Fractal distribution of floaters on a fluid surface and the transition to chaos for random maps

Physica D 53 (1991) 102-124

North-Holland PHYSlCABl

Fractal distribution of floaters on a fluid surface and the transition to chaos for random maps

Lei Yu’, Edward Ott’,2 and Qi Chen Laboratory for Plasma Research, University of Maryland, College Park, MD 20742, USA

Received 14 January 1991

Revised manuscript received 13 May 1991

Accepted 10 April 1991

Communicated by U. Frisch

The long-time spatial distribution of particles floating on the surface of a confined fluid whose flow velocity has

complicated time dependence is considered. It is shown that this distribution can be either a fractal or else can clump at

several (or one) discrete points. The transition from the latter type of distribution to the former occurs when the Lyapunov

exponent characterizing the particle motion passes through zero from negative values to positive values. The characteristic

features of this type of transition are investigated using random maps. It is shown that near the transition there are

extremely intermittent temporal fluctuations in the particle cloud, and their scaling with a parameter is elucidated.

1. Introduction and motivation

In this paper, motivated by the problem of

particle convection in a confined fluid, we are led

to the study of random dynamical systems. In

particular, we will consider a random map of the

form

X n+, =F(x,,n,p), (1)

where x can be a vector, p is a system parameter,

and the map F is chosen randomly at each iterate

n according to some rule. We suppose that the

fluid velocity u(x, t) has smooth large scale spa-

tial dependence, but complicated time depen-

dence. Such situations can, for example, occur as

a result of fluid instabilities at low Reynolds

numbers. If we sample the particle position at

‘Also Department of Physics, University of Maryland, Col-

lege Park, MD, USA.

*And Department of Electrical Engineering, University of

Maryland, College Park, MD, USA.

discrete evenly spaced times t = 0, T, 2T,. . . , the

problem reduces to that of eq. (1) where F at

time n is, in principle, determined by integrating

the equation of particle motion from time nT to

time (n + 1)T. If T is larger than the effective

correlation time of the Eulerian fluid velocity,

then the maps F(x, n, p) are essentially uncorre-

lated for different II values. We note that the

complicated time dependence of the Eulerian

fluid velocity v(x, t) may be due to the chaotic

dynamics of the fluid. However, in this paper, we

wish to reserve the word “chaos” as a description

of the behavior of the motion of particles in the

fluid. Thus, to distinguish these two, we speak of

“complicated” time dependence of the fluid and

“chaotic” behavior of the particle distribution.

In addition to the n dependence of the map F,

we also restrict ourselves to the case where F is

not area preserving. This situation can arise, for

example, if the convected particles are floating on

the upper surface of the fluid. In particular, al-

though fluid is typically incompressible, V * Y = 0,

0167-2789/91/$03.50 0 1991- Elsevier Science Publishers B.V. All rights reserved

L. Yu et al. /Floaters on a fluid surface 103

a floating particle responds only to the horizontal component of o, which we denote v I. Since V, .v I = - tb,/&z, where t is the vertical direc- tion, floating particles are not convected in- compressibly. For example, the simple steady incompressible flow, v = -(x/L)u,x, - (y/L)uOyO + 2(z/L)u,z,, yields an attractor at the fixed point x =y = 0 for floaters on the surface z = 0 of a fluid in the region z < 0. Further discussion is given in section 2.

The main focus of our concern is the following situation. We imagine that we sprinkle floating particles on the surface of a confined fluid with complicated time dependence and follow the motion of the particles with time. After waiting a long time, we take a snapshot of the particle distribution on the fluid surface. We call this distribution a snapshot attractor [l, 21. What are the characteristics of this distribution? We shall see [l-3] that the particles either clump at a single point (or several points), or else, depending on the parameters (e.g. p in eq. Cl)), the particles can be distributed on a fractal of dimension between one and two. In particular, as p is varied, a transition can occur in which the long-time particle distribution changes character from being distributed on a fractal to being concentrated at a single point (or several points). We shall be con- cerned with this transition. This problem is very similar to the problem of characterizing the transition to chaos for deterministic dynamical systems in which there are a small number of often observed routes (e.g. period doubling, intermittency, crises). Similar to the case of deterministic systems, one can define Lyapunov exponents to characterize the stability of random systems (see section 2 for the definition). The meaning of these exponents again is the divergence (or con- vergence) rates between infinitesimally neighboring trajectories. The geometry of a snapshot attractor depends on whether the largest Lyapunov exponent is positive or negative. In particular, if the largest Lyapunov exponent is negative, since the distance between any two nearby trajectories in the phase space contracts under the dynamics,

the initial cloud shrinks and eventually settles down to a single point (or several points) whose location jumps randomly in time. On the other hand, when the largest Lyapunov exponent is positive, although the initial cloud shrinks finally to a zero-volume set (the map F is on average volume-contracting), since the neighboring trajectories diverge exponentially, the final snapshot attractor does not consist of a single point (or several points), but is a fractal. Indeed, Ledrappier and Young [3] have shown that the dimension of the snapshot attractor is given by the Lyapunov dimension introduced by Kaplan and Yorke [41, which, in this case, is greater than one. Therefore, generically, if the largest Lya- punov exponent is positive, the snapshot attractor is a fractal set. We say that the random map is at the transition point if its largest Lyapunov exponent is zero, and we call the behavior of the floating particles chaotic if the largest Lyapunov exponent is positive.

In fig. 1, we show schematically the dependence of the Lyapunov dimension of the snapshot attractor on its largest Lyapunov exponent. Below the transition point, the dimension is zero; above the transition point, by the formula for the Lya- punov dimension (section 31, the dimension is greater than one; at the transition point the di-

D

1 I h

Fig. 1. Schematic plot of the dimension of a snapshot attrac-

tor versus its largest Lyapunov exponent.

104 L. Yu et al. /Floaters on a fluid surjace

mension approaches one from the fractal side. Therefore, the dimension is discontinuous at the transition point. Just above the transition point, the transition dynamics falls on an approximately one-dimensional attractor. This suggests that the transition dynamics for one dimensional random maps might serve as a paradigm for the transition dynamics of higher dimensional random systems. Our numerical experiments confirm this.

The main conclusion of this paper is that, as the transition point is approached, the time dependence of the fractal snapshot attractor exhibits a remarkable and extreme form of intermittency. We investigate the scaling of this temporal intermittency with a system parameter near the transition.

This paper is organized as follows. In section 2, we discuss the motion of particles floating on the surface of a confined flowing fluid with complicated time dependence and introduce the random Zaslavsky map as a model for such situations. Then, in section 3, we numerically investigate the characteristic dynamical behavior occurring near the transition point. We present numerical results for the scaling of the largest Lyapunov exponent, the average attractor size, and an intermittency index of the snapshot attractor with a system parameter. A special numerical technique is introduced to overcome the artificial clumping of the snapshot attractor above the transition due to finite computer precision. In section 4, we investigate the same problem for the one-dimensional random circle map. The results of section 4 confirm the proposition that the essential transition behaviors in the two-dimensional case are present in the one-dimensional model. Based on this result, in section 5, we introduce simple random one-dimensional map models for which it is possible to obtain a complete analytical solution. We study the scaling of the attractor in both chaotic and nonchaotic regimes near the transition. These theoretical results are in good agreement with the numerical results presented in section 3 and section 4. We summarize the main conclusions of this study in section 6. A prelimi-

nary version of this work has been reported in ref. [5].

2. Random Zaslavsky map

In this section we consider a specific example. We emphasize, however, that our subsequent arguments indicate that the main qualitative behavior found near the transition to fractal particle distributions is model independent.

Consider an incompressible flow, i.e. V. v = 0. If uz = (YZ, then we have aXuX + ?$L’~ = -_(y. The trajectories of tiny particles floating on the z = 0 surface of the fluid are thus described by an area contracting flow,

dx dY -_=L’ -==c’ dt x’ dt y’ (2)

In particular, any initial area A,, evolved under the flow contracts exponentially with time A(t) = A, e-“‘. As an example, we take u I = L’,x(, + u,yo to be the sum of three components u i = u i , +

uL2+uL3, where vI, satisfying V-u,, = -(Y is a contractive component,

u-,1= -UYY,,,

v i 2 is a shear flow,

and v,~ models the effect of a vertical flow with complicated time dependence. To facilitate the calculations to follow we take u 13 to have the form

n=-x

where 8(t) is a delta function. (Since u is periodic in x, we may roughly think of x as an angle in an annular cylindrical geometry, with y playing the role of the radial variable.) Note that V . u i 2 =

V’V,X = 0. Solving eq. (2) we get the position of a particle just after the time t = (n + 1)T in terms


of its position just after the time t = nT,

X n+l = x, + l - CUT

i a Py,) mod 2rr, (3a)

Y n+1= e -aTyll=ksin(x,+,+f3,), (3b)

where ~9, = O(nT>. If we think of O(t) as behaving chaotically with time, then e,, can be considered a random variable, and eqs. (3) are hence in the form of (1). Eqs. (3) with 0, zero are often called the Zalavsky map [6]. If we consider a strip 1 yJ < K, where K, > k(1 - ebolT)-‘, then on one iterate of the map this strip is mapped into a nar- rower band I yl < K, <K,. Thus any long term behavior is confined to ( yl < k(l - e-uT)-l. Note that the Jacobian determinant of the map (3) is J s e_aT, and thus on each iterate, the map contracts areas by the factor epaT. Also note that, if we normalize y to (Y/P, the map can be shown to depend on only two independent parameters rather than the four parameters ((Y, /3, k and T) appearing in (3). Thus we can, for example, hx the values of T and p without loss of generality. Subsequently, in our numerical work we will take T= 1 and p = 1.

We consider the case in which the random variables 0, are independently and evenly distributed in [O, 2~). (We numerically verify at the end of section 4 that results obtained for this case are qualitatively similar to those with other distributions of 0,.) It is instructive to introduce a new variable u,+r =x,+r + 0, in (3). In this case (3) becomes

u,+r = ( ~,+(l-e-~~)~y~++~ mod2T,

P 1

Y,+~ = e -aTyn + ksin(u,+,).

Thus, if 0, is evenly distributed in [O, 2a), so is

4&e,-en_,> mod 2~. Hence the randomness in 0, causes the u,, to be random and independent from each other and uniformly distributed in [O, 2~). This is convenient for analysis because the largest Lyapunov exponent can be calculated from the product of statistically independent ran-

dom matrices without consideration of the actual particle orbits. The largest Lyapunov exponent is the growth rate of a generic infinitesimal vector. More specifically, it is the growth rate of the length of a typical tangent vector (6x,, 6y,),

6X n+, =6x, + l- e-“Tp8yn, ff gy,+, = e -aT6yn+kcosun+16x,+1,

which depends only on the uniformly distributed random variables u, + , and not on the details of the trajectory (x,, y,). Thus St, = M,M,_, . . . Ml St,,, where 85 = (6x, 6~) and Mj are independent random matrices,

/ \ 1

1 _ e-aT

(Y P lVj =

k cos en epaT + ’ -z-aT@k cos uj+l

The largest Lyapunov exponent is then given by

for almost any initial position (x0, yO), almost any StO, and almost any realization of the random sequence u,. The other exponent is then A, = A, - aT (since det Mj = eeaT for all j). Proofs of the existence of Lyapunov exponents for random matrices are discussed in ref. 171.

Here we take the word chaos, as applied to the random maps (e.g. eq. (l)), to have the same meaning as in the case of strange attractors of nonrandom deterministic systems. That is, when the system is chaotic, two nearby points on typical trajectories separate exponentially with time, or equivalently, the largest Lyapunov exponent is positive. As indicated by Romeiras, Grebogi and Ott [ll, snapshot attractors appear to be fractal in random dissipative systems if the system is chaotic. If we have particles floating on the surface of a flowing fluid as described above, and if nearby floating particles separate exponentially in time,

106 L. Yu et al. /Floaters on a fluid surface

a

0.0 0.4 0.8 1.2 1.6 2.0 11 a, 1 z

k a

Fig. 2. The largest Lyapunov exponent versus parameters (Y

and k for the random Zaslavsky map with p = T= 1.

Fig. 3. Largest Lyapunov exponent versus (Y with k = 0.5 and

p = T = 1. The dashed line is an analytic approximation to A,

obtained by using microcanonical ensemble method (appen-

dix A).

then we expect a fractal pattern to form on the

surface. The numerically computed dependence

of the largest Lyapunov exponent for eq. (3) on

the parameters (Y and k (with /3 = T = 1) is shown

as a contour plot in fig. 2. The solid curve in fig. 3

gives the numerically computed largest Lyapunov

exponent A, as a function of (Y for k = 0.5. The

dashed curve in fig. 3 is the result of an approxi-

mate analytical calculation of A, given in ap-

pendix A; this result is in good agreement with

the numerical computation. Note that the Lya-

punov exponent apparently crosses 0 smoothly

and with a finite nonzero slope. Examples of the

snapshot attractor are shown in fig. 4. These

pictures are made by sprinkling a large number of

initial conditions in the rectangle shown and then

plotting their locations at a single time occurring

after a large number of applications of eqs. (3).

Figs. 4a-4d correspond respectively to attractors

with successively smaller largest Lyapunov expo-

nents (A,) and fractal dimensions: A, = 0.10, D,

= 1.53 for fig. 4a; A, = 0.076, D, = 1.35 for fig.

4b; A, = 0.051, D, = 1.21 for fig. 4c; A, = 0.031,

D, = 1.11 for fig. 4d. (D, is defined in section 3.)

Fig. 4e shows a case where D, = 0 and A is

negative (A, = - 0.06). In this case all initial con-

ditions apparently clump at a single point which

Xl

i

i

jumps randomly from place to place. Fig. 4e

shows the location of this point at several succes-

sive times (n = 700-705).

To see how, as time goes on, an initial cloud of

particles finally concentrates on a fractal, one

might naturally start with a particle cloud C,

uniformly distributed in x and in I yl 5 y,,,, and

let it evolve under the random map (3), C, = T,C,,

C, = T,T,C,, . . . , C, = T, . . . T,T,C,, etc, where

T T,,... 1,“‘, are a specific sequence of realiza-

tions of the random map. Typically, if this is

done, as the cloud contracts, the next cloud im-

age is not contained in the previous one, and the

snapshot attractor changes its shape and location

randomly on each new iterate. In order to graphi-

cally illustrate the contraction of the cloud to a

fractal, we eliminate this random jumping behav-

ior of the snapshot attractor by making a differ-

ent sequence of cloud images: C’, = T,C,,, C; =

T,T,C,, . . . , CA = T, . . . T,,C,,, etc. Thus image CL

has the same last n - 1 random map applications

as the image Cl, _ ,. That is, we make the length of

the time series longer by one time unit by adding

an additional random map at the beginning of the

sequence, rather than at the end. This has the

effect of eliminating the large random changes in


Y

-3 L 0

1.0

i

-0.2 t

t -".40-

/ (4

71 2lC

X

0.2

-0.2

-0.6

X

0.6 I 0105

I /

Fig. 4. Snapshot attractors for random Zaslavsky map at: (a) a = 0.09, k = 0.5; (b) (Y = 0.14, k = 0.5; Cc) a = 0.19, k = 0.5; Cd) (Y = 0.24, k = 0.5; (e) (Y = 0.5, k = 0.5, and we plot the point snapshot attractors at consecutive steps n = 700-705.


Fig. 5. Overlay image of three snapshots of the particle distribution evolved from a uniform cloud for the random Zaslavsky map

at (Y = 0.09, k = 0.5, using the numerical method described in the text. The initial cloud is uniformly distributed in I yl s 15. The

snapshots are taken at t = 22, t = 28, and t = 70.

the cloud that occur when each new random map

is applied at the end of the time sequence. Note

that under this modification, Cl, tends to be a

subset of CL_ i. Fig. 5 shows an overlay of three

cloud images at successively larger times gener-

ated by this method. At t = 22 the cloud occupies

the red, green, and yellow regions. At t = 28 the

cloud occupies the green and yellow regions. At

t = 70 all the points in the cloud have concen-

trated in the yellow region. Thus as time goes on,

the cloud image contracts to a smaller and smaller

area, and eventually limits on a fractal image.

Remark. In (3) we have taken 0, to be random

so as to model a chaotic time dependence of the

tluid flow velocity. If instead we were interested

in a fluid flow which was periodic in time,

v(x, t> = 4x, t + T), then the locations of floating

particles at times t = nT + t, would be described

by a nonrandom stroboscopic map. In such a

situation, the transition to chaos (and a fractal

particle distribution) with variation of a system

parameter follows the usual routes commonly

found for two-dimensional maps. (A model of

such a situation would be the flow resulting in (3)


but with 19(t) and e,, replaced by a constant phase.) If one considers the distribution of particles which eventually evolves from an initial cloud, then a period doubling bifurcation from period N to period 2N, for example, would be observed as a transition to a situation where the particle distribution clumps at 2N points from a situation where it clumped at N points.

3. Dynamics in the transition region for the

random Zaslavslq map

Ledrappier and Young have proved that the Lyapunov dimension is equal to the information dimension of the measure of the snapshot attractor for a random diffeomorphism [3]. The Lya- punov dimension is defined as [4]

D,=k+*’ k+l

where A, >A,> . . . 2 A, are the Lyapunov exponents for a d-dimensional dynamical system, and k is the largest number such that the sum in the numerator is positive. Here we deal with the case d = 2. According to (41, D, = 0 if A, < 0, and

Dr=l+&

if A, > 0 and A, + A, < 0. (The latter condition always holds in our case, A, + A, = --(YT.) Eq. (5) implies that near the transition point, as A, approaches zero from positive values, the dimension of chaotic snapshot attractors approaches 1. The dimension then jumps to 0 when the system becomes nonchaotic. This is illustrated in fig. 1. Because the change in dimension is from one to zero at the transition point, one might expect that one-dimensional random maps will capture the basic features of the transition. (To test this we will consider the random circle map in the next section as an illustration.)

For the case we investigate numerically in this section (i.e. the random 8, are independent and

uniformly distributed in [O, 2~11, whenever we make a picture of the snapshot attractor it appears to be one connected piece (i.e. there is one attractor), and when the distribution tem- porarily clumps in a small region it does so in only one such small region, and not several (clumping is common near the transition point). Therefore, it makes sense to define a size of the snapshot attractor. We define the (vertical) size as the root mean squared deviation of the y coordinate of the points on the attractor from their average. To compute the size, we sprinkle initial points uniformly in a region of the phase space. Suppose after a number of iterates, they concentrate on a snapshot attractor, and the y coordinate of the ith particle is y’; then the size of snapshot attractor is defined as

where N > 1 is the number of points, and

(y)=$ cyi. i=l

(In defining the size s, we use y rather than X, because x is an angle variable and this makes a reasonable definition using the spread in x some- what awkward.) In fig. 6, we plot this size averaged over 10000 time steps as a function of the largest Lyapunov exponent for the random Zaslavsky maps.

Our analytical results, obtained later in the paper, suggest that the time average attractor size approaches zero linearly with A,. The time average sizes obtained numerically shown in fig. 6 are roughly consistent with this, but we note that the statistics becomes particularly bad near the transition A, = 0. This is also to be expected, as we shall discuss later. In particular, if we average over longer time, the statistical fluctuations in fig. 6 will decrease, and this has been verified by examining similar plots averaged over less than


0.30 I I I /

0.25

0.20

(S)" 0.15

0.10

0.05

0.00

the fraction of the time the attractor has size of

order 1. We plot the intermittency index versus

the largest Lyapunov exponent for the random

Zaslavsky map in fig. 8. This index apparently

diverges when the largest Lyapunov exponent ap-

proaches 0.

I , I 1

0 0.005 0.010 0.015 0.020 0.025

Note that, near the transition, at particular

time intervals, the size can become extremely

small. This poses a problem for the numerical

computation, since we find that the size can con-

tract so much that the resolution (imposed by the

computer roundoff) can be insufficient to distin-

guish different points x’ in the small region where

they clump. Thus roundoff can result in the com-

puter assigning exactly the same coordinates to

all the x’ if, at some time in the evolution of the

snapshot attractor, the region where x’ clumps is

smaller than the roundoff. When this happens,

the discreteness of the computer arithmetic causes

all subsequent iterates of the points x’ to be the

same! To avoid this artifact, we add a very small

amount of noise (different for each point x’> to

our computation of the orbits from the map. We

have confirmed that, when the noise is sufficiently

small, the results depend little on the noise level

(see figs. 7c, 7d).

Fig. 6. Time average size of the snapshot attractors versus

the largest Lyapunov exponent for the random Zaslavsky map

with k = 0.5 and an artificial noise level lo-“.

10000 steps. The reason for this is that when the

transition point is approached, the time depen-

dence of the snapshot attractor can be very inter-

mittent. To see that this is so, we plot the size of

the snapshot attractors versus the time step n for

a realization of the random map (figs. 7a and 7b).

Fig. 7b shows s versus n with s plotted on a

linear scale, while fig. 7a shows the same data

with s plotted logarithmically. Note that in fig.

7a, the attractor size s is often extremely small

(e.g. it spends roughly half its time in the range

s I lo-“!) with occasional intermittent bursts of

order 1. To characterize this intermittency, we

define an intermittency index,

z= (s2>tl <sx ’

where (. >, denotes a time average. If s is a

constant (no intermittency), then I equals to 1.

For the case of our snapshot attractors near the

transition point the averages (s2 >, and (s), in

(6) tend to be dominated by rare events when the

attractor bursts to have a size of order one. Thus

in this case the index I is roughly the inverse of

4. Random circle map

As noted in section 3, near the transition point,

the dynamics falls approximately on a one-dimen-

sional set. This leads us to suspect that the basic

transition phenomena found in two dimensions

will also be present in one-dimensional random

maps. To test this we consider the following sim-

ple one dimensional circle map:

X fItI =[x,,+Ksin(x,+B,_,)]modl, (7)

where x is an angular variable, and the tin are

random variables uniformly distributed in [O, 2n).

Eq. (7) can be simply obtained from the random

0.6

0.4 s

0.2

0.0 0


n (x103)

4 6 8

n (x103)

! 10

10

S

10

10

IR) 10

1-O A

4 6

n (x103)

.I

8

b)

I

(d

10 4 6 8 :

n (x103)

Fig. 7. The size of snapshot attractor versus time step for the random Zaslavsky map with k = 0.5 and (Y = 0.30 and different levels

of artificially added noise: (a) lo-” (linear plot); (b) IO-‘” (logarithmic plot); Cc) lo-‘*; Cd) IO-“.

5

4

3

I

2

1

0 (b)

0.005 0.010 0.015 0.020 0.025 0.001 0.01 0.1

hl hl

Fig. 8. Intermittency index versus the largest Lyapunov exponent for the random Zaslavsky map at k = 0.5: (a) linear plot; (b) log-log plot.

Zaslavsky map by taking the limit aT - CC while

keeping K = (a/P)k fixed. In this case eq. (3b)

yields

Y,, = k gin( x,, + 0,,_ ,>, (8)

which when inserted in (3a) gives (7). The slave

variables y,, allow us to define a vertical size of

the attractor using the definition as in the previ-

ous section. That is, if we sprinkle initial point

uniformly on the circle and let them evolve, after

a long time, they will distribute on a snapshot

attractor. The vertical size of the attractor is then

computed as the root mean square deviation of y

from its mean position.

To calculate the Lyapunov exponent, it is again

useful to introduce u,, =x,, + 19,,_ ,, in which case

ld,, obeys

U ,* + I =(~,,++,,+Ksinu,)mod2~.

Thus we again have that the quantities u,, may be

regarded as independent identically distributed

random variables with uniform distribution in

[0,2~). Using this information the Lyapunov ex-

ponent may be expressed as an average over u,

A= /

21rln(l +Kcosul~ 0

= ln(iK), K> 1,

= ln($(l + -)I, K< 1.

We note that, except at K = 1, the Lyapunov

exponent is a smooth function of the parameter

K and is negative for K < 2, positive for K > 2

and crosses zero linearly at K = 2 (the transition

point)#‘.

In what follows we report numerical experi-

ments on the one-dimensional model, eq. (7). The

salient point is that these results are essentially

the same as for the two-dimensional case. When

#‘One-dimensional maps of the form x,+, = f,Cx,) =x, + E,(x,) where E, are random functions, have been studied

previously by Deutsch [g]. Depending on the amplitudes of

the random function E,, he finds the following three regimes

K < 2, a uniform cloud of initial conditions, after

many map iterations, eventually settles down to a

single point which, if examined at successive iter-

ates, jumps randomly in its location on the unit

circle. On the other hand, when K > 2, the at-

tractor has a finite size which fluctuates with

time. Similar to the two-dimensional case, as the

transition is approached from K > 2 these fluc-

tuations become more extreme, and we observe

that, near the transition point, the attractor size

jumps sporadically back and forth over several

orders of magnitude. The oscillation of the at-

tractor size slightly above the transition point is

clearly seen in fig. 9, where we plot the attractor

size as a function of time. To obtain this figure,

we sprinkle a large number of initial points uni-

formly in 0 to 2~r, iterate the map, and, for each

iterate, calculate the size defined in the previous

section using eq. (8) for y. (We choose k = 1 in

our calculation.) We observe that the size has

bursts from very small values to values of order 1.

When the Lyapunov exponent is near zero, there

are two regions of almost equal extent in x where

the local tangent map is contracting and expand-

ing. Thus as the distribution jumps between these

two regions, the attractor size contracts and ex-

pands with time.

Although, for individual realizations of the ran-

dom dynamics, the attractor size fluctuates vio-

lently near the transition, its average value is well

defined. In fig. 10, we show the attractor size

averaged over the time interval II = 0 to n = T as

a function of T. The time required before the

where the nth-iterated map g,,(x) = f,,Cf,_ ,(. .f,(x). )) for large n exhibits qualitatively different behaviors: (1) the func-

tion g,, is monotone and consists of large flat steps; (2) the

function g,, is non-monotone, however, it mainly consists of

large flat steps except on a set of zero measure where it varies

wildly; (3) the function g,, varies wildly everywhere. In the

case of the random circle map, these correspond to the

following three parameter regimes: 0 < K 5 1; 1 <K 5 2: K > 2. In particular, the transition from Deutsch’s regime 2 to

his regime 3 is our transition from A < 0 to A > 0. A.S.

Pikorsky 191 has also investigated the Lyapunov exponent for a

similar map.


0.8’

0.6

s

0.4

0.0 0 2 4 6

n cx103:

8

Fig. 9. Attractor size versus time for the random circle map at K= 2.1. The artificial noise level is IO-“‘: (a) linear plot;

(b) logarithmic plot.

0.15 :

(S)” 0.10 ’

1L /-------------

0.05

0.00 L_A. , A~_ I.Lu _-_I 0.0 0.5 1.0 1.5 2.0

n (x104)

Fig. 10. Average attractor size as a function of time at K =

2.1. The artificial noise level is 10V’4.

average stabilizes can be very long when the map

is near the transition point. For K = 2.1, or A =

0.05, the required time is of the order of lo4

iterates. This time apparently tends to infinity as

the transition point is approached from above.

(As in the two-dimensional case, due to limitation

of finite computation time, this is the primary

cause of statistical error in our estimate of the

average attractor size near the transition.) As in

the two-dimensional case, small levels of noise

were added to eliminate artificial permanent

clumping. To check if the small numerical noise

we added in the simulation affects the attractor

size, we ran our calculation for several different

noise levels. Again we found that varying the

level of the small added numerical noise over

several orders of magnitude (lo-“’ to 10-14) has

little effect on the average attractor size.

We also investigate the probability distribution

of the snapshot attractor size s. Letting

P(llog s’I)dllog ~‘1 be the probability that [log SI is

between Ilog $1 and Jlog s’l + dllog ?I, we find that

P(Jlog $1) has the typical form shown in fig. 11. To

obtain P(llog sl) in fig. 11, we generate a long

time series (of order lo5 iterates) for [log sl and

evaluate the fraction of those that fall in each bin

of a uniformly spaced partitioning of the llog sl

axis (intervals of length Allog SI = 1 were used).

From fig. 11, we see that for a wide range of s

values (Jlog SI r lo), log P decreases linearly with

llog $1 or

P - exp( -cullog sl), (9)

where CY is a constant.

Having investigated the case where the phase

angle in eq. (7) is uniformly distributed in [O, 27r),

P lo-”

1p 0 20 40 60

En ( l/s )

Fig. 1 I. The probability distribution I-‘(llog vi) ot attractor

size .Y for the random circle map at K = 0. I.

we now consider a case where the phase angle

undergoes a random walk,

where 0 < E < 2~ and t,, is uniformly distributed

in [O, 1). Letting CL,, =x,, + H,,_ ,, cq. (7) becomes

cl ,I+ I = (u,, + E(,, + K sin u,,) mod 2~.

When E = 0, this is a deterministic system, and

the transition to chaos follows the usual period

1

0

-1

h -2

-3

-4

-5

3)

2 4

K

doubling cascade. On the other hand, when E =

2~, the system is equivalent to (7). For E between

0 and 2~. the random walk in H,, products corrc-

lation (i.e. the average (costs,, + H,,)cos(x,,+ , +

0 ,,+ ,)) f 0). For this case, we observe that typi-

cally the transition behavior is similar to the fully

random case E = 2~r. That is, the transition oc-

curs at A = 0; near the transition point, the

Lyapunov exponent scales linearly with the pa-

rameter (this is shown in fig. 12a for t = 0.1 with

k = 3.570 at the transition); and the snapshot

attractor has extremely intermittent temporal os-

cillation (fig. 12b). However, one important dif-

ference from the fully random case is that. when

passing through the transition point, the snapshot

attractor generally changes from several points to

the same number of disjoint chaotic bands. For

the cast shown in fig. 12 tt = 0.1). below the

transition point, the snapshot attractor consists of

two points. At K = 3.6, above the transition point.

the attractor consists of two disjoint bands in .I-.

The observation of two disjoint bands at the

transition. in this lower amplitude noise case. is a

manifestation of the effect of noise on period

doubling cascades [lt)]. In fig. 12b, WC plot the

size of the attractor as a function of time for one

of the two bands. Indeed, the attractor size varies

very intermittently with time.

U.8

0.6

(b)

n (x10”)

Fig. 12. Plots of (a) the Lyapunov exponent and (b) the attractor size versus time for the caw where the phase undergoing a

random walk with t = 0.1.


5. Soluble models in one dimension

5.1. Simple contraction expansion model

In the previous section we have verified that a one-dimensional map captures the basic two- dimensional transition phenomena seen in section 3. Thus we are motivated to seek a simple soluble one-dimensional model which will enable us to better understand what we have seen in our numerical experiments. We consider the following random one-dimensional process:

X ,,+,=Nx, mod1

with probability p,

1

(lOa>

X n+l =xx?l

with probability q = 1 -p, ( 1Ob)

where N > 1 is an integer. The Lyapunov exponent of this map is

h=plnN+qln&=2(p-$)lnN.

A is a linear function of the parameter p, and the transition occurs at p = +. Eqs. (lOa> and (lob) capture the two features which appear to be essential ingredients for the universal properties of the transition to chaos for random maps: first, eqs. (lOa) and (lob) are a multiplicative random process which incorporates both expansion (eq. (lOa>> and contraction (eq. (lob)); second, there is a limit to the attractor size (the mod 1 in (lOa)). For our two-dimensional model, the random Zaslavsky map (eqs. (3a) and (3b)), the limitation in attractor size in x results from the fact that x is taken modulo 27r, while the limitation in y comes from our previous discussion which showed that the snapshot attractors must lie in the strip lyl5 k(1 - ePaT)-‘.

Suppose that at n = 0 we take the distribution to be uniform in [O, 1). Then, at any subsequent time, the distribution will be uniform in some interval [O, X,,) where X,, can take values

(1, l/N, 1/N2,. . . , l/N’, . . .). For the purposes of this section we take X,, to be the attractor size (in terms of the size s defined previously, X = 26s). Thus we can use 1 as a discrete label for the attractor size. Let P(I,n) be the probability that the attractor has size l/N’ at time n, then PC/, n) satisfies the following equation:

P(l,n) =pP(l+ l,n- 1) l tqP(l- l,n- 1)

for 1 L 1, (lla)

P(O,n) =pP(l, II - 1) +pP(O,n - 1). (lib)

Note that a size l/N’ (I 2 1) can be realized by either multiplying a size l/N’+ * by N (with probability p) or by dividing a size l/N’-’ by N (with probability q); thus we have (lla). But a size 1 can only be realized by multiplying a size 1 or l/N of the previous step by N; hence we have (lib). This can also be expressed by writing I, = log,X,l, in terms of which eqs. (10) give

1 n+,=~n+l with probability q,

1 n+, = (1, - l)u(l,) with probability p,

where 40) = 0 and u(l,,) = 1 if ln = 1,2,3,. . . .

Thus we have a random walk on 1 which is asymmetric if p # $.

5.2. Behavior in the chaotic regime

We solve eqs. (11) in appendix B. The important result is that, for p > i (i.e. the Lyapunov exponent is positive), P(1, n) asymptotes to a unique stationary distribution P(1) as n + CQ. This stationary solution is given by

(12)

where? =(2-p-‘)and V= -ln(p-‘-1)which is the form given in (9). For p close to 4, v = n = 4(p - i), and the distribution has an e-folding width i(p - iI_’ which diverges as p -+ 4. Eq. (12) can be checked by substitution into the stationary difference equation (obtained by omitting

the n dependence in eqs. (II)),

P(I) =@(I+ 1) +@(I- 1) for 1s 1.

P(0) =pP( 1) +pP(o).

The average size of the attractor is

2-l/P = 1 -(l/N)(l/p- 1).

Note that for p > i, close to the transition, the

size 1 /N’ decreases very rapidly with 1 compared

to the decrease with I of the stationary distribu-

tion P(I) (which has an e-folding width in I of

approximately $p - +>V’). Thus we see that near

the transition the main contribution to the above

average is from small 1 and hence the average is

dominated by rare events such that the size of the

attractor is of order one. Furthermore, (I) a

(In s) diverges near the transition as (p - +)-I.

Hence, although the attractor size occasionally

bursts to a size of order one, for most of the time,

the attractor size is exponentially small near the

transition. To demonstrate this, using eq. (12) we

note that the fraction of time f the value of 1 is

greater than 1,, is determined by j;P(l)dl= f, yielding I,, = (l/v)ln(T/fv). For example, for f =

i, Ip-pCI< 1, and s= l/N’, we have that for

half the time, s satisfies the condition

s < exp[ -K/(p -PC)], (13)

where K = (In N)(ln 2)/4. The intermittency in-

dex defined in section 3 is

(l/N2’>, (l/N'): =

2- l/P 1 - (W2)(l/a - 1) ii

2-l/P 2

1 - (l/N)(lh - 1) ’

Thus 1-(/.,-$’ near p = i, and we see that

the intermittency index diverges as the Lyapunov

exponent approaches zero (i.e. p approaches +)

with a critical exponent of - 1.

Let 3 denote the “bursting time” defined as

the time during which the attractor size is less

than some threshold value. In appendix B we

consider the probability distribution of bursting

times P(A) for our model, eq. ( 11). We are able

to obtain a closed form solution for P(d). The

main result is that for large 3, this distribution

has an exponential tail,

P(il) - exp( -53).

where the characteristic time between bursts di-

verges as l/l - lp -pJ* near the transition.

To see under what conditions our addition of

noise (as done numerically in sections 3 and 4)

does not affect the time averaged attractor size,

we note that, if the noise amplitude is of order

l/N”, the size of the snapshot attractor cannot

shrink much below this value. Thus P(l) is essen-

tially forced to zero for 1 Z+ q. If, however, CJ is

large compared to the width of P(I), vq ZP 1 (or

q > i(p - i)F’ near the transition), then P(I)

will be little changed for 1 values of the order of

one. Since these low 1 values dominate the sum

CTZC,,N-‘P(I) = (N-l), the time averaged attrac-

tor size will be little changed for vq Z+ 1. (We

have verified the above arguments by solving our

model with a boundary at 1 = q past which proba-

bility transport is blocked.)

Note that our assumption of an initially uni-

form distribution in [0, 1) is not restrictive, since,

for p > i and any smooth initial distribution, a

large time asymptotic snapshot distribution will

become uniform in some interval [O, N-‘), I=

0,1,2 ).... To see this, we note that the stable

invariant measure for the nonrandom map

X ,,+ I = Nx,, mod 1 is uniform and, furthermore,

that almost all realizations of random process

(10) contain sequences where (lOa) is applied

arbitrarily many times in a row.


0.8

0.6

s

0.4

J -200 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

n (x104) n (x104)

Fig. 13. The attractor size versus time for the simple contraction-expansion model with N = 2 and p = 0.499: (a) linear plot;

(b) logarithmic plot.

5.3. Behavior in the nonchaotic regime near the transition

We also solve eqs. (11) in appendix B for

p < i. No stable finite solution P(I) exists in this

case. This is consistent with the fact that when

contraction dominates expansion, for almost all

realizations, the size of the attractor eventually

goes to 0, corresponding to 1= +m. In figs. 13a

and 13b, we plot the attractor size versus time for

the simple contraction-expansion map for the

parameter below and near the transition point on

both a linear and a logarithmic scale. Initially, the

variation of size exhibits intermittency-like behav-

ior. However, after a certain time n, the attractor

size becomes small and remains small even after.

This is because, as time goes on, there is less and

less probability for the attractor size to be of

order 1. The asymptotic probability distribution is

estimated in appendix B. When n is large and p is close to but less than t, we have for fixed 1

P( I, n) - (2fi)” - e-“‘/2-p)2n.

Therefore, the typical time for the attractor size

to be small and stay small is

% N 1 1 1

ln(2fi) N (+ -P)~ = (P,-P)~’ (14)

This time diverges as p -+ $.

When the attractor size becomes small, as

shown below there is a nonzero probability that

the attractor size never gets back to 1. We have

calculated the probability Q&I that the attractor

never attains size 1 if it initially has size N-‘0. In

our simple contraction-expansion model, this is

equivalent to starting the random walk of a parti-

cle at I= I, and computing the probability that

the particle remains in 1 > 0 for all time. This is

also equivalent to evolving a particle probability

distribution function from an initial delta func-

tion at 1 = 1,) and imposing the boundary condi-

tion that the distribution function is zero at I= 0

(i.e. particles are immediately removed as soon as

they get to the boundary 1 = 0). The total proba-

bility in 1 > 0 at t = m is then Q(l,). Using (lla),

and the following initial condition and the bound-

ary condition,

fYl,O) = %,r,p ( 15a) P(0, n) = 0, ( 15b)

a detailed calculation for this model is carried out

in appendix B. The result is

Q(I,,) = c P,)I,,(I,m) = 1 - (p/q)“‘= 1 - ePt’“. I=0

( 16)

Near the transition, the decay exponent is < =

4(+ -p) = 4(p, -p). Thus, consistent with the

time evolution of the attractor size in figs. 13a

and 13b, the smaller the initial size (i.e. the larger

I,,), the less chance there is for the attractor size

to get back to 1.

5.4. Model with distribution of expansion rates

To take another point of view, we can rewrite

the evolution equations (11) in a diffusion form

on l-space,

S,, P + ~6, P = D6;P,

where

(17)

D=+(p+q)=& i--A,

6,,P(I,n) = P(l,n + 1) - P(l,n),

6,P(l,n) = +[P(I+ 1,n) -P(/- l,n)],

&?P(l,n) =P(l-t 1,n) +P(I- 1,n) -2P(I,n).

Written in this form, we see that the evolution

equation of the probability distribution is basi-

cally a discrete version of the differential equa-

tion describing the evolution of a density which

undergoes diffusion and convection in l-space.

Here D is the diffusion coefficient, and 1% repre-

sents the convective drift velocity. Note that I‘ < 0

(~1 > 0) for p > i (p < $>. Thus again we see that

the underlying dynamics of the map is an asym-

metric random walk in l-space. When p is greater

than and close to pc = i, the width of the station-

ary distribution becomes large compared to one.

Thus, to describe the relaxation of an initial dis-

tribution of large width to the stationary distribu-

tion, we can use continuum approximation to the

discrete evolution equation (17). Hence in this

case P(I, 17) satisfies the partial differential equa-

tion

( 1x1

Actually we believe that (1X) applies more gen-

erally than just for the model equations (10). To

see this, one can generalize the model and con-

sider a map with a distribution of the expansion

rate,

X ,I+ I = a,, X,, mod 1, ( 19)

where LY has a probability distribution F(a). The

Lyapunov exponent is

A = i

F(cr)Inrudcr.

Again consider an initial distribution in [O. 1). At

any subsequent iterate, points will be distributed

in the interval [0, X,,) (the distribution will not, in

general, be uniform), where X,, = 1, and X,,

evolves according to the rule

X lItI = I if N,~X,, 2 1,

= ai,, x,, if CY,,X,~ 5 1 .

Again we regard X,, as a measure of the size of

the snapshot attractor. Let h,, = -In X,,. Then

for X << 1

h ,?+I= -In cy,, + h,,.

This is an additive random process. Let P(h, n) be the probability distribution function of h at

time n. Near the transition, we expect, as for the

simple contraction-expansion model, that the

variation of P(h, n) with h is slow compared to a

typical step in h-space (given by In (Y,,). Hence


the continuum approximation again applies,

l3P 2

;i;;+cg=D~, (20)

where L’ = -A is the average drift in h and

D = /<ln a - A>2E’(a> d (Y. Since A is small near

the transition, D = j da F(a)(ln aj2.

When A > 0, similar to the calculation in ap-

pendix B, one can show that P(h, n> asymptotes

to a unique stationary distribution,

P(h) = $C-(A/o)/J.

Again we evaluate the average size

which is proportional to the Lyapunov exponent,

and,

(X2) = ~~le-z/f$e~‘“/““‘dh,

1 A =7j~ forA<l.

The intermittency index is

I= (x”) 1 D >>l -E _- (x)2 2 A .

Thus the intermittency index I again diverges as

A - ’ near the transition.

When A < 0, for finite h, the solution of (20)

asymptotes to zero exponentially as time tends to

infinity,

p( h, n) - e-([~*/4D)n.

The typical time scale for the decay is

40 1 n0 -7--.

u- A2

Thus this time again diverges as 1 p - pclp2 as in

(14). By solving the diffusion equation (20) under the

initial and boundary condition (1.51, we obtain the

probability that the attractor size newr attains 1

if it initially has size s = eP’“,

Q( lo) = 1 _ e’-“/D’[~~ = 1 _s[./D. (21)

This is consistent with the result for the simple

contraction-expansion model, eq. (16).

6. Conclusions

In this paper we have attempted to study the

qualitative behavior of the evolution of a distribu-

tion of particles floating on the surface of a

confined fluid whose velocity u has complicated

time dependence. We have argued that noncon-

servative two-dimensional random maps are good

models for this situation. Our analysis has thus

concentrated on discovering the properties of dis-

tributions of points as they evolve under succes-

sive applications of random maps. Thus our

results, although motivated by the problem of

particles floating on a confined fluid surface, may

have more general interest. Indeed we believe

that the basic mathematical questions raised in

this paper will appear in other physical contexts

[ll, 121.

Our principal results are the following:

- There is a transition with parameter varia-

tion from a time asymptotic fractal distribution

(I, >p,) to a time asymptotic distribution where

the particles concentrate at a single point (or

several points) in space (p <p,).

- Near this transition there is a tendency for

extremely intermittent temporal fluctuations in

the particle distribution. Between intermittent

bursts the attractor size is exponentially small,

s - exp[ -K/(p -p,)l. Furthermore, the distribu-

tion of the intermittent bursting time A has an

exponential tail, P(A) N e-c(p-p~)*d, for large A. - For p near pc on the chaotic side (p >p,>

the time averaged attractor size scales as (s), -

(p -I),); the intermittency index scales as I _

(p -p,)-‘.

- For p near pc on the nonchaotic size ( p <p, 1

the attractor size initially fluctuates intermit-

tently, as in the chaotic case, but eventually damps

to zero in time typically of the order of I p - pclpz.

Acknowledgement

This work was supported by the Office of Naval

Research (Physics), Defense Advanced Research

Projects Agency, and Department of Energy (Sci-

entific Computing Staff, Office of Energy Re-

search).

Appendix A. Microcanonical ensemble

approximation for calculating the largest

Lyapunov exponent of random Zaslavsky map

The Lyapunov exponents are given by the

eigenvalues of the product of the following ran-

dom matrices

I 1

M(qJ =

, k cos B,, e paT +

where (0,, II = 1,2,3.. . } are independent identi-

cally distributed random variables with probabil-

ity density p(0) = l/2 57, for 0 E [0,2rr). We will

estimate the largest Lyapunov exponent using the

microcanonical ensemble approximation [13]. The

largest Lyapunov exponent is defined as

y = lim I lG,v St01

N-m Xhr ]&$“I ’

where G, = n,“=,M<0,), and se,, is a generic

vector. It has been proved that y has the same

value for almost ecery realization [71, y = A,.

Hence A, can be written as

where the angle brackets denote an ensemble

average. We now state the result of ref. [13] and

apply it to our problem. Suppose 0 has a proba-

bility distribution p(0) (jp(r9)de = 1). Then the

largest Lyapunov exponent is

A = d0(p(0)lnp(H) +P(B)lnh[E(H)] /

-dWn[WQl} (A.1)

where the functional A,[xl is the largest eigen-

value of lM(0) x(o)d@, and x(0) is the solution

of the following functional equation:

64x1 “Wgx =P(Q) A[xl. (‘4.2)

For our problem the eigenvalue of jM(H) x(0)d0

is

+&(2-I)x,,+M+ 4(1-r)xi ( )

(A.3)

where r = 1 - em”’ characterizes the dissipation,

w = [(l - e-L’T>/~]pk is the amplitude of the

random variable, XJX] = /x(B)dtl, and x,[xl =

jx(8)cos 8 do. Since by eq. (A.3) the quantity A is

a first order homogeneous function of x,, and x,,

we have

6A aA 6x,, ilA 6x, aA aA -= 6x ~6x+ax,X=~+as,cos”7

(A.4)

where we have utilized SX,,[X]/~X = 1 and

Sx,[x]/Sx = cos 8. From the homogeneity of

(A.3) we see that ah/&, and ah/an, depends

only on the ratio h -x,/x0, and thus the largest


k

Fig. 14. Comparison of numerical result (solid line) and mi-

crocanonical ensemble approximation (dashed line) for the

largest Lyapunov exponent versus parameter k at a = 0.

Lyapunov exponent is determined when h is

determined. To accomplish this, we use eq. (A.31

to write

(A.51

Using (A.4) in (AS) and integrating both sides

with respect to 0, we find that h is the solution of

the polynomial equation

3w2h4 + 8(2 - r)wh3 + (4r2 + 2w2)h2 - w2 = 0.

(A.6)

Once h is determined, x(0) follows from (AS),

which when substituted in (A.l) yields the Lya-

punov exponent A,. In carrying out this proce-

dure it is necessary to determine which of the

four possible roots of the polynomial for h is the

relevant one.

In fig. 14 we plot the numerical Lyapunov

exponent and the one predicted by the above

approximation. They seem to agree well with

each other. Even when the parameter k is large,

the prediction agrees well with the numerical

result as shown in fig. 14.

Appendix B. Solution for eq. (11)

Let

I- 1

Q(f,n> = CP(i,n). WI i=O

Then P(f, n> = Q(l + 1, n) - Q<l, n). Eq. (lla) is

then equivalent to

Q(l,n + 1) =pQ(l+ 1,n) +qQ(l- 1,n) (B.2)

for 12 2, and (lib) becomes

Q(l,n + 1) =PQ(‘L~>. P.3)

If we let Q(0, n) = 0, then eq. (B.3) is the same

equation as (B.2) for 1 = 1. Thus, by our introduc-

tion of Q in (B.l) our problem reduces to that

of solving (B.2) with the boundary condition

Q(0, n) = 0. Note from the definition of Q in eq.

(B.l) that, since P(I,n) is a probability, we must

have that Q(m, n) = 1. Our problem thus reduces

to that of solving (B.2) in 1> 0 subject to an

initial condition Q(l,O> and the boundary condi-

tion Q(0, n) = 0. We wish to extend the domain

of (B.2) to the entire range --03 < 1 < 03. To do

this we use the method of images. That is, we

choose an initial condition Q(l, 0) in I < 0 such

that with the already specified Q(l,O> in I> 0 the

solution of (B.2) in the whole domain - ~4 < 1 < 00

satisfies the condition Q(0, n) = 0. To proceed we

assume that such an image approach is possible

(subsequently we shall show how to specify the

required image initial condition in 1 < 0). If we

view I as sites on one-dimensional discrete space,

eq. (B.2) represents an additive cellular automa-

ton. We define a characteristic polynomial (or

generating function, cf. ref. [ 141)

A(x,n) = 5 Q(l,n)x’. (B.4) I= --m

Because XT= _-Q(l k 1, n) x’ =x ’ ‘A(x, n), multi-

plying A(x, n) by x ” shifts the values of sites I

to sites 1 i 1. So the time evolution of the cellular

automaton is represented by multiplying the

characteristic function by a dipolynomial PY’ +

q.x >

A(x,n+l)=(px~‘+qx-)A(x,n). (BS)

This is an additive process (that is, the distribu-

tion resulting from the sum of two initial distribu-

tions, A(x,O) + B(x,O), after y1 steps is just the

sum of their final distributions A(x. n) + B(x. II )).

Hence it suffices to consider the evolution of

generating functions x’, j = 1,2,3.. to describe

the evolution of any initial distribution. Ignoring

the boundary condition for the moment, we

write the generating function at step ~2 as

A,(x,n) = (px-’ +qx)Y. (B.6)

When n <j or the difference of n and j is an odd

number, the term containing x0 is absent from

(B.6), and hence Q(0, n) = 0; when n > j and is

an even number the coefficient of the x0 term in

A;(x, n) is

where C;,’ is the binomial coefficient. Defining

C”J” = 0 if p is odd or if 0 <p/2 I n is violated,

tte above equation for Qj(O, n) now holds for any

j 2 0. Now let us consider the evolution of .rPk

where k is positive. This is an initial distribution

on sites with negative 1. By replacing j by -k, we

obtain the coefficient of the x0 term of the gener-

ating function at step n,

g_,(O,n) = C(n~k)/2P(n-k)/Zq(n+k)/2 I?

Setting k = j and noticing that Ccn+‘)12 =

C(“P’)/z, we have Q,(O, n>/Q_,(O, n) 1 (p/q)‘,

which is independent of n. Using this fact,

we obtain an initial distribution which keeps

Q(0, n) = 0. The initial condition on its generat-

ing function is

/4,(x,0) =x’- (p/9)‘Y’,

and thus its generating function is

A,( x,n) = (px ~‘+4_1.)“[x’-(p/q)‘x~‘]. (B.7)

By expanding the polynomial A,(x, n) we tind the

contribution of an initial distribution concen-

trated on site j to site 1 at step ?I,

Q,(,,n) = p-;y t+/)/Z _ C/l,” -t IV?)

Note that Q,(l, n) = 0 if n - I + j is odd. Also

note that the first term contributes only if

II ~ jl I n; while the second term contributes only

if /I + jl I tz.

Now we look at the evolution of the initial

distribution Q( j,O) = a,. Note that lim,,, a, = 1

because CT_, P(I,O) = 1. The quantity Q(l, n) is

the sum of contributions from all N, weighted by

Q,U. n ),

Q(l,n) = &Q,(W

(B.9)

Letting j = 2k - II + 1 in the first sum and j =

2k - n - I in the second sum, eq. (B.9) becomes

Q(l,n>= ; ~2hh+lcPk4N-A X=[(n-/+ I)/21

-(UP)’ c u,k_,,_,c,~P”q”-“, k=[(n+/+ I)/21

(B.10)

where we have used C” = CHPk I, I, . Note that

C,lP% 'lek is the k th term of the expansion of

L. Yu et al. /Flouters on a fluid surface 123

(p + LJ)~ = 1. Thus EkC,kpkqflmk = 1. Further-

more, if n is very large, Ctp”q”Pk has a sharp

peak near k =pn. The width in k of this peak is

R(G). In particular, for large n the width is

large compared to one but small compared to the

range of k over which the sums are performed.

Thus, if p > i, the peak of CipkqnPk will fall

within the limits in k over which the sums are

performed, both sums approach 1 for large n.

(We use a 2p,1_,1+, + 1 as n becomes large.) Then

we have a stationary distribution

Q(l) = 1 - (q/d. (B.ll)

This is equivalent to saying that P(I, n) ap-

proaches a stationary distribution as n -+ co,

P(f) = Q(l+ 1) - Q(j)

= (1-4/PH4/P)’ = (2 - l/p)( l/p - 1)‘. (B.12)

Thus we obtain the principal result of this ap-

pendix: any initial distribution P(I,O) evolves to

the unique stationary state given by (B.12).

To calculate the probability distribution P(A)

for the bursting time A, we consider a snapshot

attractor with a threshold size N-‘. Then P(A) is

the probability that the attractor size remain be-

low N-j until after A steps. This is equivalent to

evolving an initial probability distribution of size

Q(l,O> = Sj, with (B.2). The requirement that the

attractor size stays below N-j leads to a bound-

ary condition Q( j, n> = 0. Because an attractor

with size N- (j+‘) has probability p to get back to

size N-‘, we have P(A) =pQ( j + 1, A). Thus we

adopt the solution to (B.2) and (B.3) directly

(with a shift of the boundary from the 0th to the

jth site) and obtain

P(A) = (C;/;-’ - C,“/;-‘)( pq)“‘. (B.13)

Note P(A) = 0 for odd A because the size can

not get back to the initial size in odd steps. For

large A, using Stirling’s formula,

P(A) = Ad& (4pq)3’2 - e-<-‘,

(B.14)

where i= -lnG=2(p-+)‘as p-i.

When p < $, the peak of C,fpkq”-” does not

fall within the k limits over which the two sums

are performed. In this case, P(f, n) approaches 0

as n tends to infinity for all finite 1. This is

because, as time goes on, the probability weight

becomes more and more concentrated at I = + co,

corresponding to the attractor being at the point

x = 0 (l/N’+ 0 as I + m). To see how the initial

distribution finally relaxes to a single point, we

look at the behavior of Q(r, n> for large n. Using

Stirling’s formula we have

k/n I-k/n t1

cipkqn -k _ (+)f:“(lq_ k,n)l-k/”

= [F(k/n)]“,

therefore, for large n,

(B.15)

Q<l, n) - ; c;pkqr!-h

k=[n/2]

- i [F(k/n)l”. k=[n/21

(B.16)

When n + 03, the sum is dominated by the largest

F(k/n), which equals 2\lpq. Thus we have, aside

from a prefactor depending on I,

Q(l, n> - (zfi)“,

and hence

P(l, n) - (2fi)‘l. (B.17)

That is to say the probability that the attractor

has size l/N’ decays as (26)“.

The solution for eq. (lla> under the initial and

boundary condition (19 is given by (B.81,

(n-/t/,,)/2 P,,,(L n> = c,, P (II--l+/,])/2 (n+/-/,,)/2

4

- ( P/%PC, (n-I-l,,)/2

P (rl-l-I,,)/2 (N+/+/,,)/Z

4

Therefore the probability that the attractor size

nelvr attains 1 if it initially has size N-‘ll is

Using the same argument as in obtaining eq.

(B.ll), in the limit II + m, when p < i, both

peaks of the two binomial distribution terms of

P,,I(l,m) fall within the summation ranges of I,

giving

P(b) = 1 - ( PhP.

References

[I] F.J. Romeiras. C. Grehogi and E. Ott, Phys. Rev. A 41

(1990) 784.

121 L.Yu. C. Grebogi and E. Ott, in: Nonlinear Structure in

Physical Systems, eds. L. Lam and H.C. Morris (Springer,

New York, 1990) pp. 223-231.

[3] F. Ledrappier and L.-S. Young, Commun. Math. Phyh.

I17 (IYXX) 52’).

[4] J.D. Farmer, E. Ott and J.A. Yorke. Physica D 7 (19X3)

15.1.

[S] L. Yu, E. Ott and Q. Chen. Phya. Rev. Lett 6.5 (1090)

2Y35.

[6] G.M. Zaslavsky, Phys. Lett. A 60 (197X) 145.

[7] J. Cohen. II. Kesten and C.M. Newmann, eds.. Random

Matrices and Their Applications, Comtemp. Math. Vol.

SO (American Mathematical Society. Providence. RI.

I YX4).

[X] J.M. Deutsch, Phys. Rev. Lett. 52 (lY84) 1230; J. Phyq. A.

IX (1985) 1440.

[9] AS. Pikorsky, in: Nonlinear and Turbulent Processes in

Physics, eds. R.Z. Sagdeev et al. (Harwood, New York.

1984) Vol. 3, p. 1603.

[IO] J.P. Crutchfield and B.A. Huherman. Phys. Lett. A 77

(1980) 407.

[ll] J.M. Ottino. Kinematics of Mixing: Stretching Chaos and

Transport (Cambridge Univ. Press, Cambridge, lY89).

1121 E. Ott and T.M. Antonsen, Phys. Rev. A. 39 (IYXY) 3660

[I31 J.M. Deutsch and G. Paladin, Phys. Rev. Lett. h2 (10x9)

695

[I41 0. Martin, A.M. Odlyzko and S. Wolfram. Commun.

Math. Phys. Y3 (1984) 219.

Documents

Fractal distribution of floaters on a fluid surface and the transition to chaos for random maps