p o l y g o n a l
w i t h n o n z e r o c o n s t a n t s K . and n o n n e g a t i v e i n t e g e r s L 7
corresponds t o t h e p a i r s of
E-
l i n e i n t h e Newton d i a g r a m . The v a r i o u s s l o p e s -l/A ( h ) E Q - , h = l , .
. . ,H,
o f Newton’s p o l y g o n a l l i n e t h e n y i e l d t h e c o r r e c t r e l a t i o n b e t w e e n a and E by
(3.5)
a = €6,
6
=
+E
A ( h ) , h=l
,.. . , H ,
and l e a d t o H normal r e p r e s e n t a t i o n s o f
( 2 . 3 ) a i n t h e form
with appropriate positive rationals u (h)
.
The “Newton p o l y n o m i a l “
P t ( r ) i s t h e c o e f f i c i e n t o f t h e lowest E-power h t h u s n o monomial. IV.
i n P ( r , E , + E A ( h ) ) and
M A I N RESULT
A ) Under t h e h y p o t h e s e s Hl,H2 and H3 t h e above
H normal forms ( 3 . 6 I E
c a n b e d e r i v e d from ( 2 . 3 ) a . A n y b i f u r c a t i n g i n v a r i a n t ( k + l ) - t o r u s o f t h e form ( 2 . 1 3 ) w i l l b e o n e f o r o n e o f t h e s e B) I f
+ Ph
representations
(3.6)
.
o r P h h a s a s i m p l e p o s i t i v e zero p t h e n t h e r e e x i s t s a n i n -
v a r i a n t ( k + l ) - t o r u s Ma f o r ( 2 . 3 ) , o f t h e form
I (r,$,z,a): r=$(+,a),
z = & ( e ( $ , a ) ,$,a)
,
$ET~+’}
and 5 are g i v e n by m(h)) F ( $ , a ) = I a l m ( h )( P + O ( I ) ) , 2(P(+,a)t$,a) = o(lal w i t h m ( h ) = m / ( l + A ( h ) ) . The s t a b i l i t y p r o p e r t y o f Ma w i t h i n t h e c e n t e r f o r a > o o r a < o , r e s p e c t i v e l y . The f u n c t i o n s
s t a b l e m a n i f o l d f o l l o w s from t h e s i g n o f aP’ C)
(p)
.
ar h I f m i n H2 i s odd t h e n t h e r e e x i s t s a t l e a s t o n e h~~1,...,H]so
t h a t t h e “Newton s l o p e ” - l / A ( h )
i n ( 3 . 5 ) c o r r e s p o n d s t o an odd o r d i -
n a t e segment i n t h e Newton d i a g r a m of
(3.4). I f -l/A(h)
+
i s such a
s l o p e t h e n a t l e a s t o n e o f t h e p o l y n o m i a l s Ph o r Pi p o s s e s s e s an odd o r d e r p o s i t i v e z e r o . Such z e r o s c a n “ i n g e n e r a l ” b e r e d u c e d t o s i m p l e z e r o s ( c f . [ 5 , S e c t i o n V]:The o n l y e x c e p t i o n c a n be c a u s e d b y t h e n e e d o f i n f i n i t e l y many r e d u c t i o n s t e p s ) .
On the D)
Tk + 7k"Bifurcation Problem
45
I n t h e p a r t i c u l a r c a s e m=l t h e r e e x i s t s j u s t t h e Newton s l o p e
+
- l / A ( 1 ) = - 1 / ( 2 ~ ~ - 1 )I.n t h i s c a s e e i t h e r PI or P; h a s t h e s i m p l e p o s i t i v e z e r o P = / K ~ / K 1~ /\2 Q 0 , and one h a s a u n i q u e b i f u r c a t i o n o f an i n v a r i a n t (k+l)-torus e i t h e r f o r a>o o r f o r a
E) The number of p o s i t i v e z e r o s o f a l l t h e Newton p o l y n o m i a l s toget h e r d o e s n o t e x c e e d min(m,Eo). T h u s , " i n g e n e r a l " t h e r e are a t most m i n ( m , i ) b r a n c h e s of b i f u r c a t i n g ( k + l ) - t o r i . 0
V.
INVARIANT MANIFOLDS AND T
k+Tk+1
BIFURCATION
W e have a l r e a d y s k e t c h e d t h e p r o o f of p a r t A i n S e c t i o n 1 I I . P a r t s C , D and E f o l l o w from a t h o r o u g h d i s c u s s i o n of t h e Newton d i a g r a m
f o r ( 3 . 4 ) . H e r e , w e w i l l j u s t g i v e an o u t l i n e how t h e t h e o r y of i n -
+
v a r i a n t m a n i f o l d s c a n be u s e d t o p r o v e p a r t B. W e s u p p o s e t h a t Ph possesses a simple p o s i t i v e zero p,denote
+ Ph
s i m p l y by P and d r o p
t h e dependence on h i n t h e s e q u e l . I f A d e n o t e s
y/6
i n lowset terms
w e i n t r o d u c e t h e new p a r a m e t e r v by c = v 6 t o a v o i d f r a c t i o -
(y>0,6>0)
n a l powers of t h e p a r a m e t e r . With t h e t r a n s l a t i o n r+r+p s y s t e m (3.6)&
i s o f t h e form
c (5.1),,
6
v
60
( P ' ( p ) r + o ( l ) )+ v6~(/zj), 6 = w + O(v ), 6 = B ( $ ) z + li O ( \ Z l ) + O ( v N 6 ) .
=
I n o r d e r t o r e d u c e t h i s n-dimensional
system (5.1)
!J
t o a (k+2)-di-
m e n s i o n a l s y s t e m w e modify t h e r i g h t - h a n d s i d e s o it h a s t h e n e c e s s a r y g l o b a l p r o p e r t i e s . Because o f
4
= o
+
6
O(p ),
(2.4) t h e system
= B($)z
a l s o a d m i t s a n e x p o n e n t i a l dichotomy ( c f . [ 1 2 ] o r [ 4 1 ) . T h u s w e c a n d e f i n e t h e G r e e n ' s f u n c t i o n f o r t h e m o d i f i e d s y s t e m which e n a b l e s u s t o r e d u c e t h e n - d i m e n s i o n a l s y s t e m t o a ( k + 2 ) - d i m e n s i o n a l one on a c e n t e r manifold
A f u r t h e r averaging transformation
$+$+v
6
V(r,$,u)
-where f o r t h e
r e c u r s i v e computation of V t h e d i o p h a n t i n e c o n d i t i o n H l i i is used once more- g e n e r a t e s a weakly c o u p l e d s y s t e m (5.3)
r = v
6U
$ = w
+
-
(P'(p)r+G(l)), Oo(rlv)
+ O(v
6U+1
1
on t h e c e n t e r m a n i f o l d ( 5 . 2 ) . S i n c e
Q0
d o e s n o t depend on $ t h e n o r -
mal f l o w d o m i n a t e s t h e t a n g e n t i a l f l o w f o r ( 5 . 3 ) . Thus t h e r e e x i s t s
46
D. Flockerzi
an invariant manifold (5.4)
{r=:($,u)=o(p)
:
k+ 1 GET 1
on the manifold (5.2). All in al1,we thus have arrived at the desired (k+l)-torus Ma with the properties stated in Section 1V.B. We note that the various scalings (3.1) and (3.5) amount to
so that the functions
f and 5 of (2.13) and Section 1 V . B aregivenby
$ ( $ , a ) = am(h) ( p + > ( $ , p ) )
,
z(r,$,a) = am(h)2(rfG,u).
REFERENCES 1. A.Chenciner and G.1ooss:Bifurcations de tores invariants,Archive Rat .Mech .Anal.69 ( 1979)109-1 98. 2. A.Chenciner and G.Iooss:Persistance et bifurcations de tores invariants,Archive Rat.Mech,Anal.71 (1979)301-307. 3. D.Flockerzi:Existence of Small Periodic Solutions of ODE'S in IR2 , Archiv der Mathematik 33(1979)263-278. 4. D.Flockerzi:Weakly Nonlinear Systems and Bifurcation of Higher Dimensional Tori,in:H.W.Knobloch and K.Schmitt,Equadiff 82,Springer Lecture Note in Math. 101 7 ( 1983) 185-1 93. D.Flockerzi:Generalized Bifurcation of Higher Dimensional Tori, 5. University of Wurzburg preprint 97(1983),(to appear in the JDE). 6 . D.Flockerzi:Resonance and Bifurcation of Higher Dimensional Tori, University of Wiirzburg preprint 102 (1983). 7. R.A.Johnson:Measurable Subbundles in Linear Skew-Product Flows, Illinois J .Math. 23 ( 1 979 ) 183-1 98. 8. R.A. Johns0n:Analyticity of Spectral Subbundles ,JDE 35 (1980) 366-387. 9. R.A.Johnson:On a Floquet Theory for Almost-Periodic Two-Dimensional Linear Systems,JDE 37(1980)184-205. 1o.K.R.Meyer:Tori in Resonance,Univ.of Minnesota preprint 13(1983). 11.D.Ruelle and F.Takens:On the Nature of Turbulence,Comm.Math.Phys. 20( 1971)167-192. 12.R.J.Sacker and G.R.Sell:A Spectral Theory for Linear Differential Systems,JDE 27(1978)320-358. 13.G.R.Sell:Bifurcation of Higher Dimensional Tori,Archive Rat.Mech. Anal.69(1979)199-230. Dietrich Flockerzi Math.Institut der Universitat wiirzbwg Am HUbland,P8700 Wzburg,West Germany.