Apparent fractal dimensions in conservative dynamical systems

Volume 118, number 7

PHYSICS LETTERS A

3 November 1986

APPARENT FRACTAL DIMENSIONS IN CONSERVATIVE DYNAMICAL SYSTEMS Giancarlo B E N E T T I N Dipartimento di Fisica dell'UniversitY, Via Marzolo 8, 35100 Padua, Italy

Danilo CASATI Corso di laurea in Fisica dell'Universita, Via Celoria 16, 20133 Milan, Italy

Luigi G A L G A N I Dipartimento di Matematica dell'Universitd, Via Saldini 50, 20133 Milan, Italy

Antonio GIORGILLI Dipartimento di Fisica dell'Universit~, Via Celoria 16. 20133 Milan, Italy

and Luisella S I R O N I Corso di laurea in Fisica dell'Universith, Via Celoria 16, 20133 Milan, Italy

Received 6 June 1986; revised manuscript received 25 September 1986; accepted for publication 26 September 1986

For conservative dynamical systems, the invariant sets which are in a sense the analog of the strange attractors of dissipative systems, namely the closures of homoclinic orbits of hyperbolic points, are known to have in general integral dimensions. We show however, working numerically on a particular model, that actual numerical estimates for perturbations of an integrable system will necessarily exhibit an apparent fractal dimension, which will be the effective one to all practical purpose. A simple scheme of interpretation is also given.

1. Introduction Many works were recently d e v o t e d to the p r o b l e m o f computing fractal d i m e n s i o n s in dissipative d y n a m i c a l systems [ 1-11 ]. In the case o f conservative systems one finds instead rather few works: in a hamiltonian system o f three degrees o f freedom [ 12 ], already studied numerically in order to detect Arnold diffusion, a non-integral d i m e n s i o n was observed [ 13 ], while in t w o - d i m e n s i o n a l m a p p i n g s the analog o f attractors, namely the so-called chaotic orbits, were just assumed to have d i m e n s i o n two [14,15] ( a n d thus called "fat fractals"). In fact, in the case o f conservative systems there is a diffuse opinion that the H a u s d o r f f dimension o f the " a t t r a c t o r s " should be an integral n u m b e r [ 16 ], a n d

this is supported, at least in the case o f two-dimensional mappings, by a t h e o r e m p r o v e d by Newhouse [ 17 ]. Let f b e an area preserving d i f f e o m o r p h i s m o f a compact connected orientable two-manifold ~//, and let p be a hyperbolic fixed (or p e r i o d i c ) p o i n t o f f , then one considers the set H ( p , f ) o f transverse homoclinic points o f p, a n d its closure H ( p , f ) is in a sense the analog o f the " a t t r a c t o r s " o f dissipative systems. N o w Newhouse proves that for a "generic" d i f f e o m o r p h i s m f, each set H ( p , f ) has H a u s d o r f f d i m e n s i o n two. Thus in such case the question o f the d i m e n s i o n seems to be settled. Things however a p p e a r to us to be m o r e complicated, even in the particular case considered a b o v e o f a two-dimensional mapping. Indeed, consider for example the case o f a family fu 325

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of mappings depending on a real parameter/t such that for p = 0 one has an integrable case with invariant one-dimensional curves. Then, how does it come about that for "almost" all ~t, small as one wants, there appear invariant sets of dimension 2, with an integral j u m p in the dimension? And how is this reconcilable with continuity? Stimulated by the deep analogy of such problem with that of equipartition of energy in classical statistical mechanics for perturbations of integrable systems, where exponentially long times were observed (see below), we decided to investigate carefully the problem of the fractal dimensions o f " a t t r a c t o r s " in a family of two-dimensional area preserving mappings.

3 November 1986

~3

%.o

0.1

0.2

0,3

0.4

0.5

Fig. I. The orbit of mapping (l), for an initial point near the hyperbolic fixed point, forlt=O.04, at a macroscopic scale. Number of iterations = 50000.

2.Numerical results We consider the family o f mappings [ 18 ]

x'=x+y

(modl),

y'=y-~sin[2n(x+y)]

(modl),

(1)

of the torus T 2, depending on the real parameter ~t. For # = 0 one has an integrable case, with invariant curves y = c o n s t ; for # ¢ 0 one has two fixed points, namely the origin (0, 0) and the point (½, 0), The latter being hyperbolic f o r / t > 0 . Taking an initial point near its unstable manifold and computing a great number n of iterates (n = 50000) we find for two different values o f the parameter ( # = 0 . 0 4 and /~=0.12) the points plotted in figs. 1 and 2 respectively. Clearly, for the greater value of~t the points appear visually to be scattered in a two-dimensional region, while they might appear to lie somehow on a line for the smaller value of/~. The two cases become however more similar if, in the latter case, one looks at a smaller scale (see fig. 3). This rather simple remark is the clue to our result: in order to find an effective dimension 2 one has to look at sufficiently small scales, which decrease as ,u decreases; thus, if one is given a minimal scale o f observation ~, there will correspondingly be a value ~(e) of the parameter such that for lower values of/~ the apparent or effective dimension will be 1. Moreover, the scales decrease exponentially fast with 1/#, and the required number of iterates correspondingly increases exponentially fast. 326

Before giving evidence for this, let us first report the results of a "naive" computation of the dimension, for/t in the interval (0.01,0.12 ), using the boxcounting method (see ref. [ 19] for an efficient procedure). Having fixed a value o f / t in such interval and having computed the corresponding orbit up to a number n ~
¢2' qD

%.o

0.1

0.2

0.3

0.4

Fig. 2. S a m e as in fig. 1, f o r / z = 0 . 1 2 .

0.5

Volume 118, number 7

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3 November 1986

• }J= 0.09

106

,

o P= 0.06

,,,,'"

p=0.03

,"

o p

z_ E

,* 10 4

,

,,"

, ,

,

....

2/" 10 2

d

t

t i tlIH[

102

I

I J

103

~HI

I

10 ~

1/E

0

. . . . . . .

t2. ~po

~

O. ~92

~

O. dPd

Fig. 5. Curves log .N~ versus log( 1/~) for three values of/~.

O. 496

O. d98

O. 500

Fig. 3. Same as in fig. 1, at a smaller scale. Number of iterations= 130000, in order to obtain that the number of points appearing in figure be roughly equal to the number of points appearing in fig. 2.

should in principle define the dimension d by log N ( c ) d=lim ~ 0 log(l/e) '

(2)

i.e. by the limit slope, as l / c - - , ~ , o f the curve log~r( c ) versus log(l/e). In practice one usually takes for d the slope o f a straight line interpolating the points o f l o g N ( c ) versus log(l/e), and the results thus found for the estimated value o f d versus/z are summarized in fig. 4. As one sees, the estimated dimension is 1 up to / z ~ 0 . 0 3 and then rises smoothly towards 2. In order to understand how the apparent

dimension smaller than 2 comes about, let us go back to the curves of log ?¢(c) versus log(i/e) and investigate more carefully whether one is really authorized to interpolate them with straight lines. Three such curves are reported in fig. 5, for three values of #. One has the appearance of straight lines (with different slopes) at the small and at the high value of/z, but at the intermediate value one has clearly a concave curve; so it is evident that the limit slope as I/c--,~, which is the relevant one by definition, should be larger than the slope of the interpolating straight line. Let us elaborate on this. For any pair of nearby values of c one can get the slope s o f the segment giving the corresponding linear interpolation, thus obtaining an estimate o f the slope s versus 1/c. The results for several values of/z are reported in fig. 6.

2.0

2.0 •

° 1.8

•

°

•

.0.090 u 0.(17'5

°

°

"

o 0.060 oi

•

1.6

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,

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•

0.0~0

1.5 1.~ A 0,030

*

1.2

1.0 i lJ~,l

f

0.0

i

I

0.02

~

I

i

I

0.04

~

i

j

I

l

I

0.06

i

I

0.08

i

I

I

I

0.10

i

I

l

I

0.12

I

F

i

~ iiiiiI

i

~

i

Illllh

1111

I

0.1~

P

Fig. 4. Estimated dimension d as a Function o f the perturbative parameter #.

102

103

10 ~

1/E

Fig. 6. Slopes of the curves log N, versus log(l/E) of fig. 5 for several values of/~. 327

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o, ~,

* •

,

o

10 3

10 2

, ,°~ ",~ ° ,I ' A,

'

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3 November

1986

dimension 2 of the "attractor", and such a scale decreases exponentially fast with 1/p. As a consequence, if one works with a minimal scale e, there will exist a finite value/t of/~ below which the apparent or effective dimension will be 1, while intermediate values will occur for slightly larger values oflt. This remark might in particular explain the nonintegral dimension reported in ref. [ 13 ].

10 L:

,.

LETTERS

1.60 1.70 1.80 1.85

3. A simple scheme of interpretation I

I

I

I

I

I

I

10.

I

I

I

L

20.

30.

1/~ Fig. 7. T h e v a l u e s o f log( 1/~ ) r e q u i r e d to get s l o p e s, as a f u n c t i o n

of 1/#, for several values ofs.

The basis remark is that, even for a set with a smooth boundary, the number N, of cells required to cover the set is given by N , = a / e + b/e e ,

One clearly sees that, as p decreases, smaller values of e (smaller scales) are required to find any given value of s in the interval (1, 2). This is made quantitative in fig. 7, which gives the value of 1/e (in logarithmic scale) needed in order to get the slope s, as a function of 1/#, for several values ofs. Clearly, for each s one has an exponential law of the type 1/e =A (s)e "('/" .

(3)

For the coefficient /t0 we find ~to=0.395+0.060 (three standard deviations); the functions A(s), determined empirically in the range [1.4, 1.93] of the values o f s, is reported in fig. 8. So, it is clear that any value of the parameter p requires a different scale e in order to display the

the first term being due to the boundary, and the second one to the bulk. The boundary term is a correction which disappears as e ~ 0 , but is important for finite values of e; in our case it should be particularly important, because the considered set appears to be constituted of a bulk around the hyperbolic fixed point, and by thin wings. So we assume that N, is given by (4), where a and b are unknown positive functions of/t. Then for the slope d log N, d log(I/e)

S=

one gets ble

s=l+

- -

a + b/e "

~0.

From this, by inversion, one gets exactly a function of the empirically guessed form (3), which gives the minimal scale needed to exhibit dimension s, namely

30.

~

to

(4)

1

20.

as-1 _

e

(5)

b2-s"

/

By comparison with ( 3 ) one not only determines the ratio a/b as function of/t, but also finds the form of the function A ( s ) , namely

10.

0.0 i

I 1 .d

i

I i

I i 1.6

l

i

I i 1.8

i

a / b = 7e ~'°/u ,

(6 )

S

Fig. 8. T h e f u n c t i o n A ( s ) o f f o r m u l a ( 3 ) : d o t s , e m p i r i c a l p o i n t s ; d a s h e d line, t h e o r e t i c a l c u r v e g i v e n b y ( 7 ) f o r 7 = O. 5.

328

A(s)=7

]

s-1 2-s'

(7)

Volume

118, number

PHYSICS

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r..

lo-'

“. . .,

.

10-Z

‘.

‘.

2 i;

10-3

10-L

Fig. 9.The function b(p) defined by (8). Dots, empirical points, using the known values of NC:; dashed line, linear interpolation

(in semilogarithmic scale). with an arbitrary constant y. The agreement with the experimental curve is excellent for example with 7 ~0.5 (fig. 8, dashed line). So one gets l/e21 ,

N, =:b(p)[(ylt)eW@+ ,fL()ZO.39, yzo.5

)

(8)

with a still undetermined function 6(p). By fit with the known values of N, in the interval [ 0.04, 0.121 for p (fig. 9)) one finds b(p) z We-““@

,

(9)

with N*=2.19?0.21 and ~,=0.3698-tO.O135. In fact, consistency would require pLo=p, =,L?, which is true within the errors; so the law for N, should be N, %:NY(yle+e-@Ve2) ~~0.5.

~~0.38,

, (10)

By the way, one can also directly determine the functions a(p), b(p) by lit with formula (4), and one thus obtains the form (lo), with N*=2.06-tO.21, ji=O.38&0.01 and ~~4.3.

4. Conclusions Our basic perturbative observation, ing to eq. (3

result is thus that, for any value of the parameter p, one has a typical scale of depending exponentially on l/p accord) , which must be reached in order to see

LETTERS

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3 November

1986

the dimension 2. Correspondingly, as the number of iterations required to invade a certain scale E is expected to be of the order of (l/~)~, it follows that the number of iterations needed to display the dimension 2, also increases exponentially with 11~. Our conclusion turns out to be in a qualitative agreement with a very interesting analytical result obtained at Barcelona by Fontich i Julia and Simo [20]. These authors investigate the separation between the stable and unstable manifolds at a homoclinic point for families of conservative diffeomorphisms near the identity, and show in particular that, in the analytic case, such separation is exponentially small with the parameter [ 201. This explains essentially the form (6) for the ratio a/b, which is at the basis for the interpretation of our results. By the way, the correction due to the boundary might be effective also in the dissipative case. As a comment, we mention that an indication for exponential diffusion times in a conservative fourdimensional mapping was given by Kaneko and Bagley [ 2 1 ] ; this should be a priori be expected as due to Arnold diffusion in the spirit of Nekhoroshev’s theorem [22,23], which does not apply to twodimensional mappings. In the two-dimensional conservative case, a relevant phenomenon is instead the limitation to transport due to the existence of “cantori” described by Mackay, Meiss and Percival [ 241. Finally, we would like to stress the conceptual relevance of the exponential law thus found for the scale (or the number of iterations) needed to display the dimension 2, which is known to be the true one for the complete orbit. In our opinion one should clearly distinguish between the proporties which are true for the complete orbit, defined by an infinite number of points, and the properties that can be observed for finite realizations of such orbits: in the latter case some apparent properties are to all purposes the effective ones, in the sense that they will be changed only when the finite realizations will be prolonged to an exceedingly higher number of points, so that they are in fact very stable. Such situation is very similar to that occurring in connection with the problem of energy equipartition. Considering for example a chain of equal harmonic springs (with angular frequency o) on a straight line, coupled by a repulsive potential between for example the right extremes of the springs, one 329

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finds numerically [25] that the time required to obtain equipartition among the translational and the vibrational degrees of freedom increases exponentially w i t h t h e f r e q u e n c y ~o. T h u s o n e m i g h t h a v e i n principle equipartion of energy (the analog of dimension 2 for the closure of the homoclinic orbits in the conservative diffeomorphisms), but the time n e e d e d to d i s p l a y e q u i p a r t i t i o n m i g h t b e so l o n g ( m o r e t h a n t h e age o f t h e u n i v e r s e in s o m e c o n c r e t e examples), that a situation of apparent nonequipart i t i o n at s o m e f i n i t e t i m e s m i g h t in a s e n s e b e t h e e f f e c t i v e o n e to all p r a c t i c a l p u r p o s e s . By t h e way, j u s t t h i s a n a l o g y led us t o guess t h e e x p o n e n t i a l l a w f o r t h e scale n e e d e d to d i s p l a y d i m e n s i o n 2 in t h e case o f c o n s e r v a t i v e d i f f e o m o r p h i s m s .

References [1 ] B.B. Mandelbrot, The fractal geometry, of nature (Freeman, San Fransisco, 1982). [2] J.D. Farmer, E. Ott and J.A. Yorke, Physica D 7 (1983) 153. [3] D.A. Russel, J.D. Hanson and E. Ott, Phys. Rev. Lett. 45 (1980) 1175. [4] Y. Termonia and Z. Alexandrowicz, Phys. Rev. Lett. 51 (1983) 1265. [5] P. Grassberger, Phys. Left. A 97 (1983) 224. [ 6 ] P. Grassberger and 1. Procaccia, Physica D 9 (1983) 189. [7] P. Grassberger, Phys. Lelt. A 107 (1985) 10l. [8] R. Badii and A. Politi, Phys. Rev. Lett. 52 (1984) 1661.

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[ 9 ] G. Paladin and A. Vulpiani, Lett. Nuovo Cimento 41 ( 1984 ) 82. [ 10] J. Guckenheimer, Cont. Math. 28 (1984) 357. [ 11 ] R. Gonczi, Measure of the fractal dimension of a strange attractor by a discretization method, Preprint ( 1986 ). [ 12] G. Contopoulos, L. Galgani and A. Giorgilli, Phys. Rev. A 18 (1978) 1183. [ 13 ] M. Pettini and A.Vulpiani, Phys. Lett. A 160 (1984) 207. [ 14 ] D.K. Umberger and J.D. Farmer, Phys. Rev. Lett. 55 (1985) 661. [15] C. Grebogi, S.W. McDonald, E. Otl and J.A. Yorke, Phys. Lett. A 110 (1985) 1. [16] E. Ott, Rev. Mod. Phys. 53 (1981) 655. [17] S.E. Newhouse, Soc. Math. France, Ast6risque 5l (1978l 223. [ 18 ] G. Benenin, C. Cercignani, L. Galgani and A. Giorgilli, Lctt. Nuovo Comento 28 (1980) 1; 29 (1980) 163. [ 19 ] A. Giorgilli, D. Casati, L. Sironi and L. Galgani, Phys. Lett. A 115 (1986) 202. [20] E. Fontich i Julih, Mesura del trencament de separatrius cn families de difeomorphismes amb punts Hiperb61ics, Thesis. Universitat de Barcelona (1985 ) ch. 6, theorem 6.4, p. 128: C. Sire6 and E. Fontich i Julia, On the smallness of the angle between split separatrices, to be published in: Proc. Colloque Geometrie sympletique et mdcanique (Montpellier, 1984). [21] K. Kaneko and R.J. Bagley, Phys. Lett. A 110 (1985) 435. [22] N.N. Nekhoroshev, Usp. Mat. Nauk 32 (1977) [Russ. Math. Surv. 32 (1977) 1]; Tr. Sem. Petrows. No. 5 (1979) 5. [23] G. Benettin, L. Galgani and A. Giorgilli, Cel. Mech. 37 (1985) 1. [24] R.S. Mackay, J.D. Meiss and I.C. Percival, Physica D 13 (1984) 55. [25] G. Benettin, L. Galgani and A. Giorgilli, Exponential law for the equiparlition times among translational and vibrational degrees of freedom, Preprinl.