Modeling fractal entrainment sets of tracers advected by chaotic temporally irregular fluid flows using random maps

i

PHYSICA ELSEVIER

Physica D 110 (1997) 1-17

Modeling fractal entrainment sets of tracers advected by chaotic temporally irregular fluid flows using random maps Joeri Jacobs a,,, 1, Edward Ott a,l,2, Thomas Antonsen a,1,2 James Yorke b a Institutefor Plasma Research, University of Maryland, College Park, MD 20742, USA b Institutefor Physical Science and Technology, Universityof Maryland, College Park, MD 20742, USA

Received 3 February 1997; received in revised form 29 April 1997; accepted 9 May 1997 Commmticated by J.D. Meiss

Abstract We model a two-dimensional open fluid flow that has temporally irregular time dependence by a random map ~n-I-1 = Mn (~n), where on each iterate n, the map Mn is chosen from an ensemble. We show that a tracer distribution advected through a chaotic region can be entrained on a set that becomes fractal as time increases. Theoretical and numerical results on the multifractal dimension spectrum are presented. Keywords: Chaos; Incompressible fluid flow; Dimension spectrum: Chaotic scattering

1. Introduction The transport of passive scalar particles in fluid flows has been studied, both because of its intrinsic value in explaining phenomena like combustion and mixing, and because of its illustrative effect as a visualization of dynamics in phase space. For a two-dimensional flow, the incompressibility of the flow leads to a structure of the equations of advected particle motion, that is equivalent to Hamiltonian dynamics in phase space. The cases of chaotic particle transport in both closed [1-14] and open [15-20] flows have been investigated extensively. In the case o f open flows, these problems can often be cast in a form that makes them mathematically equivalent to chaotic scattering of a free particle by a scattering potential. In particular, as in scattering of a free particle, we consider cases where the passive scalar particles start in a region where the flow is rather simple, then encounter some region where they are transiently entrained, the scattering region, and finally end up in a region with a simple flow again. Furthermore, the incompressibility of the flow implies Hamiltonian dynamics of advected particles. The scattering is said to be chaotic if, while the particle is entrained, it experiences transient chaos. In particular, while entrained, a * COrresponding author. Tel.: (301) 405 1658; fax: (301) 405 1678; e-mail: [email protected]. 1 Also at: Department of Physics, University of Maryland, College Park, MD 20742, USA. 2 Also at: Department of Electrical Engineering, University of Maryland, College Park, MD 20742, USA. 0167-2789/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PH S01 67-2789(97)00122-X

2

J. Jacobs et al./Physica D 110 (1997) 1-17

slightly perturbed orbit exponentiates away from the original orbit so that the two orbits can end up at very different positions in the final region. In past studies of passive transport in incompressible open flows, the fluid flow has most often been taken to be time periodic. The chaotic motion will then be organized around a nonattractive invariant chaotic set of the stroboscopic surface of section map. Particle trajectories stay in the scattering region forever and never reach the final "regular" region. Therefore, points which lie close to the stable manifold of this set will follow this chaotic motion for a long time before moving off to regular motion in the asymptotic region, i.e. they exhibit chaotic transients. In this paper, we study a flow situation analogous to chaotic scattering but where the incompressible flow is not time periodic and varies in a temporally irregular manner. We will model the flow by an area preserving two-dimensional random map: ~n+l = Mn (~n), where the index n of the map indicates that, on each iterate n, the map is randomly chosen according to a given probability distribution function. Here, (n = (Xn, Yn) is a two-dimensional vector giving the particle position at time t = n T , where T is a convenient sampling time. We may think of M~ ((n) as being obtained from a flow field v(~, t) by integrating the equation d ( / d t = v((, t) from t = n T to t = (n + 1)T with initial condition ( = (n' ff the flow is time periodic, v((, t) ----v(~, t + T), the map is the same for each time interval of duration T. In the situation of a temporally irregular flow, however, this is not the case, and the map function itself varies irregularly with n. The dynamics generated by random maps has been the subject of theoretical study, and random maps have been previously used to model irregularly time varying flows [22]. On the experimental side, the authors of Refs. [21,22] have also used random maps to discuss and analyze their results. These experiments dealt with the fractal patterns, called "snapshot attractors", formed by a scum of floating particles on the surface of a fluid which was undergoing temporally irregular motion. In this paper, we will study the case where a uniform flow in the x-direction is combined with a temporally irregular vortex flow. The strength of the vortex will be chosen to vary randomly as n increases according to some probability distribution. We show that, as in the nonrandom case, the scattering is chaotic. In particular, the position where a particle "exits" the scattering region as a function of the position where it entered that region, is singular on a Cantor set of initial position values. A difference with the time periodic case is that in the random case, there will not be a nonattracting chaotic invariant set around which the dynamics is built. A set E is invariant for a map M of a time periodic flow if M ( E ) = E. Although there are no invariant sets for the temporally irregular flow case, we introduce sets, analogous to the invariant set and its stable and unstable manifolds. Numerically, we demonstrate that these sets are fractal. One kind, the entrainment set, can be observed in an experiment [23]. The latter set is analogous to the unstable manifold of a chaotic saddle for a nonrandom map. Another basic difference between the cases of temporally irregular and time periodic flows is that KAM surfaces are typically present in the time periodic case and occupy some positive measure of (-space. In contrast, for the temporally irregular case, there is no analog to K A M surfaces. The entrainment set is defined as follows. Suppose we sprinkle a very large number N of initial points in the fluid at some time. For the purposes of our discussion it will sometimes prove useful to choose the origin of time so that the initial time at which the sprinkling is done is negative. We denote this initial time ni. With ni < 0 imagine that we randomly generate the map sequence.

Mni, M(ni+l), M(ni+2) . . . . . M-I, MO, M1 . . . . . Mnv-1, Mnf,

(1)

and use it to map the sprinkled points forward in time. We keep track of those points that, once having entered a certain chosen region of interest, have not left it by the final time nf. The collection of these points at time nf is an approximation to the entrainment set. and we call the region of interest the scattenng region.

J. Jacobs et al./Physica D 110 (1997) 1-17

3

We have in mind a flow in which, far upstream and far downstream, the flow is uniform and in the positive x-direction. Initially upstream particles are swept towards the scattering region (the vortex in our example) where they execute possibly complex orbits, then leave, and are swept downstream. The scattering region referred to above should be chosen to include the region where particles are entrained. Suppose we now repeat the above procedure keeping nf fixed, but doing the sprinkling at an earlier initial time n~ = ni - no, where no > 0. To generate the map sequence from time n~ to time nf, we first randomly generate the sequence Mn I, M(ni+l ) . . . . .

M(ni-1)-

(2)

We then append this initial sequence in front of our previously generated sequence (1). Using the sequence of maps, (1) and (2), we evolve the N sprinkled particles from n~ to nf, and we again keep track of those that have not left the scattering region by time nf. This new set will necessarily be a subset of the previous one (in the limit N --+ c¢). Imagine that we repeat this process by sprinkling our points at earlier and earlier times, while keeping the realizations of the map at overlapping times constant. We obtain a nested sequence of sets which we believe in typical situations approaches a fractal set of measure zero in the limit n~ --+ - o o . We call this limiting set the entrainment set for time nf. This set is not invariant because it moves around irregularly as nf increases. (In the time periodic case where all the Mn are the same, this set would be independent of nf.) We numerically demonstrate in Section 2 that, by continuing to sprinkle points at earlier times, the entrainment set converges toward a fractal set. Experimentally, this situation is equivalent to distributing dye in an upstream region and looking at the pattern of dye in the entrainment region after most of the dye has advected through the entrainment region. Based on our results, we predict that the resulting pattern will exhibit fractal structure on smaller and smaller scale as we observe the pattern at a later time. In Section 3, we will also establish the existence of other fractal sets that are the analogs of the stable manifold and the chaotic invariant set for a nonrandom map. We call these sets the pre-entrainment set and the intermediate entrainment set. As we shall see, howevel, these sets do not have an observable counterpart in the dye experiment. In Section 4, we will derive an equation for the dimension spectrum of the entrainment set in terms of the decay time (the characteristic time for a typical particle to leave the scattering region) and the distribution function for finite time Lyapunov exponents. Box-counting estimates of the fractal dimension spectrum Dq of the entrainment set will be shown to be in good agreement with the prediction of this equation.

2. The entrainment set

The map we consider can be thought of as a model for a two-dimensional fluid flow that is composed of two sequentially applied parts. During the first phase of the fl0w, there is a uniform flow v0 in the positive x-direction. After that, all points are rotated by a vortex flow around the origin by an amount that depends on the distance to the origin [24]. The randomness of the map arises because we make the strength of the vortex different on each iterate. Our model is a modification of the model of Stolovitzky et al. [24]. Our model differs from the Stolovitzky et al. model in that we make the map random and take the core of the vortex to be soft (i.e. we do not allow the rotation angle to diverge: for points near the origin), If we represent the points in the plane by complex numbers, z = x + iy, the map reads: Zn+l

=

Zn

{ (1 - exp(-lzn'2)) } exp iKn [znl 2 + v0,

(3)

where Kn is a real number, randomly chosen at each iterate n with uniform probability distribution between O:3K and K. (In [24] the argument of the exponential in (3) is iKIzn 1-2, where K- is a real constant.) We believe that the

J. Jacobs et al./Physica D 110 (1997) 1-17

10

k

L

t

10

8

(b)

8

you

i~'

your

i! I ! 6

4

7.80

7.98

8.~5

8.32

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.y in

8.250 yin

8.275

8.300

"0

(c) j

am

8

6I 8.2200

8.22"0

8.2220

8.2230

j._

Fig. 1. Scattering function (a) and consecutive blowups (b)-(c).

results we find for this example can be regarded as indicating general qualitative phenomena common to a large class of temporally irregular open flows. We choose for the scattering region the square with side 2x0 centered at the origin of the scattering region (we work with x0 = 6, K = 5.8 and v0 = 1). We start with a large number of initial conditions upstream at x = - x 0 . For each of these initial conditions (with coordinates ( - x 0 , y)), we find the y-coordinate y ----y] at which they first map into the region x > x0. Note that for x >_ x0, the y-coordinate is approximately constant as x translates to the right. This exiting y-position as a function of the initial y-position is called the scattering function and is plotted in Fig. l(a) for one realization of the random sequence Kn. In Figs. l(b) and (c) we show consecutive blow-ups of the same graph to show that there is a Cantor set of initial y-values on which the scattering function is singular. We say the scattering function at y is singular if there exists an interval of scattering values, such that the scattering functions assumes all of these values in any y interval including the singular point, no matter how small that y interval is. In Fig. 2(a), we plot the time it takes for a point at initial position ( x0, y) to first arrive in x _> x0, and blow-ups of this plot are shown in Figs. 2(b) and (c). If one compares Figs. 1 and 2, one can see that at the same Cantor set, at which the exiting position as a function of initial condition is singular, the time to travel through the restraining region becomes infinite.

J. Jacobs et al./Physica D 110 (1997) 1-17 25O

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(o)

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(b) 1:

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iiiiiiii i iiiii

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50 o 8.2200@

I

8.22075

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8.22150

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Fig. 2. R e s i d e n c e t i m e s in scattering r e g i o n as a function of Yin.

We then obtain a picture o f the entrainment set in the following way. A t time ni -~ - 5 0 , we sprinkle a large number o f initial conditions uniformly in the square with side 12.0 centered at the origin (the scattering region). We iterate all o f these points forward and keep those that have not yet left the scattering region at time nf = 50. The positions of these points at time 50 are plotted in Fig. 3. To illustrate the fact that this set is indeed an approximation o f a fractal set up to some length scale, consider Figs. 4(a)-(c). In the first of these figures, we sprinkle points at time ni = 40, and plot their locations if they have not left the restraining region at time nf = 50. In each of the consecutive plots, we do the same, but starting at successively earlier times (ni = 30, 20). In this process, the realizations o f the random maps at times that are common with the previous picture are kept the same. We see that each time, the new set is a subset of the previous one, and develops structure on smaller and smaller length scales. This supports our Claim that, in the limit where the initial time ni goes to negative infinity, one obtains a fractal set of Lebesgue measure zero. Thus, for dye in a fluid flow that can be modeled by our map, the dye left in the entrainment region would concentrate on a set that approximates a fractal to finer and finer scale as time goes on (or until the scale is so fine that microscopic diffusion cannot be neglected).

J. Jacobs et al./Physica D 110 (1997) 1-17 .

_

4

Y

entrainment o

set

50 --2

-2

I

I

0

2

I

4

6

X Fig. 3. Entrainment set.

3. The intermediate and pre-entrainment sets We now introduce two other fractal sets, analogous to the stable manifold and the invariant chaotic set for a nonrandom map. Using the random map (3) with the same parameters as before, in Fig. 3, we plotted the points at time nf = 50 that were sprinkled in t h e scattering region at time n i . = - 5 0 and do not leave that region between those times. If we now plot the initial conditions (at time - 5 0 ) that form the entrainment set at time 50, we get the picture in Fig. 5. Again, this set is an approximation to a fractal set in the sense that, as we demand that the initial conditions at time ni = - 5 0 stay in the restraining region until time nf = - 4 0 , - 3 0 , - 2 0 , again by leaving the random realizations of the map in the overlapping time ranges the same, the structure of the set develops to smaller and smaller scale. This process is shown in Figs. 6(a)-(c). In the limit where the points at ni = ' - 5 0 are required to stay in the restraining region forever, we claim that a mathematical fractal is obtained, and we call this set the p r e - e n t r a i n m e n t s e t at time - 5 0 . We think of it as analogous to the stable manifold of a chaotic invariant set of a nonrandom map. To emphasize that the pre-entrainment set keeps on developing structure at smaller length Scales, consider Fig. 7 (a), which is a blow-up o f Fig. 5. Remember that these are the points at time - 5 0 that have not left the'scattering region at time 50. In Fig. 7(b), we plot the same region, but only show those points that have still not left the scattering region at time 100. This set, in contrast to its analog for nonrandom maps (i.e. the stablemanifold) is not invariant; in the sense that, if one repeats the process starting at time ni ----- 4 9 (and leaving all the Mn the same as before), one obtains a different set. In particular, this will be the forward iterate of the previous pre-entrainment set using the random map M-50. We also remark that the points on the line at x = - x 0 in Section 2 for which the scattering function is singular coincide with the intersection of the line x = - x 0 with the pre-entrainment set.

J. Jacobs et al./Physica D 110 (1997) 1-17 6

- -

4

Y

7 2

o

2

-2 -2

0

2

X 6

6

4 ~

o

2

X

i

Y 20

-2 2

i

iI

i

0

2

4

6

Fig. 4. Consecutive approximations to the entrainment set by sprinkling points at earlier times.

The last fractal set we introduce, is analogous to the unstable chaotic invariant set for a nonrandom map with chaotic scattering. In particular, for our random map model consider the image at n = 0 of the points that are randomly sprinkled at time n = ni < 0 that have not left at n = nf > 0. Such a set is shown in Fig. 8 for - n i = nf = 50. We claim that this set will converge towards a fractal set as ni ~ -cx~ and nf --> -boo. We call this the i n t e r m e d i a t e e n t r a i n m e n t set. Results of this process are shown in Figs. 9(a)-(d) for - n i = nf = no and successively larger no. Here again this set is not invariant because, if we plot the points at n = 1 (rather than at n = 0), the resulting set would be different.

4. Relation between finite time Lyapunov exponents, decay time and dimension spectrum of the entrainment set In this section, we consider the characterization of the fractal properties of the entrainment set using the dimension spectrum Dq [25,26] (where q is a continuous index). Imagine that we divide the scattering region into a grid of size e, and compute the fraction of points in the entrainment set that lie in each of the boxes of the grid. Let/zi denote

J. Jacobs et al./Physica D 110 (1997) 1-17

6

Y

e t;Q set --~i

--

--2

-6

~f I

I

-4

-2

I

X

0

Fig. 5. Pre-entrainment set of Fig. 3.

the fraction in the ith box computed with the number of initial points large enough that there are many points in box i. Then

1

Dq

= - -

lim

log(I (q, e)) log(l/e)

1 - q E~o

,

(4)

where I (q, e) = E

tt q .

i

Defining D1 by D1 = limq--+l Dq, we get from l'Hospital's rule: D1 = lira Y~i/zi log(/zi) ~0 log(e)

(5)

For a chaotic attractor, Badii and Politi [27] and Ott et al. [28] have expressed this dimension spectrum in terms of the distribution function of finite time Lyapunov exponents (defined subsequently). We modify this method to express the dimension spectrum of the entrainment set in terms of the distribution of finite time Lyapunov exponents and the decay time (r). The time (~) is the characteristic lifetime in the scattering region, which we can determine as follows. Sprinkle a very large number of initial conditions N(O) in the scattering region. Iterate these using the random map, and count the number N ( m ) that have not left the scattering region after m iterates. We define (r) as

(r) "1 =

lim

lim

m--+~ U(O)--+c~

i

ml°g

(N(0) ~,.-NC~ ]

.

(6)

J. Jacobs et al./Physica D 110 (1997) 1-17

9

l

Y

Y

0

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-21 -6

-4-

0

-2

2

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-6

I

-4

X

....

o nf -2

-6

-2

X

(c)

20 i

i

i

-4

-2 X

0

Fig. 6. Consecutive approximations to the pre-entrainment set by requiring the points to stay in the scattering region for longer times.

-0.5360

-0.5360

0.5365

0.5365

Y

Y

-0.5375

-0.5380 0.74000

-0.5575

-0.73967

X -0.73933

-0.73900

-0.5580 0.74000

-0.75967

X -0.73933

-0.75900

Fig. 7. (a) Blow-up of Fig. 5 and (b) same picture but requiring the points to stay in the scattering region for an extra 50 iterates.

10

J. Jacobs et al./Physica D 110 (1997. 1-17

6

I

intermediate set at n - O

4-

er

Y 2 ~,i llli ~

~

~,~..

"HI

' -~ -'

illlll

!I

0

50 -2 -2.5

n<

I

I

-1.5

-0.5

50 t

0.5

X

1.5

Fig. 8. Intermediate entrainment set of Fig. 3.

This limit should, with probability one, be independent of the randomly chosen map sequence Mn. Alternatively, we can define (r) from (6) by

N(m) ~

N(0) exp ( - ( ~ )

(7)

for large m. We now define the finite time Lyapunov exponents and the finite time Lyapunov exponent distribution. Suppose we take an arbitrary point ( and follow it from iterate ni to nf. Denote ~ni = ~,

~ni+l = Mni(~ni),

~ni+Z = M n i + l ( ~ n i + l ) . . . . .

~nf ~---M n f - l ( ~ n f - 1 ) .

Then the finite time Lyapunov exponent for an initial condition ~ is: 1

h(~, hi, nf) -- - - l o g ( ~ . ( ~ , nf -- ni

hi, nf)),

(8)

where £ is the largest of the absolute values of the eigenvalues of the matrix: nf

DMni (~) = DMnf- 1(~nf- 1)-DMnf-2 (~nf-2)"" DMni (~).

(9)

(Since each of the maps Mn is area preserving, the product of the two eigenvalues of (9) is unity.) Suppose we now sprinkle points uniformly with respect to Lebesgue measure in the scattering region at time ni and keep those points that have not left the scattering region at time nf. We then compute the finite time Lyapunov exponent of these initial conditions and hereby obtain the distribution P(h, ni, n f ) of finite time Lyapunov exponents

Ibl!

J. Jacobs et al./Physica D 110 (1997) 1-17

6l

i

,

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t

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(o):

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2

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.5

0.5

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,

n~ =

-2

I

-0.5

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-1.5

2.5

,

0.5

X

0.5

.5

i

-20

()cl

Nf

~f

Y

-25 25

J~i z z

Y

tl -2 -2.5

j

~ -1.5

~ -0.5

"<":s

~ 0.5

X

1.5

-2 -2.5

~ -1.5

~ 0.5

X

0.5

1.5

Fig. 9. Consecutive approximations to Fig. 8.

for the entrainment set. Then for large nf - - hi, the distribution function is asymptotically of the general form [29]:

P(h, hi, n f ) ~ [2zr (nf

- - n i ) ] - 1 / 2 exp(--(nf --

ni)G(h)),

(10)

where the minimum value of the function G is zero and occurs at h = h. Note that as (nf - ni) is increased, the distribution P(h, h i , n f ) becomes more and more peaked about h = h and approaches a delta function, ~(h - h), asnf--n

i ~

¢~.

The definition of the dimension spectrum Dq in Eq. (4) is a simplified version of a generalization of the Hausdorff dimension in which one calculates the partition function [30]: N

1-'q(~), {Si}, ~ ) : - ~ / z l / E / / ) ( q - l ) ,

(11)

i=1

where {Si} is a covering of the set, Ei is the diameter of Si, and 6 > Ei for each i. Now, let

Fq(D) = lira

a-+0

{

supFq(D,{Si},~)

f o r q >_ 1,

& infPq(b,{si},a) &

f o r q < 1.

(12)

J. Jacobset al./PhysicaD 110 (1997) 1-17

12

The quantity Fq([)) makes an abrupt transition from + e c to zero with variation o f / ) . This tr,'msition point /) = /)q defines the dimension spectrum/)q which is a function of the continuous index q. It is believed that for most invariant sets of dynamical systems the box-counting definition (Dq of Eq. (4)) yields the same value for the dimension spectrum/)q defined by the transition value of Fq (/)), and we will, therefore, drop the tildes o n / ) q . hnagine that we divide the scattering region up into a grid of size e, so that the diameters of all the sets in this partition {Si } are the same and can be set equal to 3. One can show [31 ] that in this case we can introduce an estimate for (11) given by the "Lyapunov partition function",

Fq(D'ni'nf)

= e x p [ (nf - n i l) ( q( - r ) 1)

{[2tD~l ( " hi' n f ) ) l ('~ hi' nf)]l-q )'

(13)

where X t and X2 stand for the two different eigenvalues of the matrix (9), the superscript e stands for entrainment set, and the angle brackets in ([k D-I ((, n i, nf)X1 (~, ni, nf)] | - q ) signify an average over ~ with respect to the natural measure on the pre-entrainment set. The limit 8 --+ 0 in (12) corresponds to (nf - hi) --~ -boo in the Lyapunov partition function (13) [311- In the case where the determinant of the map is one, Eq. (13) becomes

Fq(D, ni, nf) =_exp((nf - ni)(q-1) ) (r)

([X(~, ni, nf)Jer),

(14)

where a = (2 - D)(1 - q). Using (10) we have (t,~(~, hi, nf)l a) = ~exp[(nr - ni)crh(~, ni, nf)l) f

exp{-(nf -

ni)tG(h) - ah]} [2~(nf

dh ni)G"(h)] 1/2"

-

(15)

For large nf - ni, the dominant contribution to this integral comes from the vicinity of the minimum value of the function G(h) - ah occur3,ing at h = her given by

G'(her) -= or, where

(16)

G'(h) = dG(h)/dh.

Thereforc, we have

([)~(~, hi, nf)] o) ~ exp{--(nf --

ni)[G(her) -

oha]}.

(17)

The Lyapunov partition function becomes

Fc~(D, ni, n f ) ~ e x p l ( n f - - n i ) [ (q-l)(--~-) G(ha) +ahcr]

i.

(18)

S i n c e / ~ goes to + o o or zero as (n f - n i) ~ fyz depending on whether the term in square brackets in the exponential is positive or negative, we have that Dq ix determined by requiring this term to be zero. Hence. we obtain the following transcendental equation tk)r Dq:

(q - 1) - -----{G(her) - o'her], (r)

(19)

where

~=(2-Oq)(l

-q).

(20)

13

J. Jacobs et al./Physica D 110 (1997) 1-17

E

10

'~

I

'

~

L

I

I

I

I

[] [] []

nf- n i = 80

8

~

[]:

[]

zx

% n f - n i = 50

4

/

A •A

6 A

•A

A[] A[]

[]

A A

[] []

A A

°% 0.0

0.1

0.2

0.3

O.4

0.5

Fig. 10. Distribution of finite time Lyapunov exponents for times equal to 50 and 80.

5. Comparing the prediction for the dimension spectrum with box-counting dimension estimates Eqs. (16), (19) and (20) determine the dimension spectrum in terms of the functions G(h) and the lifetime (v). To apply Eqs. (16), (19) and (20), we must determine the function G(h). Therefore, we compute the distribution of finite time Lyapunov exponents for nf - ni = 50 and 80 (respectively sprinkling points at time ni = - 2 5 and keeping track of them until time nf = 2 5 , and from ni = --40 until nf = 4 0 ) . The corresponding distributions are shown in Fig. 10, and one can see that they indeed become more peaked as nf - - ni increases. From these histogram approximations to P ( h , n i , nf), we then construct approximations to the asymptotic G(h)-function (see Eq. (I0)): G(h) _

1 - {log[P(h, ni, nf)] + 1 log(nf - ni) + C}, nf - ni

(21)

where the constant C is determined by the requirement that the minimum value of G is zero. The approximations to G(h) obtained in this manner are shown in Fig. 11. It can be seen that the G(h)-approximations for 50 and 80 iterates agree very well with each other, confirming the apparent existence of a limiting function G as predicted by Eq. (10). In order to be able to determine the derivative of G(h) with respect to h as required to apply Eq. (16), we fit a cubic polynomial through the G-function based on the time 80 distribution function (and also make sure it does not deviate much from the fit through the G-function data based on the time 50 distribution function). The remaining quantity to be determined is (v). Based on the definition (r) in Eqs. (6) and (7) we plot log[N(n)] versus n as shown in Fig. 12, and we fit the best straight line through this graph. The inverse o f the slope of this line is our estimate of (r). We are now in a position to find Dq from Eqs. (16), (19) and (20). We proceed as follows: we pick an h in the range where the G-function can be well fitted with a cubic polynomial, and call it h,r. The slope: of the G-function at this h is then cr (see Eq. (16)). F r o m Eq. (19) we then find the corresponding value for q, and Dq

14

J. Jacobs et al./Physica D 110 (1997) 1-17

0.20

'

'

'

I

zS~

J

i

L

I

~

'

i

I

L

i

i

I

~

'

~

I

i

,

A zS~ZX

0.15

A ZX

nf - h i =

8o

t ~ I i

p i

A

0.10 nf'ni=50

~ %

0.05

0.00 ~ i

'0.05

i

-0.1

I i

i

i

0.0

I i 0.1

0,2

I ~ i

i

0.3

I i

~ I

0.4

0.5

Fig. 11. G-functions for the entrainment set of Fig. 3 based on the distribution functions of Fig. 10.

20

I

k

i

I

I

I

,

,

,

I

I

J

I

l

l

l

I

I

I

I

I

15

ln[N(n)] 10

5

0 0

I

50

100

n

I

150

200

Fig. 12. Approximate exponential decay as evidenced by the behavior of the logarithm of the points remaining in the scattering region at a function of time.

15

J. Jacobs et al./Physica D 110 (1997) 1-17

1.80

'

'

I

I

i

I

~

I

1.75

Dq

O

O

O O O O O O O O O O

Z O O O OO

©

1.70

OOooo

o~

mm

1.65

I

0.0

0.5

,

I

1.0

~

I

,

I

1.5

q

2.0

,

I

2.5

,

3.0

Fig. 13. Prediction of the dimension spectrum based on the solution to Eqs. (15)-(17) (diamonds) and dimension estimates for the entrainment set of Fig. 3 for q = 0.5, 1, 1.5, 2 and 2.5, obtained by box-counting.

for this q is then obtained from Eq. (20). Repeating this for different initial choices of h, the resulting dimension spectrum is shown in Fig. 13 for 0 < q < 3. Also shown on this plot are results of box-counting dimension estimates, according to Definitions (4) and (5), for q =- 0.5, 1, 1.5, 2 and 2.5. These estimates o f Dq are obtained by approximating the entrainment set by iterating a very large number of initial conditions, for - n i = n f = 5 0 , using the points in the scattering region at time rtf tO estimate/xi and hence I (q, e) (see Eq. (4)), and then determining the slope of a fitted straight line to a numerical plot of l o g [ / ( q , e)] versus log(e). The values for e range from 0.25 to 0.015. The fit between the data and the fitted straight line is good, as can be seen for q = 2 i n Fig. 14. It can be seen from Fig. 13 that the prediction for the dimension spectrum falls well within the error bars of the box-counting estimates.

6. C o n c l u s i o n

In this paper, we have modeled two-dimensional open fluid flows with temporally irregular time dependence by introducing a random map ( n + l = M n ( ( n ) , where on each iterate n, the map Mn is chosen from an ensemble according to some probabiIity distribution. We numerically established that an initial distribuiton of tracer particles will concentrate on a fractal set as time goes to infinity. This entrainment set should be observable in physical experiments. Other fractal sets, analogous to the nonrandom map invariant chaotic set and its stable manifold, have been introduced for the random map case. Finally, a theory for the multifractal dimension spectrum for the entrainment set is proposed and compared with box-counting estimates of these dimensions.

16

Z ~cobs et al./Physica D 110 (1997) 1-17

12

~

I

I

,

i

t

,

~

,

1

,

I

~

,

I

~

,

,

,

I

I

,

I

~

,

i

i

I

i

i

i

I

I

i

,

,

i

I

i

,

i

i

I

i

r

~

10

8

ln[I(2,e)] 6

4

2 0

1

2

3

4

F 5

ln(e) Fig. 14. Log-log plot for the dimension estimate of the entrainment set for q = 2.

Acknowledgements T h i s w o r k w a s s u p p o r t e d b y the Office o f N a v a l R e s e a r c h a n d b y the D e p a r t m e n t o f Energy. T h e n u m e r i c a l c o m p u t a t i o n s r e p o r t e d in this p a p e r w e r e s u p p o r t e d in p a r t b y a g r a n t f r o m the W . M . K e c k F o u n d a t i o n .

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