NUMERICAL INVESTIGATION OF NONUNIFORM FRACTALS

FRACTALS IN PHYSICS L. Pietronero, E. Tosatti (editors) © Elsevier Science Publishers B.V.,

453 1986

NUMERICAL INVESTIGATION OF NONUNIFORM FRACTALS

Remo B A D I I * a n d A n t o n i o P 0 L I T I

+

P h y s i k - I n s t i t u t der U n i v e r s i t a t , S c h o n b e r g g a s s e 9 , 8001 Z u r i c h , S w i t1z e r l a n d * I s t i t u t o N a z i o n a l e di O t t i c a , L a r g o E . Fermi 6 , 50125 F i r e n z e , Italy" "

B e s i d e s the concept o f d i m e n s i o n , which g i v e s a f i r s t a v e r a g e c h a r a c t e r i z a t i o n o f f r a c t a l s , a s e ­ cond r e l e v a n t q u a n t i t y , the nonuniform!'ty f a c t o r , i s here i n t r o d u c e d and d i s c u s s e d . A t h e o r e t i c a l c a l c u l a t i o n i s performed f o r the b i n a r y C a n t o r s e t i n o r d e r t o p o i n t out the i n t u i t i v e meaning o f n o n u n i f o r m i t y a s the s p r e a d among the d i f f e r e n t c o n t r a c t i o n r a t e s o f l e n g t h s . The S i n a i map i s then i n v e s t i g a t e d i n d e t a i l , a s a p r o t o t y p e f o r a map w i t h n o n - c o n s t a n t J a c o b i a n , both i n the i n v e r t i b l e and n o n i n v e r t i b l e parameter r e g i o n .

1.

2.

INTRODUCTION R e c e n t l y , t h e r e has been a g r e a t deal

terest

i n the s t u d y o f f r a c t a l

many a r e a s o f p h y s i c s .

1

them, many d i f f e r e n t d e f i n i t i o n s like quantities

characterize

two g r o u p s ,

one d e r i v i n g from p u r e l y g e o m e t r i c a l

The d e f i n i t i o n s

o f the f i r s t

to g i v e i d e n t i c a l while

results for physical

the d i f f e r e n c e s

. They can

Renyi

Κ (ε), q

under

as

= {in

/

Σ P.(e)}

i-thbox,

entropies

(1-q)

(1)

i

C o n s e q u e n t l y , the Renyi d i m e n s i o n s a r e g i v e n by

among q u a n t i t i e s b e l o n g i n g

o f the f r a c t a l

generalized

=0,l,2,...,n,

o f the

systems,

to the second one are a measure o f the degree 4-6 "nonuniformity"

introduced

ΚHα( ε )

theory.

group u s u a l l y seem

parti­

t i o n o f the phase space w i t h boxes o f s i z e ε , and

the

first

definition

o f Renyi d i m e n s i o n o f o r d e r q . Assuming a

requests,

the second b e i n g r e l a t e d to i n f o r m a t i o n

here the

d e f i n i n g Ρ · ( ε ) as the p r o b a b i l i t y 2

of dimension2-4

have been i n t r o d u c e d

DIMENSION FUNCTION AND NONUNIFORMITY For c o n v e n i e n c e , we r e c a l l

in­

sets occurring in

In order to

be r o u g h l y c l a s s i f i e d i n t o

of

DH

of

conside­

ration.

= - l i m Κ ( ε )H /

In ε

(2)

ε-Κ)

In p a r t i c u l a r ,

D 0 i s the SSD ( u s u a l l y

evaluated

by means o f the b o x - c o u n t i n g a l g o r i t h m ) , D 2 i s

I n o r d e r to b e t t e r u n d e r s t a n d and q u a n t i f y

called information d i m e n s i o n

33

3

and D 2 i s the c o r ­

t h i s concept, a continuous i n f i n i t y of dimensions ( c a l l e d d i m e n s i o n f u n c t i o n ) has been r e c e n t l y i n 4

satisfy

D q> D p (when p > q ) , the

equa­

troduced

lity

b e i n g o b t a i n e d i n the c a s e o f u n i f o r m

sets,

i.e.

such t h a t the p r o b a b i l i t y

Ρ(δ,η)

t h r o u g h the moments o f the

o f n e a r e s t - n e i g h b o u r (nn)

among η p o i n t s on the f r a c t a l .

distances δ

As the d i m e n s i o n

f u n c t i o n DF can be proved t o y i e l d , points, quantities (SSD),

tool

suitable

other

Renyi

, i t s e v a l u a t i o n p r o v i d e s an u s e f u l

f o r the s t u d y o f n o n u n i f o r m i t y

systems.

in

as s e l f - s i m i l a r i t y d i m e n s i o n

i n f o r m a t i o n d i m e n s i o n and a l l 4b

dimensions

distribution

in physical

relation

Ρ Ί· ( ε )

integral

exponent ^.

the r e l a t i o n

These q u a n t i t i e s

P^ s c a l e s as 6

« ε ° , where D i n d i c a t e s any d i m e n s i o n . As

the d i r e c t impractical

e v a l u a t i o n o f the D ^ ' s i s , i n

general,

f o r computer memory r e a s o n s and l a c k

of s t a t i s t i c a l

c o n v e r g e n c e , we f o l l o w e d a d i f f e r ­

ent a p p r o a c h . C o n s i d e r a r e f e r e n c e d-dimensional

Euclidean space, plus

point χ in a (n-1)

others,

R. Baddi, A. Politi

454

all

o f them chosen a t random, w i t h r e s p e c t to the

natural

m e a s u r e , on the s e t * . We then d e f i n e

as the d i s t a n c e between χ and i t s bour y among the δ(η)

(n-1)

6(n)

nearest-neigh­

other p o i n t s .

neral,

some a v e r a g e o v e r a l l >s η

- 1

/0.

Evidently,

points χ will

behave intro­

distribution

P ( 6 , n ) o f nn

d i s t a n c e s among η p o i n t s . Now, f o l l o w i n g R e f .

Ύ

< 6 ( η ) > = Μ (η) Ύ

= /

Y

and a l l

i n the f i x e d

s u i t a b l e entropy

D

(3)

It

properties

0 o f dimen­ (DF).

The p r e f a c t o r

However,

i t s dependence on γ i s , by d e f i n i t i o n ,

irrelevant,

is

More­

possi­ Η

Η

recovered.

T h e r e f o r e , the e v a l u a t i o n o f moments ( 3 ) a l l o w s determination

a first in t u r n ,

the

v a l u e o f γ i s chosen t o

obtain

e s t i m a t e o f the d e s i r e d d i m e n s i o n w h i c h , i s used a s a new i n p u t u n t i l

a satisfac­

t o r y a c c u r a c y i s r e a c h e d . However, t h i s i s not general

n e c e s s a r y , a s the e s t i m a t i o n o f ϋ ( γ )

some p r e f i x e d

in

in

p o i n t s g i v e s the same i n f o r m a t i o n .

Anyway, i n o r d e r to check the s t a b i l i t y

of

the

First,

distri­

S e c o n d l y , f o r some p h y s i c a l

systems,

s i n g l e p o i n t s a r e not a c c e s s i b l e but Ρ ( δ , η ) can be e v a l u a t e d , e i t h e r a n a l i t i c a l l y it

or

numerically

i n the f o l l o w i n g way: We c o n s i d e r by η unequal

b a l l s of

, n ) and then r e f i n e i t

t a k i n g more b a l l s o f s m a l l e r s i z e . For

by

percolating

probability

the

o f f i n d i n g a c l u s t e r o f s i z e δ i n a box

( o f s i z e L) c o n t a i n i n g η c l u s t e r s : Then ,one s h o u l d c o n s i d e r a l a r g e r box ( i . e . l a r g e r n) and r e n o r m a l i z e l

s h r i n k t o zero w i t h i n c r e a s i n g n . F i n a l l y , that, in general, Ρ(δ,η) w i l l tually,

o f any D^ by means o f a r e c u r s i v e

method: An i n i t i a l

P(6,n). t o the

i t t o the s c a l e L , i n such a way t h a t the 6 s

where D q i n d i c a t e s the o r d e r - q Renyi d i m e n s i o n . is

"uniformity

s y s t e m s , f o r example, Ρ ( δ , η ) would r e p r e s e n t

b l e to prove the g e n e r a l r e l a t i o n D{y = (1 - q)D } = D ,

For q = 0 , the f i x e d p o i n t r e l a t i o n

it

t o examine some

i s smooth, contrary

d i a m e t e r s δ^(ι = 1,

t h i s v a l u e o f the DF c o i n c i d e s w i t h the S S D , a t

o v e r , f o r the same c l a s s o f s y s t e m s , i t

sinthetize

has been c a l l e d

a c o v e r i n g o f the f r a c t a l

is satisfied,

l e a s t i n the c a s e o f s e l f - s i m i l a r ^ f r a c t a l s .

to a

increasing nonuniformity:

o f the n n - d i s t r i b u t i o n

interpreting

c l a s s o f s y s t e m s , to an u n e s s e n t i a l p e r i o d i c i t y 8 4 i n l n ( n ) . I t has been shown t h a t , whenever relation

i s related

i s then w o r t h w h i l e

the f r a c t a l .

w h i l e the dependence on η r e d u c e s , i n a l a r g e

a fixed-point

to 0

b u t i o n o f p o i n t s i n the E u c l i d e a n s p a c e c o n t a i n i n g

s i o n hereafter c a l l e d dimension f u n c t i o n Κ depends on both γ and n :

i s equal

and can be used to

grows towards 1 f o r

factor".

where D(y) i s a γ-dependent d e f i n i t i o n

i.e.

p o i n t x = D ' ( D 0) 4

hence, t h i s q u a n t i t y

P(
has

d i m e n s i o n s c o i n c i d e . M o r e o v e r , the s l o p e

note t h a t i t

γ = 0(γ),

p o i n t s . In Ref. 4 b , i t

the degree o f n o n u n i f o r m i t y o f f r a c t a l s , a s

4,

we compute the moments o f Ρ ( δ , η ) a s oo

the DF i n the f i x e d

0 and 1 a n d , f o r u n i f o r m f r a c t a l s ,

To be more s p e c i f i c , we

duce the p r o b a b i l i t y

i s n e c e s s a r y t o compute the s l o p e o f

been shown t h a t D ' ( Y ) i s a l w a y s bounded between

i s a nonincreasing function of η and, in ge­

as <δ(η)

method, i t

lative

i n K e f . 4b i t r . m . s . Δ(η)

would notice

s p r e a d when η g r o w s : A c ­

has been shown t h a t the

o f the d i s t r i b u t i o n ^)

1 / 2

/M

for

be­

haves a s Δ ( η ) = ( M

2 D- M o

λ: T h i s c l a r i f i e s

the meaning o f n o n u n i f o r m i t y a s

Do

* n \

re­

Ρ(δ,η)

the s p r e a d among the d i f f e r e n t c o n t r a c t i o n o f l e n g t h s , when η goes t o i n f i n i t y . sets

(i.e.

small

rates

For u n i f o r m

λ = 0 ) al 1 d i s t a n c e s d e c r e a s e w i t h

the

same s p e e d . The DF i s , i n d e e d , bounded by two

* I n the c a s e o f dynamical s y s t e m s , p o i n t s a r e g e n e r a t e d i n a s e q u e n t i a l way. For f r a c t a l s such a s p e r ­ c o l a t i n g l a t t i c e s o r polymer a g g r e g a t e s , they can be p i c k e d up a c c o r d i n g t o any r u l e , i n o r d e r t o g e t information on the s t r u c t u r e : F o l l o w i n g d i f f e r e n t r u l e s c o r r e s p o n d s t o a s s i g n i n g d i f f e r e n t p r o b a b i l i t y w e i g h t s t o the v a r i o u s r e g i o n s o f the o b j e c t .

Numerical investigation of nonuniform fractals

horizontal

asymptotes c o r r e s p o n d i n g to the f a s t e s t

( γ = -οο) and the s l o w e s t (γ = ») c o n t r a c t i o n

rates.

455

p o s i t i o n of d i s t r i b u t i o n s of uniform Cantor s e t s w i t h d i f f e r e n t d i m e n s i o n s . Note t h a t D r a n g e s b e ­ tween - l o g p x and - l o g p 2 which c o i n c i d e

the asymptotes o f the DF D(-°°) and D(°°). M o r e ­

3. FRACTAL MEASURES Let us now i n v e s t i g a t e tion Ρ(δ,η)

the shape o f the

func­

i n a s i m p l e c a s e : The u n i f o r m C a n t o r

s e t , generated by keeping two segments o f l e n g h t a and d e l e t i n g

the middle p a r t o f the u n i t

A l s o , the same p r o b a b i l i t y parts. P(6,n)

segment.

i s a s s i g n e d to the

two

For t h i s s e t , the f o l l o w i n g e x p r e s s i o n f o r 4b has been o b t a i n e d D

Ρ ( δ , η ) = 2 D 0n ( 2 6 ) o "

1

υ

χ θρ { - η ( 2 δ ) 0 } .

s e l f i n the shape o f the p r o b a b i l i t y us c o n s i d e r a b i n a r y Cantor s e t

Ρ(δ,η),

or

i n R e f . 4b s u g g e s t t h a t d e v i a t i o n s

the form (4)

are an i n d i c a t i o n o f

and i n f o r m a t i o n

from

nonuniformity.

LYAPUNOV DIMENSION IN NONINVERTIBLE MAPS

As a more p h y s i c a l example o f f r a c t a l . . c 3c,10 . • we c o n s i d e r the S i n a i map χ'

= χ + y + g cos(2*y)

mod 1

y'

= χ + 2y

mod 1

sufficiently

o f area w i l l

"non-lacunar (i.e.

dimen­

measure,

(6)

s m a l l g . A s , however,

the sum o f

s h r i n k to zero i n t i m e :

t h a t the p r o b a b i l i t y

element

T h i s means

distribution will

be h i g h l y

peaked, t h a t i s , the a t t r a c t o r i s a f r a c t a l measu­ re.

integral

S ( o , n ) f o r a r e f e r e n c e p o i n t χ to h a ­

ve a n n , chosen among ( n - 1 )

points, within a d i s -

S(&9n)=J

I n R e f . 4 b , we o b ­

tained

Ρ(δ,η)

Lyapunov exponents i s a l w a y s n e g a t i v e , an

s i o n c o i n c i d e { D x = - ( P i l n ( p ! ) + P2"In(p 2) ) / l n 2 < 1 } .

tance δ ,

higher

whose s o l u t i o n c o v e r s the whole u n i t s q u a r e f o r

This i s

We are i n t e r e s t e d i n the e v a l u a t i o n o f the probability

embedded i n

I n d e e d , some p l o t s o f

ex­

The whole u n i t segment i s f i l l e d

D0 = 1 ) , while Hausdorff

reported

centered

Similar considerations

should apply a l s o to f r a c t a l s dimensional s p a c e s :

let

probabili­

t i e s pi and p 2 ( P i + P 2 = l ) > r e s p e c t i v e l y . g

1

it­

i s a Gaussian

d i m e n s i o n w h i c h , hence, appears

to play a s p e c i a l r o l e .

, t h a t i s the s e t

p a n s i o n c o n t a i n s zeros and ones w i t h

1

a t the i n f o r m a t i o n

3 a

of p o i n t s χ ( 0 < χ < 1) such t h a t t h e i r b i n a r y

an example of f r a c t a l measure

o v e r , the w e i g h t f u n c t i o n

4.

(4)

I n o r d e r to see how n o n u n i f o r m i t y m a n i f e s t s

fractal" :

with

P(y,n)dy.

the f o l l o w i n g e x p r e s s i o n f o r the comple­

mentary d i s t r i b u t i o n

S = l-S

r-

*

_ exp C

P 9 Pg log

Γ'°

1 2/ ,1/2 S e c [-log 2 δ ]

log 2 8 V:

- - log loa

*

p. D.

e x p [-n ( 2 8 ) ° ]

where l o g i n d i c a t e s the b a s e - o f - t w o and Ρ 2 < Ρ ι · The f i r s t

(5)

^ 2/ ΧrΊ > Γ ( D - K X , ) ] dD

logarithm

term i n the i n t e g r a l

r e s u l t one would o b t a i n f o r a uniform Cantor s e t of d i m e n s i o n D

4 b

. Therefore,

is

the

ternary the

whole

e x p r e s s i o n f o r S can be i n t e r p r e t e d as a s u p e r -

Lyapunov dimension D|_ ( f u l l l i n e ) and Dimension F u n c t i o n ϋ ( γ ) ( d o t s ) v e r s u s g . The DF i s computed f o r 5 d i f f e r e n t v a l u e s of γ ( - 2 , - 1 , 0 , 1 , 2 ) .

R. Baddi, A. Politi

456

Another r e a s o n t o expect n o n u n i f o r m i t y f o r t r a n s f o r m a t i o n i s the nonconstancy o f the b i a n J = 1 + 2^g s i n ( 2 * y ) .

Infact,

G r a s s b e r g e r and P r o c a c c i a point-like

this jaco-

1 1

, fluctuations in

the

i n t h e s e systems and are r e s p o n s i b l e f o r the

dif­

ference among the v a r i o u s d i m e n s i o n s . M o r e o v e r , the i n f o r m a t i o n d i m e n s i o n Di o f the a t t r a c t o r

can

be computed i n terms o f the Lyapunov exponents a s * ϋ χ . The e q u a l i t y 12

mensional i n v e r t i b l e

maps

holds for

while,

two-di-

f o r more g e n e ­

ral

s y s t e m s , D L p r o v i d e s a good a p p r o x i m a t i o n

ϋχ.

It

i s , therefore,

the b e h a v i o u r o f t h i s

interesting

to

to

(the

values of

central

the

one c o r r e s ­

ponding to the i n f o r m a t i o n d i m e n s i o n ) , v e r s u s g . At the p o i n t g = l / ( 2 7 r )

(indicated

by a

(North-Holland,

3 . a) J . D . Farmer, Z . N a t u r f o r s c h . 37A ( 1 9 8 2 ) 1 3 0 4 , b) P. G r a s s b e r g e r and I . P r o c a c c i a , P h y s . Rev. L e t t . 50 (1983) 3 4 6 , c ) J . D . Farmer, E . O t t and J . A . Y o r k e , P h y s i c a 7D (1983) 153, d) Y . Termonia and Z . A l e x a n d r o v i t c h , P h y s . Rev. L e t t . 51 (1983) 1265, e) J . Guckenheimer and G. B u z y n a , P h y s . Rev. L e t t . 51 (1983) 1438. 4.

R. B a d i i and A . P o l i t i , a ) P h y s . Rev. L e t t . 52 (1984) 1 6 6 1 , b) J . S t a t . P h y s . 40 (1985) 725.

and,

g i v e s o n l y an upper bound

to Di = D ( 0 ) , a s e x p e c t e d . M o r e o v e r , the s y s t e m nonuniform a l r e a d y a t g = 0.1

and the

h i g h e s t s p r e a d among the v a r i o u s d i m e n s i o n s i s o b ­ t a i n e d a t the t r a n s i t i o n p o i n t g = l / ( 2 i r ) .

It

however, e v i d e n t t h a t n o n u n i f o r m i t y changes smoothly w i t h g and does n o t have a w i t h the n o n i n v e r t i b i l i t y

o f the map.

5 . P. G r a s s b e r g e r , P h y s . L e t t . 6 . H . G . E . Hentschel and I . 8D (1983) 435.

relation

is, rather

97A (1983)

227.

Procaccia, Physica

7.

C . T r i c o t , M a t h . P r o c . Camb. P h i l . S o c . 91 (1982) 57.

8.

R. B a d i i and A . P o l i t i , P h y s . 303.

vertical

the map becomes n o n i n v e r t i b l e

for larger g-values,

i s rather

2 . A . R e n y i , P r o b a b i l i t y Theory Amsterdam, 1 9 7 0 ) .

investigate

g . I n F i g u r e 1 , we d i s ­

l i n e ) and f i v e

dimension f u n c t i o n D ( y )

dashed l i n e ) ,

B . B . M a n d e l b r o t , The F r a c t a l Geometry o f N a t u ­ re (Freeman, San F r a n c i s c o , 1 9 8 3 ) .

"Lyapunov d i m e n s i o n " a s a

f u n c t i o n o f the parameter play D L (as a f u l l

1.

a s shown by

v a l u e o f Lyapunov exponents are s t r o n g

D,= 1 + λι/|λ2|

REFERENCES

Lett.

9. B . B . Mandelbrot, M u l t i p l i c a t i v e F r a c t a l s , t h i s volume.

104A (1984)

Chaos and

1 0 . Y a . S i n a i , R u s s . M a t h . S u r v e y s 4 (1972) 2 1 . 1 1 . P . G r a s s b e r g e r and I . (1984) 34.

P r o c a c c i a , P h y s i c a 13D

1 2 . L . S . Young, J . E r g o d i c Theory and Dynam. S y s . 2 (1982) 109.