Detecting strange attractors in turbulence - CRCV

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symmetry, like the case of the Couette flow, then a same type of symmetry must hold. ['l and hence for Zf~. In this case
Detecting s t r a n g e a t t r a c t o r s in t u r b u l e n c e . Floris Takens.

1.

Introduction. Since [19] was w r i t t e n , much m o r e a c c u r a t e e x p e r i m e n t s on the o n s e t of

turbulence have been made, e s p e c i a l l y by F e n s t e r m a c h e r , Swinney, Gollub and Benson [6,8,9,10]. T h e s e new e x p e r i m e n t a l data should be i n t e r p r e t e d a c c o r d i n g to [19] in t e r m s of s t r a n g e a t t r a c t o r s ,

o r they should falsify the whole p i c t u r e given in that p a p e r .

such i n t e r p r e t a t i o n s one u s e s in g e n e r a l the s o - c a l l e d p o w e r s p e c t r u m .

For

It is h o w e v e r not

at all c l e a r how to r e c o n s t r u c t tt~e " s t r a n g e a t t r a c t o r s " f r o m a p o w e r s p e c t r u m (with continuous p a r t s ) ; even w o r s e : how can one s e e w h e t h e r a given p o w e r s p e c t r u m (with continuous p a r t s ) might have been " g e n e r a t e d " by a s t r a n g e a t t r a c t o r ?

In this p a p e r I p r e s e n t

_proc_e_dure___s _t_o_deci_de_ w h e t h e r one may a t t r i b u t e certain} ..experimen_t.aj datta___as_ in t h e o n s e t" o f turbulences_ to the p r e s e n c e of s t r a n g e a t t r a c t o r s .

T h e s e p r o c e d u r e s c o n s i s t of

a l g o r i t h m s , to be applied to the e x p e r i m e n t a l data i t s e l f and not to the p o w e r s p e c t r u m ; in fact, I doubt w h e t h e r the p o w e r s p e c t r u m contains the r e l e v a n t i n f o r m a t i o n .

In o r d e r to d e s c r i b e the p r o b l e m s and r e s u l t s , m o r e detail, I shall f i r s t review the ideas of [19], e~posed by Landau and L i f s c h i t z [13], cylinders.

t r e a t e d in this p a p e r ,

in

a l s o c o m p a r i n g t h e m witl~ those

in r e l a t i o n with the flow between two r o t a t i n g

It was this s a m e e x p e r i m e n t which was c a r r i e d out to g r e a t p r e c i s i o n by

Swinney e t . a l .

[6, 8, 10].

It should be noted that the d i s c u s s i o n in [19] is not r e s t r i c t e d to this situation but should a l s o be applicable to o t h e r situations w h e r e an o r d e r l y dynamic changes to a chaotic one; s e e [8] for a d i s c u s s i o n of s o m e e x a m p l e s .

Also, our

p r e s e n t d i s c u s s i o n should be applicable to t h e s e c a s e s .

The T a y l o r - C o u e t t e E x p e r i m e n t . We c o n s i d e r the region D between two c y l i n d e r s as indicated in figure 1. In this region we have a fluid.

We study

top ~

C in

its motion when the o u t e r c y l i n d e r , the top and bottom a r e at r e s t , while the i n n e r c y l i n d e r has an a n g u l a r velocity ~ . s o m e fixed point in the i n t e r i o r of D.

p is

- C out

For

a n u m b e r of values of ~, one component

bottom

367

of the velocity of the fluid at p is m e a s u r e d as a function of t i m e .

In [19] the idea was

the following : f o r each value of ~ the s e t of ai1 " p o s s i b l e s t a t e s " is a H i l b e r t s p a c e H ~ c o n s i s t i n g of ( d i v e r g e n c e f r e e ) v e c t o r fields on D s a t i s f y i n g the a p p r o p r i a t e b o u n d a r y conditions (these v e c t o r fields r e p r e s e n t velocity d i s t r i b u t i o n s of the fluid).

F o r each [~

t h e r e is an evolution s e m i - f l o w

{r

H a - , Hi.~]tEIR,N+= {t E N i t > 0] , +

such that if X ( Hf~ r e p r e s e n t s the s t a t e at t i m e t = 0 tlaen ~?0(X) r e p r e s e n t s the s t a t e at t i m e t 0.

We a s s u m e that for all values of f~ u n d e r c o n s i d e r a t i o n ,

7 ~ c H ~ to which ( a l m o s t ) all evolution c u r v e s cptO(X) tend as t --* =. point we d o n ' t want to specify the t e r m " a t t r a c t o r " . ) a s y m p t o t i c b e h a v i o u r of all evolution c u r v e s r

t h e r e is a n " a t t r a c t o r " (At this

AQ and ~ t [ A a then d e s c r i b e the Roughly the m a i n a s s u m p t i o n s in

[19] could be r e p h r a s e d as : ~p?lAff b e h a v e s just as an a t t r a c t o r in a finite d i m e n s i o n a l differentiable dynamical system.

In m o r e detail,

the a s s u m p t i o n was that for all values

of [~ u n d e r c o n s i d e r a t i o n t h e r e is a s m o o t h finite d i m e n s i o n a l manifold M ~ a Hf~, s m o o t h l y depending on ~,

such that :

(i)

MQ is i n v a r i a n t in the s e n s e that f o r X E Mf~, {p?(X) E M ~ ;

(ii)

MQ is a t t r a c t i v e in the s e n s e that evolution c u r v e s q0?(X), s t a r t i n g outside MQ tend to Mr; f o r t - ~

;

(tit) the flow, induced in Mf~ by (p?, is smooth, depends s m o o t h l y on f~ and has an a t t r a c t o r Aa .

Some j u s t i f i c a t i o n for this a s s u m p t i o n was given by M a r s d e n [15, 16].

Apart

f r o m this we used g e n e r i c i t y a s s u m p t i o n s : if Zf~ d e n o t e s the v e c t o r field on M a which is the i n f i n i t e s s i m a l g e n e r a t o r of f a m i l y of v e c t o r fields. symmetry,

~?lMa,

we a s s u m e (Mfl, Z ~) to be a g e n e r i c o n e - p a r a m e t e r

(If h o w e v e r the p h y s i c a l s y s t e m u n d e r c o n s i d e r a t i o n has

like the c a s e of the Couette flow, then a s a m e type of s y m m e t r y m u s t hold

for Mf~, r ['l ' and h e n c e for Zf~.

In this c a s e g e n e r i c i t y should be understood within the

c l a s s of v e c t o r fields h a v i n g this s y m m e t r y ; s e e [ 1 8 ] . )

In the L a n d a u - L i f s c h i t z p i c t u r e , attractor) is quasi-periodic,

i.e. of the form

one a s s u m e s that the l i m i t i n g motion (or

368

~t(X) = fi'~(X' a l e

w h e r e ~0i

)

and a. depends on ~ and w h e r e for each ~ only a finite n u m b e r of a. is 9

non-zero.

2rrico t 27ria~2t 1 , a2e ....

1

1

One can imagine that, a s m o r e and m o r e a. b e c o m e n o n - z e r o , the motion 1

g e t s m o r e and m o r e turbulent.

Also in this last description we have a smooth finite dimensional manifold as attractor, namely an n-torus, but such attractors do not occur for generic parameter values of generic one-parameter families of vector fields.

It should be noted however

that for generic one-parameter families of vector fields there may be a set of parameter values with positive measure for which quasi periodic motion occurs; see LII].

This n - t o r u s a t t r a c t o r has topological entropy z e r o and its d i m e n s i o n is an integer.

On the o t h e r hand " s t r a n g e a t t r a c t o r s " have in g e n e r a l p o s i t i v e entropy and

often n o n - i n t e g r a l d i m e n s i o n .

Hence it would be i m p o r t a n t to d e t e r m i n e entropy and

d i m e n s i o n of a t t r a c t o r s f r o m " e x p e r i m e n t a l data".

In view of the e x p e r i m e n t just d e s c r i b e d , we have to add one m o r e point to out f o r m a l d e s c r i p t i o n , n a m e l y we have to add the function ( o b s e r v a b l e ) f r o m the s t a t e s p a c e to the r e a l s giving the e x p e r i m e n t a l output (when c o m p o s e d with ~ t ( X ) ).

In the

p r e s e n t e x a m p l e of the % a y l o r - C o u e t t e e x p e r i m e n t , this function y~:Hfi -' IR a s s i g n s to each X ~ H a the m e a s u r e d component of X(p). c o n c e r n e d , we only nave to deal with

As f a r as the a s y m p t o t i c benaviour is

y~lM~ (or with

yf~[A~).

Since M ~ depends

smoothly on ~ all M~I a r e d i f f e o m o r p h i c and s o we may drop the ~ .

S u m m a r i s i n g , we have a manifold M with a s m o o t h o n e - p a r a m e t e r family of v e c t o r fields Z ~ and a s m o o t h o n e - p a r a m e t e r family of functions y ~ .

F o r a n u m b e r of

values of fl the function yl~(g~;~(x)) is known by m e a s u r e m e n t (for s o m e x in o r n e a r M which may depend on ~'t; cpi'~denotes l~ere the flow on M g e n e r a t e d by Zfl. ffhe point is t to obtain i n f o r m a t i o n about the a t t r a c t o r ( s ) of Z ~ f r o m t h e s e m e a s u r e m e n t s , i . e . f r o m r

the

functions t ~ yf~((p'[(x)).

F o r this we shall allow o u r s e l v e s to make g e n e r i c i t y

a s s u m p t i o n s on (M, Z~, y~, x).

We shall p r o v e that under suitable g e n e r i c i t y a s s u m p t i o n s on (M, ZO, yfl, x) f~

the p o s i t i v e l i m i t s e t L+(x) of x is d e t e r m i n e d by the function y~(cP~(x)).

In our " m a i n

t h e o r e m " in s e c t i o n 4 we d e s c r i b e a l g o r i t h m s which, when applied to a s e q u e n c e

369 {ai=

Yr'(~P~Y,~, u i(x))}i r~''=l Iq s u f f i c i e n t l y big, will g i v e an a p p r o x i m a t i o n f o r the d i m e n s i o n = ~ + of L (x), r e s p e c t i v e l y f o r t h e t o p o l o g i c a l e n t r o p y of (PCr [L (x). ~fhis l e a d s in p r i n c i p l e +

to a p o s s i b i l i t y of t e s t i n g and c o m p a r i n g the h y p o t h e s i s m a d e by L a n d a u - L i f s c h i t z and R u e l t e - T a k e n s

[ i 9 ~ ; s e e the o b s e r v a t i o n at the end of s e c t i o n 4.

E13~

The author

w i s h e s to a c k n o w l e d g e the h o s p i t a l i t y of the d e p a r t m e n t of m a t h e m a t i c s

of W a r w i c k

U n i v e r s i t y and the m a n y d i s c u s s i o n s with p a r t i c i p a n t s of the t u r b u l e n c e and d y n a m i c a l systems

s y m p o s i u m t h e r e d u r i n g the p r e p a r a t i o n

2.

Dynamical systems

of this p a p e r .

with one o b s e r v a b l e .

L e t M be a c o m p a c t m a n i f o l d .

A d y n a m i c a l s y s t e m on M is a d i f f e o m o r p h i s m

~ . M -~ M ( d i s c r e t e t i m e ) o r a v e c t o r field X on M ( c o n t i n u o u s t i m e ) .

In both c a s e s the

t i m e e v o l u t i o n c o r r e s p o n d i n g w i t h an i n i t i a l p o s i t i o n x 0 E M is d e n o t e d by q~t(x0) : in t h e c a s e of d i s c r e t e t i m e t

1N and q0i = (cp)t; in the c a s e of c o n t i n u o u s t i m e t E IR and

t ~* ~ot(x 0) is the X i n t e g r a l c u r v e t h r o u g h x 0.

An o b s e r v a b l e is a s m o o t h f u n c t i o n y:M ~ IR. if, f o r s o m e d y n a m i c a l s y s t e m with t i m e e v o l u t i o n r

T h e f i r s t p r o b l e m is t h i s :

we know the f u n c t i o n s t ~ y(CPt(x)),

x E M, t h e n how c a n w e obtain i n f o r m a t i o n about the o r i g i n a l d y n a m i c a l s y s t e m (and manifold) from this. research

T h e n e x t t h r e e t h e o r e m s d e a l with this p r o b l e m .

for this paper was completed,

( A f t e r the

the a u t h o r w a s i n f o r m e d that this p r o b l e m ,

l e a s t p a r t s o f it, w a s a l s o t r e a t e d by o t h e r a u t h o r s ,

s e e i t , 171.

or at

S i n c e out r e s u l t s a r e

in s o m e s e n s e s o m e w h a t m o r e g e n e r a l w e s t i l l g i v e h e r e a t r e a t m e n t of t h e p r o b l e m i n d e p e n d e n t of t h e r e s u l t s in t h e a b o v e p a p e r s . )

Theorem

1.

L e t M b e a c o m p a c t m a n i f o l d of d i m e n s i o n m .

a s m o o t h d i f f e o m o r p h i s m and y:M -* I R a the m a p ~(@, y):M -* IR2m+1,

s m o o t h function,

F o r p a i r s (r

~0:M -* M

it is a g e n e r i c p r o p e r t y that

d e f i n e d by

~5(cp,y)(X) =- (y(x), y(~p(x)) . . . . .

y(cp2m(x))

is an e m b e d d i n g ; by " s m o o t h " we m e a n a t l e a s t C 2.

Proof.

We m a y ,

and do, a s s u m e

all e i g e n v a l u e s of (dr

that if x is a p o i n t with p e r i o d k of ~ ,

a r e d i f f e r e n t and d i f f e r e n t f r o m 1.

d i f f e r e n t fixed p o i n t s of ~ a r e in the s a m e l e v e l of y. fixed p o i n t x, the c o - v e c t o r s (dY)x, d ( y r

F o r O(r

k < 2 m + 1,

Also we assume

that no two

y) to be an i m m e r s i o n n e a r a

d(y~ 2 m )x m u s t s p a n ~fx(M).

~f~is is the c a s e

370

f o r g e n e r i c y if d@ s a t i s f i e s

the a b o v e c o n d i t i o n a t e a c h fixed point.

In the s a m e w a y one p r o v e s t h a t @(g~, y) is g e n e r i c a l l y an embedding when restricted assume

an immersion

to the p e r i o d i c p o i n t s w i t h p e r i o d g 2 m + 1.

t h a t f o r g e n e r i c ( ~ , y ) we h a v e : ~ ( ~ , y ) ,

restricted

and e v e n

So we m a y

to a c o m p a c t n e i g h b o u r h o o d V

of the s e t of p o i n t s with p e r i o d g 2 m + I is a n e m b e d d i n g ; f o r s o m e n e i g h b o u r h o o d tI of (d,y),t~(q~,y)lV

is a n e m b e d d i n g w h e n e v e r

((~,y) E tl, a r b i t r a r i l y

n e a r (~, y ) ,

(~0,y) E l/.

We w a n t to show t h a t f o r s o m e

~5((p, Y) is a n e m b e d d i n g .

co-vectors

F o r a n y p o i n t x q M, w h i c h is not a p o i n t of p e r i o d < 2 m + 1 f o r ~ , the 2m (dY)x,d(~0)x,d(~4~2) x . . . . . d(yO )x E T*(M) c a n be p e r t u r b e d i n d e p e n d e n t l y by

perturbing

y.

Hence arbitrarily

n e a r y t h e r e is y s u c h t h a t ( ~ , y ) E tl and s u c h t h a t

~5- = i s a n i m m e r s i o n . T h e n t h e r e is a p o s i t i v e r s u c h t h a t w h e n e v e r 0 < ~Xx, x ' ) ~ ~, (~9, y) J~5(~,v)(X) #_~(~,v)(X');~_ --~ J p is s o m e fixed metric on M. There is even a neighbourhood tl'c tl of (~o,y) such that for any (q~,y) ~ tl', q'(~9,y) is an immersion 9 (@,y)(X') whenever x ~ x' and p(x,x') ~ ~.

From

and _~(q~,Y)(X)

now on we also assume that each

component of V has diameter smaller than s.

F i n a l l y w e trove to show t h a t in tl' we h a v e a p a i r (~9, y) w i t h ~5(~),y) i n j e c t i v e . F o r t h i s we need a f i n i t e c o l l e c t i o n 2m M \ {iN=0 g~J(v)}, and s u c h t h a t :

(i)

for each i = I .....

(it)

for each i,j = I .....

~Ui}i_N1 of open s u b s e t s

N and k = 0 , 1 . . . . 2 m ,

N and k , i

diameter

= 0,1 .....

2m,

of M, c o v e r i n g the c l o s u r e

( ~ - k ( u i ) ) < ~;

-q~-k(u i) f~ Uj / ~ and -~s

Uj #

impIy that k = l;

(iii)

f o r ~J(x) E M \ (~J U.),

j = 0 .....

2 m , x ' ~ V and O ( x , x ' ) > s,

s e q u e n c e x , ~ ( x ) , . l . . , ~ 2 m ( x ) , x , ,~0(x - ,). . . . .

no two p o i n t s of t h e

(-p 2 m ( x ,) b e l o n g to t h e s a m e U.. 1

Note t h a t (ii) i m p l i e s , (ii)'

b u t is not i m p l i e d by

no two p o i n t s of t h e s e q u e n c e x,q~(x) . . . . .

~ 2 m ( x ) b e l o n g to the s a m e U . . i

We take a corresponding partition {k.] of unity, i . e . , l

s u p p o r t ld. and 1

i

~ l X i ( x ) = 1 f o r a l l x E M \ V.

k. is a non-negative function with 1

Consider

the m a p

of

371

~:M x M x IRN-* 1R2 m + l x IR2 m + l w h i c h is d e f i n e d in the f o l l o w i n g w a y

~'(x'x"sl ..... N aN) =(~($,~s)(x),~(~, ys)(x')), where g stands for (el ..... SN) and =

yg = y + D c.X.tt " i=l

We d e f i n e W c M x M a s W = { ( x , x ' ) E M

both x and x' a r e in i n t ( V ) t . ~, r e s t r i c t e d is t r a n s v e r s e

xM[D(x,x')

> g and not

to a s m a l l n e i g h b o u r h o o d of WX{0} in (MxM) xlR N,

with r e s p e c t to the d i a g o n a l of 1R2 m + l x IR 2 m + l

"[his t r a n s v e r s a l i t y

f o l l o w s i m m e d i a t e l y f r o m all the c o n d i t i o n s i m p o s e d on the c o v e r i n g {Ui}iN1 . transversality

w e c o n c l u d e that t h e r e a r e a r b i t r a r i l y

Y(Wx[e}) ~ A = 0-

From

this

s m a l l c E 1RN s u c h that

If a l s o f o r s u c h an ~ , ( ~ , y ~ ) ( tI' then ~ ( ~ , y e )

is i n j e c t i v e and h e n c e

an e m b e d d i n g .

T h i s p r o v e s that f o r a d e n s e s e t of p a i r s ( o , y ) ,

q~(tp,y) is an e m b e d d i n g .

Since the s e t of all e m b e d d i n g s is open in the s e t of all m a p p i n g s , d e n s e s e t of p a i r s (O,Y),

Remark.

f o r w h i c h ~((p,y) is an e m b e d d i n g .

t h e r e is a n open and

7his proves the theorem.

~Ihis t h e o r e m a l s o w o r k s f o r M n o n - c o m p a c t if w e r e s t r i c t

o u r o b s e r v a b l e s to

be p r o p e r f u n c t i o n s .

~ r h e o r e m 2.

L e t M b e a c o m p a c t m a n i f o l d of d i m e n s i o n m .

smooth (i.e.,

For pairs (X,y), X a

C 2) v e c t o r field and y a s m o o t h f u n c t i o n on M, it is a g e n e r i c p r o p e r t y that

~X, y :M ~ IR2 m + l ,

d e f i n e d by & , y(X) = (y(x), y(~01(x)) . . . . .

y(O2m(X)) is an e m b e d d i n g ,

w h e r e ~0t is the flow of X.

Proof.

The proof of this theorem is almost the s a m e as the proof of theorem i.

In

this case w e impose the following generic properties on X :

(i)

if X(x) = 0 then all e i g e n v a l u e s o f (dO1) x : Tx(M) -~ fix(M) a r e d i f f e r e n t and d i f f e r e n t

from I ;

(ii)

no p e r i o d i c i n t e g r a l c u r v e of X h a s i n t e g e r p e r i o d ~ 2 m + 1.

In t h i s c a s e q01 s a t i s f i e s the s a m e c o n d i t i o n s a s $ in the p r e v i o u s p r o o f . the proof carries

T h e r e s t of

over immediately.

1[he n e x t t h e o r e m is only included f o r the s a k e of c o m p l e t e n e s s ; b e used in the s e q u e l of t h i s p a p e r ,

it will not

372

Theorem

3.

L e t M b e a c o m p a c t m a n i f o l d of d i m e n s i o n m .

For pairs (X,y),X a smooth

v e c t o r field and y a s m o o t h f u n c t i o n on M, it is a g e n e r i c p r o p e r t y that t h e m a p O X , y : M -* IR2 m + l ,

d e f i n e d by d 2m

.

~X, y(X)

is an e m b e d d i n g . C2m+l .

. d. (y(~p . t(x)))It=0,, (y(x),

(y(~t(x))) [t=0)

dt2m

H e r e ~t a g a i n d e n o t e s t h e flow of X; t h i s t i m e ,

s m o o t h m e a n s at l e a s t

Proof.

A l s o this p r o o f is q u i t e a n a l o g o u s to that of t h e o r e m 1.

assume

that a g e n e r i c v e c t o r field X h a s the p r o p e r t y that w h e n e v e r X(x) = 0, a l l

e i g e n v a l u e s of (dX) x a r e d i f f e r e n t and d i f f e r e n t f r o m z e r o .

First we may,

and do,

Sing(X) d e n o t e s the s e t of

p o i n t s w h e r e X is z e r o ; this s e t is f i n i t e .

A s in the p r o o f of t h e o r e m 1, f o r s u c h a v e c t o r field X the s e t of f u n c t i o n s y:M -* IR s u c h that ~ X , y is an i m m e r s i o n of Sing(X), an

embedding,

Finally,

and, w h e n r e s t r i c t e d

to a s m a l l n e i g h b o u r h o o d

is r e s i d u a l .

to obtain an e m b e d d i n g f o r ( X , y ) ,

c o v e r i n g in t h e p r e s e n t c a s e .

y n e a r y, w e d o n ' t need an open

One can c o n s t r u c t d i r e c t l y a m a p Yv'V in s o m e f i n i t e

d i m e n s i o n a l v e c t o r s p a c e V, w h i c h is the a n a l o g u e of Yt' w i t h the f o l l o w i n g p r o p e r t i e s

(i)

Yo = y;

(it)

f o r x E Sing(X), the 1 - j e t of Yv is i n d e p e n d e n t of v;

(iii)

f o r x , x ' ~ Sing(X), x # x' the m a p . 2 m x j .2m : V ~ J 2 mx(M) x j2xm' (M) ] x x' 2m h a s a s u r j e c t i v e d e r i v a t i v e f o r air ( x , x ' ) in v = 0; J x(M) is t h e v e c t o r s p a c e of

2m-jets

of f u n c t i o n s on M in x; j2xm(v ) is the 2 m - j e t of Yv in x.

Using Yv one defines a m a p

]~:M

The

x M

rest of the proof of t h e o r e m

x V

RR 2 m + l

• ]R 2 m + l

i now

carries over to the present situation.

as before .

:

373

From

the l a s t t h r e e t h e o r e m s

it is c l e a r how a d y n a m i c a l s y s t e m with t i m e

e v o l u t i o n q~t and o b s e r v a b l e y is d e t e r m i n e d g e n e r i c a l l y by the s e t of all f u n c t i o n s t -* y(~t(x)).

In p r a c t i c e t h e f o l l o w i n g s i t u a t i o n m a y o c c u r : w e h a v e a d y n a m i c a l s y s t e m

with c o n t i n u o u s t i m e ,

but t h e value of the o b s e r v a b l e y is only d e t e r m i n e d f o r a d i s c r e t e

s e t [0, c ~ , 2 ~ , . . . } of v a l u e s of t; c~ > 0. o n s e t of t u r b u l e n c e [6, 8,9, 10].

This happens e.g.

in the m e a s u r e m e n t s

of the

A l s o i n s t e a d of all s e q u e n c e s of the f o r m

[y(~icz(x))}i=0 , x E M, w e only know s u c h a s e q u e n c e f o r one,

o r a few v a l u e s of x

( d e p e n d i n g on the n u m b e r of e x p e r i m e n t s ) and t h e s e s e q u e n c e s a r e not known e n t i r e l y but only f o r i = 1 .

. . . .

I~ f o r s o m e f i n i t e but b i g i'4 (in [ 6 ] ,

light we should know w h e t h e r ,

under generic assumptions,

I~ = 8192 = 213).

In this

the t o p o l o g y of, and d y n a m i c s

in the p o s i t i v e l i m i t s e t

L+(x) = i x ' E M[~t.1 -* ~ with q~t.(x) ~ x ' } 1

of x is d e t e r m i n e d by the s e q u e n c e {y(q0i, c~(X))}i= 0 . T h i s q u e s t i o n is t r e a t e d in the next t h e o r e m and its c o r o l l a r y ; in l a t e r s e c t i o n s we c o m e back to the p o i n t that t h e s e s e q u e n c e s a r e only known up to s o m e f i n i t e I~.

T h e o r e m 4.

L e t M b e a c o m p a c t m a n i f o l d , X a v e c t o r field on M with flow ~Pt and p

a p o i n t in M.

T h e n t h e r e is a r e s i d u a l s u b s e t CX, p of p o s i t i v e r e a l n u m b e r s s u c h that

f o r C~ ( CX, p, the p o s i t i v e l i m i t s e t s of p f o r the flow qOt of X and f o r the d i f f e o m o r p h i s m q0c~ a r e the s a m e .

In o t h e r w o r d s ,

is the l i m i t o f a s e q u e n c e ~ t . ( p ) , n. E ~N, n. ~ 1

f o r cz E C X , p w e nave that e a c h point q E M w h i c h

t i E IR, t.1 ~ + %

is t h e l i m i t of a s e q u e n c e ~ n . . r ~ ( P ) '

1

1

1

Proof.

T a k e q E L+(p).

C ,q

{r~>0

For a a (small) positive real number define

[~ n E iN, s u c h that O(~0n.~(p),q) < ~},

0 is s o m e fixed m e t r i c

on M.

Clearly C

is open; it is a l s o d e n s e . To p r o v e t h i s l a s t s t a t e m e n t w e o b s e r v e that f o r s,q any ~ > 0 and ~ > 0, t h e r e is a p o i n t of C in (d,d + ~) if and only if t h e r e is a e,q t E (n.~,n.(~)) with p ( ~ t ( p ) , q ) < a f o r s o m e i n t e g e r n. T h e e x i s t e n c e of s u c h t f o l l o w s f r o m the f a c t that f o r b i g n t h e i n t e r v a l s ( n . ~ , n . ( ~ b i g n, n . ( ~ + ~) > ( n + l ) . ~

+ s))

o v e r l a p (in the s e n s e that f o r

and t h e f a c t that t h e r e a r e a r b i t r a r y

b i g v a l u e s of t with

0(q~t(P),q) < ~.

S i n c e C ~ , q is open and d e n s e w e can take f o r C X , p c IR4_+ the f o l l o w i n g r e s i d u a l s e t C X , p = i, ~j=i C 1 T'qj

where

Iqj} is a c o u n t a b l e d e n s e s e q u e n c e in L+(p).

374

Corollary

5.

L e t M be a c o m p a c t

m a n i f o l d of d i m e n s i o n

c o n s i s t i n g of a v e c t o r field X, a f u n c t i o n y, a p o i n t p, For generic

such (X,y,p,~)

(more precisely

c o n d i t i o n s d e p e n d i n g on X a n d p),

embedding

of "diffeomorphic" of M

into IR 2m+l

For further {(Pi R ( P ) ] i ~--0. = c {bi }~=1

c

should mapping

reference

M, w i t h { O i . J P ) /

IR 2 m + l w i t h

be clear L+(p)

in IR2 m + l

metric

(X,y) and ~ satisfying generic

:

Y(g~( k + 2 m ) . c~(P))) ] 2 0

: it means

that there

is a smooth

to this set of limit points.

that the metric

as a sequence

with

properties

of

of d i s t i n g u i s h e d p o i n t s a r e t h e s a m e

as

{b i} a s a s e q u e n c e of d i s t i n g u i s h e d p o i n t s :

properties

the corresponding

here

bijectively

we r e m a r k

b i : (Y(tPl.~(P)) . . . . . These

quadruples,

t h e p o s i t i v e l i m i t s e t L+(p) is " d i f f e o m o r p h i c "

[(Y(~Pk, o~(p))' Y(~O(k+l). o~(P)) . . . . .

meaning

We consider

and a p o s i t i v e r e a l n u m b e r c~.

: for generic

t h e s e t of l i m i t p o i n t s of t h e f o l l o w i n g s e q u e n c e

The

m.

are the same

distances

y(~(l+2m).cc(p)))

6 IR 2 m + l

in t h e s e n s e t h a t d i s t a n c e s

in M a n d

in 1R2 m + l h a v e a q u o t i e n t w h i c h is u n i f o r m l y b o u n d e d a n d

bounded away from zero.

3.

Limit capacity and dimension. There

spaces.

are several

w a y s to d e f i n e t h e n o t i o n of d i m e n s i o n

T h e d e f i n i t i o n w h i c h we u s e h e r e g i v e s t h e s o - c a l l e d

i n f o r m a t i o n on t h i s n o t i o n c a n b e found in [ 1 4 ] .

for compact metric

limit capacity.

Some

S i n c e t h i s l i m i t c a p a c i t y is n o t w e l l

k n o w n we t r e a t h e r e s o m e of i t s b a s i c p r o p e r t i e s .

L e t ( S , p ) be a c o m p a c t s(S,g)

metric

is the maximal

space.

distance less than r r(S,r

is the minimal the r r

For a > 0 we make the following definitions

c a r d i n a l i t y of a s u b s e t of S s u c h t h a t no two p o i n t s h a v e s u c h a s e t is c a l l e d a m a x i m a l

cardinality

g-separated

set;

of a s u b s e t of S s u c h t h a t S i s t h e u n i o n of a l l

of i t s p o i n t s ; s u c h a s e t is a l s o c a l l e d a m i n i m a l set.

Note t h a t c r(S,~) ~ s(S,s) ~ r(S,~)

.............

(i)

375

~he first inequality follows from the fact that a maximal

s-separated

set is E-spanning.

T h e s e c o n d i n e q u a l i t y f o l l o w s f r o m t h e f a c t t h a t in a n ~ - n e i g h b o u r h o o d of a n y p o i n t (of g a m i n i m a l ~ - s p a n n i n g s e t ) t h e r e c a n b e at m o s t one p o i n t of a n s - s e p a r a t e d s e t .

N e x t w e d e f i n e t h e l i m i t i n g c a p a c i t y D(S) of S a s

D(S) =

i n (r(S, s ))

m2

t h e f a c t t h a t t h e l a s t two e x p r e s s i o n s or rather

S-capacity,

equivalent definition.

;

e q u a l f o l l o w s f r o m (1).

~fhe n o t i o n of c a p a c i t y ,

~fhis l i m i t c a p a c i t y is s t r o n g l y

s e e [5 o r 12~, wllich is c l e a r

L e t l/ b e a f i n i t e c o v e r i n g

( d i a m (Ui))a.

all finite covers

are

I n (s(S, s )

was originally used for s(S,s).

r e l a t e d to t h e H a u s d o r f f d i m e n s i o n ,

Da,LI = i ~

= lim j,nf

[Ui]iE I of S.

from the following

~fhen f o r a > 0

N e x t we d e f i n e D a , s a s t h e i n f t n u m of Da,1/ w h e r e t/ r u n s o v e r

of S e a c h of w h o s e e l e m e n t s

Da, t ~ [r (S, s ) . r a , r ( S , ~a) . s a ] .

has diameter

c.

Notice that

it i s not h a r d to s e e t h a t t h e r e is a u n i q u e n u m b e r ,

w h i c h is in f a c t t h e l i m i t c a p a c i t y D(S),

s u c h t h a t f o r a > D(S),

resp.

a < D(S),

l i m in[ D is z e r o , r e s p . i n f i n i t e . ~fhis l a s t d e f i n i t i o n of l i m i t c a p a c i t y g o e s o v e r in s-~0 a, g the definition of Hausdorff dimension if w e replace "each of whose elements has diameter s" by "each of whose elements has diameter ha.

For later [bl]i= 0 be some

reference

w e i n d i c a t e a t h i r d d e f i n i t i o n of l i m i t c a p a c i t y .

countable dense sequence

in S.

Let

F o r ~ > 0 we d e f i n e t h e s u b s e t Js a N

by : 0 E ]e; for [ > 0 : i s Js if and only if for all j with 0 ~ j < i and j 6 is' w e have @(bi,b j) ~ s. C s denotes the cardinality of Je" 0 s .

Cn, s, m denotes the c a r d i n a l i t y of Jn , s, m" Cn, s, m is n o n - d e c r e a s i n g approximation

in m.

F o r iq = % one would have ml i-*= m C n , s , m =C n , s

H e n c e it s e e m s

reasonable

to take C n , s , i q _ n as an

of Cn, s p r o v i d e d the d i f f e r e n c e between Cn, s,i,]_ n and say,

Cn, as189

]

379

is sufficiently small, of c a l c u l a t i n g C values for C From

s a y of t h e o r d e r

of 1 o r 2~0.

In t h i s w a y w e h a v e t h e p o s s i b i i i t y

in a c e r t a i n r e g i o n of t h e ( n , e ) - p l a n e ; a l s o o n e s h o u l d c o n s i d e r t h e s e n,8 o n l y r e l i a b l e if ~ i s w e l l a b o v e t h e e x p e c t e d e r r o r s in t h e m e a s u r e m e n t .

n,g these numerical

values for C o n e s h o u l d d e c i d e , on t h e b a s i s of t h e m a i n + n,g t h e o r e m w h a t t h e v a l u e s of D ( L (p)) a n d H(L+(p)) a r e o r w h e m e r t h e l i m i t s d e f i n i n g t h e s e

values "do not exist numerically".

If, to i n f i n i t y ,

in t h e c a l c u l a t i o n of D(L+(p)),

t h e lnirtz w o u l d h a v e t h e t e n d e n c y of g o i n g

t h i s would i m p l y t h a t r e p r e s e n t i n g

manifold is a mistake.

t h e e v o l u t i o n on a f i n i t e d i m e n s i o n a l

If on t h e o t h e r h a n d t h i s I i m i t w o u l d go to a n o n - i n t e g e r ,

w o u l d b e e v i d e n c e in f a v o u r of a s t r a n g e

attractor.

Namely,

a s we h a v e s e e n in

s e c t i o n 3, f o r a C a n t o r s e t C w e m a y h a v e D(C) a n o n i n t e g e r , h a v e in g e n e r a l a C a n t o r s e t iike s t r u c t u r e ,

If t h e e x p e r i m e n t a l

e.g.

d a t a do n o t c l e a r l y

o f D(L+(p)) a n d H(L+(P)) to e x i s t a n d to b e f i n i t e , Ruelle-Takens

picture are

this

and strange

attractors

see [3].

i n d i c a t e t h e l i m i t s in t h e c a l c u l a t i o n then both the Landau-Lifschitz

to b e r e j e c t e d a s e x p l a n a t i o n of t h e e x p e r i m e n t a l

and the

data.

Final remarks. 1.

It d o e s n o t s e e m

the "inf'

dimensional

limit

for differentiable dynamical

systems

a n d " s u p " in t h e d e f i n i t i o n of l i m i t c a p a c i t y a n d e n t r o p y c a n be o m i t t e d .

they can omitted,

2.

to be k n o w n w h e t h e r ,

one has a better

and deterministic"

t e s t on t h e v a l i d i t y of t h e a s s u m p t i o n s

If

"finite

: also the first limit has "to exist numerically".

Y o r k e p o i n t e d o u t to t h e a u t h o r t h a t h e a n d o t h e r s h a d m a d e c a l c u l a t i o n s of capacities

attractors,

in r e l a t i o n w i t h a c o n j e c t u r e

see [7].

His calculating scheme

on L y a p u n o v n u m b e r s

and dimension for

is different from ours and probably faster.

The calculations indicate that the computing time rapidly increases

with dimension,

which

probably also holds for our computing scheme.

3. become more

It s h o u l d b e n o t i c e d t h a t t h e d e f i n i n g f o r m u l a s

for dimension and entropy

a l i k e w h e n w e w r i t e t h e m in t h e f o l l o w i n g f o r m inC

D(L+(p)) =

n-'~lim

(liem~nf (__.r

H(L+(p)) = Is~r~ ( l i n m s u p

r

lnC (_n _ . ~s))n,e n .

.

380 lnC If we denote n - l nn,~

by Z ( n , - I n r

and r e g a r d both n and - l n r a s continuous v a r i a b l e s

one ~an s e e f r o m a few e x a m p l e s (Anosov a u t o m o r p h i s m s on the t o r u s and h o r s e s h o e s ) that often lir~ ~ Z(~,/~) e x i s t s for all p o s i t i v e T, f o r m i n g a o n e - p a r a m e t e r f a m i l y of "topologically i n v a r i a n t s " c o n n e c t i n g e n t r o p y with l i m i t capacity.

It would be i n t e r e s t i n g

to i n v e s t i g a t e the e x i s t e n c e of t h e s e l i m i t s for m o r e g e n e r a l a t t r a c t o r s .

T h i s m i g h t be

connected with the above mentioned c o n j e c t u r e of Yorke.

References.

1.

D. Aeyels, G e n e r i c o b s e r v a b i l i t y of d i f f e r e n t i a b l e s y s t e m s , Dept. of S y s t e m D y n a m i c s , State Univ. Gent.

2.

R. Bowen, E n t r o p y of group e n d o m o r p h i s m s and h o m o g e n e o u s s p a c e s , A . M . S . , 153 (1971), 401-414.

3.

R. Bowen, On Axiom A d i f f e o m o r p h i s m s , Regional C o n f e r e n c e S e r i e s in M a t h e m a t i c s , 35, A . M . S . P r o v i d e n c e , 1977.

4.

M. D e n k e r , C. G r i l l e n b e r g e r , & K. Sigmund, E r g e d i c t h e o r y on c o m p a c t s p a c e s , L e c t u r e Notes in M a t h e m a t i c s , 527, S p r i n g e r - V e r l a g , Berlin, 1976.

5.

H. F e d e r e r ,

6.

P.R.

7.

P. F r e d e r i c k s o n , J . L . Kaplan & J . A . Yorke, Xhe d i m e n s i o n of the s t r a n g e a t t r a c t o r f o r a c l a s s of d i f f e r e n c e s y s t e m s , p r e p r i n t , June 1980, U n i v e r s i t y of M a r y l a n d .

8.

J.P.

9.

j.P. Gollub, & S . V . Benson,

I0.

j.P. Gollub & H . L . Swinney,

G e o m e t r i c m e a s u r e theory,

Springer-Verlag,

preprint,

A p r i l i980,

Trans.

Berlin, 1969.

F e n s t e r m a c h e r , J . L . Swinney & J.P. Gollub, D y n a m i c a l i n s t a b i l i t y and the t r a n s i t i o n to ehaottc T a y l o r v o r t e x flow, Journal Fluid Mech. 94 (1979) (1) 103 -128.

Gollub, The o n s e t of t u r b u l e n c e : convection, s u r f a c e waves, and o s c i I l a t i o n s , in S y s t e m s f a r f r o m E q u i l i b r i u m , Proc. Sitges Int. School and S y m p o s i u m on S t a t i s t i c a l M e c h . , Ed. L. G a r r i d o , J. G a r c i a , L e c t u r e Notes in P h y s i c s , S p r i n g e r - V e r l a g , Berlin, to a p p e a r . T i m e - d e p e n d e n t i n s t a b i l i t y and the t r a n s i t i o n to t u r b u l e n t convection, p r e p r i n t , P h y s i c s Dept, H a v e r f o r d College, H a v e r f o r d , Pa. 19041, USA. O n s e t of t u r b u l e n c e in a r o t a t i n g fluid, Phys. Rev.

L e t t . 35 (1975), 927-930. 11.

M. H e r m a n , M e s u r e de L e b e s g u e et n o m b r e de rotation, in G e o m e t r y and Topology, ed. J. Palls and M. do C a r m o , L e c t u r e Notes in M a t h e m a t i c s 59._~7, S p r i n g e r - V e r l a g , Berlin, 1977.

12.

W. H u r e w i c z & H. W a l l m a n , D i m e n s i o n theory, P r i n c e t o n U n i v e r s i t y 1948, P r i n c e t o n , N.J.

Press,

381

13.

L. Landau & E. L i f s c h i t z ,

14.

R. Ma~6, On the d i m e n s i o n of the c o m p a c t i n v a r i a n t s e t s of c e r t a i n n o n - l i n e a r m a p s , p r e p r i n t IMPA, Rio de J a n e i r o , 1980.

15.

J.E.

Marsden, The Hopf b i f u r c a t i o n for n o n - l i n e a r s e m i g r o u p s , 79 (i973), 537-541.

16.

J.E.

M a r s d e n & M. M c C r a c k e n , The Hopf b i f u r c a t i o n and its a p p l i c a t i o n s , m a t h . s c i . 19, S p r i n g e r - V e r l a g , Berlin, 1976.

17.

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