The transition to chaotic attractors with riddled...

ELSEVIER Physica D 76 (1994) 384-410

The transition to chaotic attractors with riddled basins Edward Ott a, J.C. Alexander b, I. Kan c, J.C. Sommerer d J.A. Yorke b,e

a Laboratory for Plasma Research, Institute for Systems Research, and Departments of Electrical Engineering and of Physics, University of Maryland, College Park, MD 20742, USA

b Department of Mathematics, University of Maryland, College Park, MD 20742, USA c Department of Mathematics, George Mason University, Fairfax, VA 22030, USA

d M.S. Eisenhower Research Center, Johns Hopkins Applied Physics Laboratory, Laurel, MD 20723, USA e Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA

Received 29 September 1993; revised 3 February 1994; accepted 7 February 1994 Communicated by P.E. Rapp

Abstract

Recently it has been shown that there are chaotic attractors whose basins are such that any point in the basin has pieces of another attractor basin arbitrarily nearby (the basin is "riddled" with holes). Here we consider the dynamics near the transition to this situation as a parameter is varied. Using a simple analyzable model, we obtain the characteristic behaviors near this transition. Numerical tests on a more typical system are consistent with the conjecture that these results are universal for the class of systems considered.

1. Introduction

Recently a novel type o f behavior has been demonst ra ted to occur for certain chaotic systems with a simple kind o f symmetry [ 1,2]. In particular, there exist attractors for which all points in its basin o f attraction have pieces o f another attractor basin arbitrarily nearby. That is, i f r is any point in the basin, then, for every c, however small, there are displacements 6, where 161 < ~, such that the point r + 6 is in the basin o f another at t ractor 1 , and the set of these points has nonzero phase space volume (i.e., the set has positive Lebesgue measure) . In other words, for any fixed r in the basin, a ball of radius c about the point r has a positive fraction o f its volume in a basin different f rom the basin that r is in. In such a case the basin is said [1,2] to be "r iddled".

While there is no universally accepted defini t ion of an attractor, as we discuss subsequently, the appropriate, defini t ion for our purposes is that o f Milnor [4 ], which basically defines an at tractor as a compact set which has a dense orbit and which attracts a set o f initial conditions whose total phase space volume is not zero (i.e., the basin o f at traction has positive Lebesgue measure) . Thus there

1 This situation is to be distinguished from the case where the basin is a solid volume with a fractal boundary (e.g., see Ref. [31).

0167-2789/94/$07.00 (~ 1994 Elsevier Science B.V. All fights reserved SSDI 0167-2789 (94)00062-U

E. Ott, et al. / Physica D 76 (1994) 384-410 385

exists some bounded region of phase space such that, if an initial condition is randomly chosen in the region with uniform probability density per unit phase space volume, then there is a nonzero probability that the resulting orbit approaches the attractor as time tends to positive infinity.

As a physical example [2] exhibiting a riddled basin, consider the motion of a point particle subject to friction and sinusoidal forcing in a two-dimensional potential V (r) where r = (x ,y ) and V(r) = (1 --X2) 2 + (X + ~ ) y 2 :

dEr/dt 2 = - T d r / d t - V V ( r ) + J~ sin(tot)x0, (1)

where x0 is a unit vector in the x-direction and 7, f0, to and ~- are parameters of the problem. We shall be particularly concerned with how the dynamics change with variation of the parameter 2 ~.

The phase space of this problem is five dimensional with coordinates x, dx /d t , y, dy /d t , and 0 = (tot) mod 21t. From the symmetry of the potential, the problem is invariant with respect to y ~ - y , and, as a consequence, y = d y / d t = 0 specifies a three dimensional invariant hyperplane in the full five dimensional phase space. In this invariant plane, points are specified by the three coordinates x, d x / d t and 0 = (tot) mod 2~, and the dynamics is described by

d2x /d t 2 + 7 d x / d t - 4x ( 1 - x 2) = J~ sin tot, (2)

which, for appropriate values of the parameters (7, J0 and to), has a chaotic attractor in the invariant plane.

The set comprising the chaotic attractor for dynamics restricted to the invariant plane will also be a chaotic attractor in the full five dimensional phase space of Eq. (1), if there is a set of points of positive five dimensional Lebesgue measure that are attracted to the invariant plane. In Ref. [2] it is shown that this indeed can occur (depending on the value of ~) and that the basin is riddled. Since the chaotic set in the invariant plane is an attractor, the two Lyapunov exponents associated with infinitesimal perturbations from the plane y = d y / d t = 0 are both negative for almost every orbit on the attractor. On the other hand, as explained in Refs. [1,2], riddling occurs because there is a natural measure 3 zero set of orbits on the attractor for which infinitesimal perturbations from the plane grow exponentially. For the example Eq. (1), these perturbations are indicative of initial conditions arbitrarily close to the y = d y / d t = 0 attractor that yield orbits moving off toward lY] = o¢ as t -~ c~. (We regard [y[ = oc as a second "attractor" in what follows.)

This behavior is due to the term (x + ~ ) y 2 in the potential. When x + ~ > 0, there is a y2 potential well focusing orbits onto the invariant plane; when x + ~ < 0, the well becomes an antiwell. For an appropriate range of X values, typical orbits on the attractor sample both the well region and the antiwell region, but experience net attraction to the plane. On the other hand, a natural measure zero set, dense in the attractor, has nontypical orbits which spend an abnormally large fraction of their time in the defocusing region, and infinitesimal out-of-plane perturbations from these orbits are repelled from the plane. This behavior is reflected in the structure of the attractor basin: For any initial condition in the basin of the y = 0 attractor, there are points arbitrarily close to it that are in the basin of the [Yl = c~ attractor.

As ~ is reduced, the defocusing tendency is increased, and numerical experiments show that one of the Lyapunov exponents for perturbation of typical orbits from the plane eventually becomes

2 Throughout this paper, for simplicity, we consider the dynamics in the invariant manifold to be independent of the bifurcation parameter (as in Eqs. (1) and (2)) . 3 The natural measure of a set may be thought of as the fraction of the t ime that a typical orbit on the attractor spends in the set.

386 E. Ott, et al. / Physica D 76 (1994) 384-410

positive when ~ drops below some critical value Xc. At this point the formerly attracting set is no longer an attractor. Thus, as ~ increases through xc an attractor is born and it has a riddled basin.

One of the questions addressed in this paper is what happens as a system parameter is varied from a situation where there is no attractor with a riddled basin (e.g., for the above example,

< xc) to a situation where there is such an attractor (e.g., ~ > Xc). Thus we are interested in the universal critical behavior at the transition to a riddled basin. Our approach will be to present a simple model where this transition can be analyzed in a fairly complete way. The results from the analysis of the model suggest several conjectures for the universal critical behavior which we then test numerically in other systems. As a side product, the model serves the pedagogical purpose of furnishing an understanding of the structure of riddled basins in a particularly simple setting.

This paper is organized as follows. In Section 2, we give the model. Section 3 reduces the solution of the model to a random walk problem and argues that the diffusion approximation is justified near the critical point where the transition occurs. Section 4 presents and discusses results of the diffusion approximation solution of the model. The universality of these results is then tested in Section 5 using numerical experiments on the differential equation (1). The solutions of the model yielding the results in Section 4 are presented in Section 6 along with further discussion of the validity of the diffusion approximation. Conclusions are presented in Section 7. The Appendix outlines how our analyses and results can be applied to a model which has two basins riddled in each other (in this situation the basins are said to be intermingled) [ 1 ].

2. The model

Our model 0_<y_< 1 by

{ (1/a)Xn Xn+l = (1/fl)(Xn - ~ )

TYn fOrXn < a, Yn+t = dYn fOrXn > a,

system is a two-dimensional noninvertible map given in the region 0 _< x _< 1,

forxn < c~, (3a) forxn > a,

(3b)

where fl = 1 - a . Also we assume that the y motion involves both expansion and contraction: y > 1 and 0 < 8 < 1. In the region y > 1 we imagine that the form of the map (which we need not specify here) is such that, for initial conditions with y > 1, orbits move off to an attractor somewhere in y > 1. Thus, if the orbit ever lands in y > l, it never returns to y < 1.

For Eqs. (3) the interval I = {x,y[ y = 0,0 < x < 1} is invariant. (The set I may be regarded as analogous to the y = d y / d t = 0 invariant plane for Eq. (1).) By virtue of Eq. (3a), almost every initial condition in I, has a chaotic orbit whose time average converges to a uniform probability density on I. For almost every point picked at random in I the Lyapunov exponent for the dynamics on I is

hll = a ln (1 /a ) + f l ln(1/ f l ) > 0. (4)

The set I will be a chaotic attractor in the full two dimensional space if its Lyapunov exponent for perturbations in y,

h± = a ln7 + fl lnd, (5)


y Xn+l

/ / L / i + / /

.... 013 (~2 (X. ff,+(X[~ 1 : x o . ~ . Xn

Fig. 1 Fig. 2

Fig. 1. A region that is part of the y > 1 basin extending upward from (x,y) = (0, 0) is shown crosshatched, and its first two preiterates are shown shaded.

Fig. 2. The map (3a).

is negative. This depends on the parameters, a, 7, and ~ (recall that lny > 0 and ~n~ < 0); for

Iln~l a < ac - lny + IlnOl' (6)

I is an attractor. We shall be concerned with what happens for h± small (~ near ac). This model has the key ingredients of the example Eq. (1) and other examples [ 1 ] exhibiting r idded basins: an invariant manifold on which the dynamics is chaotic (e.g., I for Eq. (3)) , and the existence of regions where orbits are both repelled from the invariant manifold (x < a for (3)) and attracted to it (x > a for (3)).

In order to see that when h± < 0, the basin for the y = 0 attractor of Eq. (3) is riddled, consider an orbit that starts in 0 < x < a and eventually enters y > 1 without ever first leaving 0 < x < a. Using Xn+l = c ~ - l X n and Yn+l = YYn, we see that such orbits originate from the crosshatched region in Fig. 1. In addition, the preiterates of these initial conditions are also in the y > 1 attractor basin. Fig. 1 shows the original region (crosshatched) and its first two preiterates (shaded). We see that the first preiterate of the region touches the x-axis at the first preiterate of x = 0 (namely, x = a) , and the two second preiterates of the region touch the x-axis at the two second preiterates o f x = 0 (namely, x = a2 and x = a + ap ) . Continuing in this way, we generate more and more, ever thinner pieces of the y > 1 attractor basin extending upward from y = 0 to y = 1. Since the preiterates of x = 0, y = 0 are dense in I, the basin of the y > 1 attractor is dense in the region y < 1. (See Ref. [1] for a similar construction.)

As we will see in Section 4, however, there is a positive area in 0 < x < 1, 0 < y < 1 for which initial conditions yield orbits going to the y = 0 attractor. (More generally, this is implied by h± < 0.) Hence the basin of the y = 0 attractor is riddled.

As mentioned in the introduction, there are various definitions of the word attractor, and our y = 0 attractor satisfies the definition advocated by Milnor [4]. We caution the reader, however, that another common definition of an attractor requires that there exists some neighborhood of the attractor such that all initial conditions in the neighborhood tend toward the attractor. This requirement is clearly violated as we see from Fig. 1, which shows that there are pieces of the y > 1

388 E. Ott , et al. / Phys ica D 76 (1994) 3 8 4 - 4 1 0

~=0 (y = 1)

Fig. 3

y . ~ o o

(y : O)

Xn+l

/ o~ o~ 2 1

Fig. 4

• . X n

Fig. 3. Schematic of the random walk.

Fig. 4. Another possible choice for the map in x.

attractor arbitrarily close to y = O. Indeed, if we examine the attractor neighborhood, 0 < y < e, for small e, then it follows from Section 4 that a fraction of order c~ ( r /> 0) of the area of the neighborhood is occupied by the y > 1 basin.

3. Random walk

The map in x, Eq. (3a), is illustrated in Fig. 2. This map has a uniformly distributed natural measure in 0 < x < 1. Furthermore, if we pick an initial condition xl at random with uniform probability in 0 < x < l, then the generated orbit,

X I , X 2 , X 3 , . . . , X n , • . . ,

is such that the probability that Xn lands in (0, a) is a and the probability that it lands in (a, 1 ) is fl = 1 - a , independent of the prior history of visits to these intervals by Xn-1, Xn-2, Xn-3 . . . . .

Now referring to the map in y, Eq. (3b), we see that, for a randomly chosen Xl, on each iterate the y coordinate will experience expansion by 7 with probability c~ and contraction by J with probability ft. (This is the content of Eq. (5)).

Let

y = l n ( 1 / y ) , y = l n y , ~ = l n ( 1 / J ) .

Assume we choose a specific initial Yl and then choose xl at random in (0, 1 ). This initial condition generates an orbit whose y values are denoted Y l , Y 2 . . . . . Yn . . . . . The situation is now as shown schematically in Fig. 3. Given Yn, the value of Yn+ 1 is obtained by taking a step of size ~ to the right with probability fl or a step of size y to the left with probability a. Thus we have a random walk. If the orbit ever reaches y < 0 (i.e., y > 1 ), it moves to the y > 1 attractor. Thus, if this occurs, then the initial point (xl,Yl) is in the y > 1 basin. If, on the other hand, as n --* +c~, Yn --* +o~ with Yn > 0 for all n, then the initial point ( x l , y l ) is in the basin of the y = 0 attractor. (This random walk is known as the gambler's ruin problem.)

We now introduce the diffusion approximation to simulate the random walk. The basic parameters of the diffusion approximation are the average drift per iterate, v = (Jy), where J y is the increment


in y in one time step, and the diffusion per iterate, D. In the case of a random walk with uncorrelated steps,

D =

In the above (---) denotes average over the initial random value of Xl. For our model we have

v = f l S - ay, (7a)

which is the same as the negative of the perpendicular Lyapunov exponent, v = - h ± (cf. Eq. (5)) , and D = ½ [ f l ( ~ - v) 2 + a ( - y - v) 2] which yields

D = ½afl(~ + y)2. (7b)

Let P (y, Yl, n ) be the probability distribution function for y (given that xl is randomly chosen on the horizontal line segment y = Yl, 0 < x < 1 ). Considering n to be approximated by a continuous variable (appropriate for n >> 1 ), P(Y, Yl, n) obeys the drift-diffusion equation,

OP OP oEP O---ff + Vrtry = D--oy 2 . (8)

Since we imagine initial conditions to all start at y = Yl, we have

P(Y, Yl,0) = 5 ( Y - Y ~ ) , y~ > 0. (9)

Since any orbit which crosses y = 1 (y = 0) is lost to the y > 1 attractor, we have

P (0 ,Yl ,n ) = 0. (10)

We require two conditions for the validity of the diffusion approximation, Eq. (8):

(i) Many steps must be taken for an orbit to reach y = 1, which is so if

y~ >> 1. (11)

Thus y~ must be close to the x-axis. (ii) The average drift on one time step, v = (~y), must be small compared to the actual step in

Y,

(JY) << Y,~. (12a)

The necessity for this condition and further discussion of it will be given in Section 6.2. Regarding y and 3 to be of order one, condition (12a) becomes

Ih±l << 1. (12b)

In other words, a must be near ac. Thus, the diffusion approximation is only valid near the transition to an attractor with a riddled basin.

At this point it is worthwhile to re-emphasize an important property of the diffusion approximation. Namely, it depends on only two parameters, v and D. This is in contrast to the original model Eq. (3) which depends on three parameters a, y and ~. There is an infinite set of these three parameters which give the same v and D, and hence the same diffusion description. Fur- thermore, our choice of (3) is only the simplest choice illustrating the derivation of the diffusion


approximation for this problem. For example, we could replace Eq. (3a), given in Fig. 2, by the map illustrated in Fig. 4; we could then correspondingly replace Eq. (3b) by multiplications of y by three different factors for x in (0, al ), (cq, c~2), (c,2, 1 ). In this case we again obtain, in exactly the same way, the drift-diffusion equation, Eq. (8), characterized by only two parameters (v and D), even though the detailed model is now specified by five parameters (al, a2 and the three y multipliers). Partly motivated by this, we conjecture that, in general, there is universal behavior in the vicinity of the transition to an attractor with a riddled basin, and that this behavior is governed by the drift diffusion equation and its two parameters D and v. Here by universal we mean that the same phenomena occur and are quantitatively governed by D and v for a large class of systems independent of their individual details. Numerical tests of this are reported in Section 5.

In this connection it is useful to have a more general way of thinking of the diffusion coefficient D. In particular, we can regard D as characterizing the decrease with increasing time of the dispersion of finite time estimates of the Lyapunov exponent h±: Imagine that we sprinkle many initial values of Xl uniformly over the attractor, and then estimate h± for each of the resulting orbits by using the first n iterates of the orbit. For a given Xl the estimated h± will be

np (xl) h±(n, x l) = n,~(xl) ln7 + ln~, n n

where n,~(xl ) denotes the number of times the orbit starting at x = xl fell in (0 ,a) , and n~ (xl) denotes the number of times it fell in (a, 1 ). With probability one, the randomly chosen xl results in an orbit for which

lim no np - - = a , l i m - - = f l , n--.o~ n n-.o~ n

so that

h± = lim -hz(n, xl) . / ' /---* OO

However, for n finite there are fluctuations of h± about h±. This is illustrated in Fig. 5, which shows the probability distribution function F(h±, n), at time n. For n ~ +o¢, we have that ('h±) = f h±F('h±,n)d'h± ~ h±, and the dispersion about the mean, given by ((~±)2)1/2 =

[ f (~x _ (~±))2Fd'~± ]1/2, approaches zero as l/±/'if:

o r

((~h.l_)2) 1/2 ~_ 2V~,

D = l im ½n((c~h±)2), t t ----~ OO

(13a)

(13b)

which follows from h± = (Yn - Yl )/n. Hence we see that the basic phenomena that we can treat using Eq. (8) are those due to

fluctuations in the finite time Lyapunov exponent for attraction to the invariant set (y = 0). Insofar as these fluctuations are diffusive, we thus expect that the results will be independent of the particular dynamical system. As discussed in more detail in Section 6.2, this is so sufficiently near the transition to an attractor with a riddled basin.

E. Ott, et al. I Physica D 76 (1994) 384-410 391

F (h_L, n)

I ~ h_L h,L

Fig. 5. Schematic of F ('h±, n ) versus h± .

4 . C r i t i c a l b e h a v i o r

We now outline the use of Eq. (8) to obtain several results for the behavior near the critical transition point a = ac.

The first question we address is the following. Say we consider the horizontal line segment y = Yl, 0 < x < 1. What fraction of the length of this line is in the basin of the y > 1 attractor? If Xl is chosen randomly in (0, 1 ), then the probability that ( x l , y l ) lies in the y > I basin is also the fraction of the line length in the y > 1 basin. Thus we can apply Eqs. (8 ) - (10) .

Let P(y , y l ,S) be the Laplace transform of P(y , y l , n ) with respect to the (continuous) time variable n,

OG

/ P (Y, Yl, s) = e-S" P d n .

0

Eqs. (8 ) - ( 10 ) yield

D d 2 - p / d y 2 - v d - P / d y - s P = - 6 (y - Yl ) (14)

with the boundary condition P(0 ,Yl ,S) = 0. Solving Eq. (14) for -P (y ,~ l , S ) is straightforward (Section 6.1 ). The probability of a random walker never having reached y = 0 is

O(3

1 - P. = limfP(y, yl,n)dy, 0

where P, is the fraction of the line in the y > 1 basin. Using the properties of Laplace transforms, 1 - P, can be expressed as

OO

1 - P, = l i m s / - P ( Y , Y l , s ) d y . s----~0 J

0

Inserting the solution for P(Y, Yl,S), we obtain (Section 6.1) 1 - P , = I - e x p [ - y l q ] where q = v / D . Hence

P. ,,~ y~, q = v / D , (15)

which is one of our main results. Thus the measure of the y > 1 basin increases as a power law in y with exponent ~ = v / D .

392 E. Ott, et al. I Physica D 76 (1994) 384-410

The result (15) gives the total length occupied by the basins on the y = Yl line, but it says nothing about the arbitrarily fine scaled riddling of the y = 0 attractor basin. In the language of Ref. [6], the basin of the y = 0 attractor is a "fat fractal'. The scaling characterizations of arbitrarily fine scaled fat fractal structures are discussed in Refs. [6,7]. Here we utilize the uncertainty exponent [3,7] characterization of the fine scale riddling: Imagine that we choose a point at random on the y = Yl line. Now choose a second point with y = Yl and x chosen at random with uniform probability density in the interval of length 2e centered at the first point. We ask, what is the probability that one of these two randomly chosen points is in the y = 0 basin and the other is in the y > 1 basin? We denote this probability (p), and we define the uncertainty exponent as

q~ = lim In(p) ~-~o lne " (16)

Thus for small e

(p ) ~

We can think of (p) as the probability of making an error when we predict which basin (xl,yl) is in if Xl has a measurement uncertainty e. The calculation of ~b for our model is given in Section 6.3. The result is

q~ = h2 /(4Dhll). (17)

We now ask, what is the effect of noise on the y = 0 attractor? Since noise can kick an orbit initially on the attractor into one of the arbitrarily nearby pieces of the y > 1 attractor basin, we expect that all orbits initially on the y = 0 attractor or in its neighborhood will eventually wind up on the y > 1 attractor. To be more specific, assume that on each iterate we add a random noise vector On so that

rn+l = M(rn) + &,

where rn = (Xn,Yn), M(r) is the model map Eq. (3), ~ is randomly chosen with uniform probability density in the disc 1~1 < u, and u denotes the noise level. Due to the noise, the orbit never gets much closer to the x-axis than y ~ u. Thus we crudely model the effect of noise on our random walk model by placing an impenetrable boundary at the location,

y . - I n ( l / u ) > Yl-

The condition of zero probability flux into the y = y . boundary is

(vP(Y, Y l , n ) - D dP(y 'y ' 'n)) = 0 . (18) d y y=y~

The question we now ask is what happens if many initial conditions are sprinkled around in the vicinity of the attractor? We find that for large time the fraction of particles N(n) remaining in y < 1 decays exponentially

N(n) ~ exp- (n / (~ ) ) ,

where the characteristic decay time (T) is given by the solution of a certain transcendental equation. For example, for rluqln(1/u) << 1 (r/ = v/D)


(z) ,~ D v - q (19) ~)2

This result and the transcendental equation from which it comes are derived in Section 6.4, as follows. The Laplace transform of P(Y, -Y l ,n ) for the problem given by Eqs. (8) - (10) with the additional boundary condition (18) and Yt taken as Yl = Y, - 0+ is solved for P(y , y l , s ) . The Laplace transform of N ( n ) is N(s ) = fo-ff(~,~l,S)d~. -N(s) is then examined, and it is found that its only singularity is a pole at a real, negative value of s whose location, denoted s = sp, is obtained as the solution of a transcendental equation. The long-time decay is then (z) = - l i s p .

If we consider Eq. (3b) to be modified by the addition of a small term f (x, y) with f (x, 0) # 0 such that the invariance of the line y = 0 is broken, then we expect the result to be similar to that of adding noise. For example, Eq. (19) should apply with v replaced by a value characterizing the typical magnitude of f (x, 0).

As a final question, we ask what happens when the parameter is varied through the transition, with the consequence that hA becomes (slightly) positive. In this case the chaotic invariant set on y = 0 is no longer an attractor. Rather, initial points placed very near y = 0 typically experience a chaotic orbit for some time, but then eventually move off toward the y > 1 attractor. (This is called a chaotic transient.) As in the case of noise, the number of points initially sprinkled near the attractor whose ensuing orbits are still in y < 1 at some later time decays exponentially with time. The exponential decay time for this case is (Section 6.5)

4D (T) = V2 , (20)

and follows from solving Eqs. (8) - (10) with v negative (recall v = -hA ). Eqs. (15), (17), (19) and (20) are the main results of this paper. We conjecture that they

are universal in the sense discussed in Section 3, and we test this conjecture by comparison with numerical results in the next section.

In terms of the quantity a¢ in Eq. (6) we can express v as

v = --hA = (ac-- a ) l n ? .

Thus v and hA vary linearly as the parameter a varies through the value ac. Such linear dependence is expected generally (e.g., see the dependence of hA versus X plotted in Section 5.2 for the case of particle motion in the potential of Eq. ( 1 ) ). Thus, if C denotes some generic parameter of the system, then typically, in the vicinity of the transition, v = --hA ~ ( C - C~). Eqs. (15), (17) and (20) then yield,

q ~ (C - Cc), (21)

g? ,.~ (C - Cc) 2, (22)

(T) ,,~ (C - Cc ) -2 . (23)

With respect to the decay time for the chaotic transient (Eq. (23)), we comment that a similar phenomenon occurs in crisis transitions. There, an attractor is destroyed by collision with its basin boundary and is replaced by a chaotic transient characterized by an exponential decay time. In that case, it has been shown [8 ] that, for crises of typical smooth two-dimensional invertible maps, the variation of the decay time with a generic parameter C is as

(T) ,,~ ( C - Cc) -¢,


and the critical exponent ~ depends on properties of the system. For the riddling transition we find the same behavior, Eq. (23), but with a universal exponent of ~ = 2.

5. Numerical experiments

The dynamics resulting from Eq. (1) were discussed qualitatively in Section 1. Here we will demonstrate quantitatively that the attractor in the invariant hyperplane of Eq. (1) has a riddled basin, and that the system exhibits critical behavior as described in Section 4. That Eq. (1), a five-dimensional continuous-time system, exhibits the same critical behavior as the simple two- dimensional map of Section 2, strongly supported the universality of the critical behavior conjectured in Section 4.

We analyzed Eq. (1) in terms of a stroboscopic Poincar6 section corresponding to its forcing period; i.e., we consider the times tot mod 2~ = 0, yielding a four-dimensional discrete-time mapping taking a phase-space point ( x ( t ) , y ( t ) , V x ( t ) = d x / d t , vy ( t ) = d y / d t ) at time t = tn = 2nrt/to into a different phase-space point (x (t + 2n/to), y (t + 2n/to), vx (t + 2n/ to) , Vy (t + 2n/to) ) one forcing period later. The remainder of this section concerns the parameter values u = 0.05, J~ = 2.3, to = 3.5, with ff varying. The dynamics of Eq. ( 1 ) was simulated using a fixed-step-size, fourth-order Runge-Kutta numerical integration, with 60 time steps per forcing period. The results are somewhat inaccurate in the sense that they are different in detail from results obtained with a variable-step-size method such as the Bulirsch-Stoer integrator. However, the qualitative behavior was the same as that produced by Bulirsch-Stoer in test cases (usually at slightly different parameter values). Our results may therefore be interpreted as reflecting the dynamics of Eq. ( 1 ) with precise but badly calibrated parameters and measurements.

Figs. 6, drawn for ff = 1.9, shows two dimensional slices through the (four-dimensional) riddled basin of the y = 0 attractor. Each initial condition on a grid [Vx(0) = vy(O) = 0, with x (0 ) and y (0) given by the horizontal and vertical axes, respectively] was followed numerically until clearly heading to lYl -- ~ or until very near the y = vy = 0 attractor. A black dot was plotted on grid points leading to lYl = ~ while no dot was plotted on grid points yielding orbits tending toward y = vy = 0. As can be seen from the expanded views, regions appearing at a given resolution to be attracted to the invariant plane actually contain initial conditions attracted to tYl -- oo. Although the observed probability of being attracted to the invariant plane increases as one approaches it in phase space, we always found a non-zero probability of being attracted to lYl = oo. This fact, coupled with the numerically determined Lyapunov spectrum of typical orbits in the y = Vy = 0 attractor (hi = hll > 0, h2,3,4 < 0) establishes that the basin of the y = 0 attractor is riddled.

Note also [particularly in Figs. 6a and 6d] the wedge-shaped clouds of black dots (initial conditions in the basin of infinity) that extend down toward the y = 0 plane. These appear similar to the wedge-shaped regions in Fig. 1, which lead away from the unstable fixed point x = y = 0 of Eq. (3), and from its preiterates. As mentioned in Section 1, we have found atypical periodic orbits in the invariant plane of Eq. ( 1 ), which spend a disproportionate amount of time in the defocusing region of the scalar potential, and which are unstable to out-of-plane perturbations. Such orbits have two positive and two negative eigenvalues. The eigenvalues are paired. One positive and one negative eigenvalue each correspond to eigenvectors in the invariant plane. Another opposite-sign pair corresponds to a pair of eigenvectors transverse to the invariant plane. Some curve in the two-dimensional unstable manifold of the periodic orbit may be "heteroclinic orbit to infinity,"

1.2-

E. Ott, et al. I Physica D 76 (1994) 384-410

0.66 -' ....... ~.~,~-

I 1

395

[o$~

0.8- 0.64-

~, 0.6 - ~, 0.63 -

0.4 - 0.62 -

0.2- 0.61 -

0.0- 0.60-, 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.50

(a) Xo (b) 0.51 0.52 0.53 0.54 0.55 0.56

xo

0.626' . . . . . . 0 . 0 6 -

~9~1 [oXo~I

m i 0.620 -, ~ 0.00 -

0.540 0.541 0.542 0.543 O.f~14 0.545 0.546 0.18 ( c ) xo (d )

I I I I I

. . . . . . . . . ¢: " ~ , : ~ , . , . , ~ . . . . . . . . : . . . . . . . . . . . . . ...,.

:.~...~.. ~.~,:~,~..~,. .~ ~:'..~., ~.:" ~, :.~":,%~ - .~! =~ ,.:: :~.. =:

' .~. ,,~,':~ '. £.~'i ~ . ' . ~:~:~.i : ")~'!'"!-':.~: "~:... ' - . : , : • ! ' . , . ? ~ : ~ . : ,.,..;,~..:. ~.:,~... '~:~.,..:.....~-.?..,,. ~..',~..~..

.:..':. "-.'.~.~._;~.'.~.~.;~-/---- ,,~:" . . ' , .?.,. : :~, ,~, . . : ; . / , . : " :.: ~,.~:,:. - ' : : - : . ~ . , : , ~ , - ~ . 'O f . ~ . . . , . : q " ' . ! . . ? - . ~ : : . ' , ~ : . : . ? . . ~ . . / . ' . ~ .!

~. '~' , : : . . . : ' . . : . '~,~.~.:~:~.~ . ' ~ .~. -= ' : . : .~ : .k . . , . . ~ ' : / " , . : >~7" . ". " ' . . . . . ' : :~.-:..~..'.~.".:-:~2i,'~." • ..~"~:'~..&.: • " , : . . .. -~ .-'~- ' . . . ' . ~- ::~ '=' ,:~ : ~ : i ! ~ ) ~ , ' : . ' - / , . : ' : :.:~. : : ~:~ ~! :, ~ '.!'...:?.: i , ;~.F- .~g '~ ' . : ~ ~;....~,.....: % . : : , ,:. : . .....~ ,. •

• .' .:" .".,~1;, '. : ~ . . . . , .= . . : . . . • ' - " . . ' ~ :., " "

!.!~: .?:::~: " . . " i : . . . . . .

0.19 0.20 0.21 0.22 0.13 0.24 XO

Fig. 6. Destination map for initial conditions in a two-dimensional slice of phase space of Eq. (1). Black dots correspond to initial conditions attracted to y = +o~. White dots correspond to initial conditions attracted to the invariant plane y = vy = 0. (a) 1024 x 1024 grid. (b) 1024 × 1024 grid giving expanded view of top inset in (a). (c) 1024 x 1024 grid giving expanded view of inset in (b). (d) 1024 x 1024 grid giving expanded view of bot tom inset in (a) (black dots enlarged for visibility). We chose a single initial condition in each cell of a grid in the phase space, and numerically iterated it forward in time until the trajectory was either definitely in the repelling region of the potential (taken to be [y] > 20, Ivy] > 100, a n d y . vy > 0), or else very close to the invariant plane (]y[ < 10 -8, Ivy[ < 10 -9, and y . vy < 0). The detail, but not the general, riddled character of the picture, changes if different initial conditions are chosen at random in each grid cell, or if different computers (with different round-off algorithms and precision) are used to do the computations. Similar riddled pictures are obtained for arbitrary slices through the phase space.

providing a path for initial conditions arbitrarily near the invariant plane to be repelled from the attractor in the plane (Fig. 7a). Furthermore, we picture each point on this curve as having a two dimensional stable manifold corresponding to the two stable directions of the periodic orbit. Thus, the curve and the stable manifolds of its individual points constitute a three dimensional manifold on which initial conditions proceed toward ]y[ = oo (provided they do not lie precisely on y = vy = 0). See Fig. 7b which schematically represents this three dimensional manifold as a


(a) y, Vy

k~ Orbit (U) ~ - heteroclinic

to infinity

Periodic The attractor --~ \ orbit (P) in the /) ) /attractor

(c) v

The y=vy=O plane

An intersection

the att ctor

~ \ A t t r a c t o r

Stable M ~

(d)

4D H y p e r w e d g e ~ ~

Fig. 7. (a) The attractor and a periodic orbit (P) of the map where P has an unstable manifold (U) going away from the invariant hyperplane y = vy = 0. (b) Stable manifold of U. (c) The attractor in the y = vy = 0 plane showing an intersection of the stable manifold of P with the attractor. (d) Four dimensional hyperwedge of points going to lyl = c~ associated with the stable manifold segment S through P.

two dimensional manifold. Considering the dynamics restricted to y = vy = 0 (i.e., the dynamics of Eq. (2)), we assume that, if we start at the periodic orbit and follow the periodic orbit's stable manifold curve, it intersects the strange attractor at some typical point on the attractor (Fig. 7c). Since the orbit, backward from such a typical point on the attractor is dense in the attractor, the stable manifold curve is dense in the attractor. This curve is also the intersection of the previously mentioned three dimensional manifold with the y = vy = 0 plane. Thus the three dimensional manifold containing points going to lYl = ~ is dense in the basin of the y = vy = 0 attractor. We view this three dimensional manifold as analogous to the vertical lines in Fig. 1 that extend upward from the (0,0) fixed point and its preiterates. Here the fact that preiterates of (0,0) are dense in the attractor of Fig. 1 is analogous to the statement that points on the y = vy = 0 stable manifold curve of the periodic orbit are dense in the attractor. Orbits starting near a leaf of the three dimensional surface of points going to infinity will stay near the leaf for some time, but typically are repelled away from the leaf. If a point starts close enough to the leaf that it stays near the leaf for a long enough time, it can attain large lYl values and then move off toward lYl = ~ . (This is analogous to our hypothesis that the region y > 1 in Fig. 1 is completely occupied by the basin of the attractor in y > 1.) Points nearer the plane y = vy = 0 must therefore start closer to the leaf, if they are to attain large enough lY[ values that they subsequently move toward lYl = c~. Thus we envision that points that are near y = vy = 0 and in the basin of lYl = c~ lie

E. Ott, et al. / Physica D 76 (1994) 384-410 397 0.15 . . . . , . . . . , . . . .

100,O00-iterate '...... estimates

0 . 1 0 . .~. ~%J '"k

0.05

.~ " \ ".~.

0.00 /~.. transition: ~c = 1.7887

-0.05

-01 . . . . 115 . . . . 210 . . . . 2 5

Fig. 8. Variat ion o f normal Lyapunov exponent h± with the parameter X in the potential governing Eq. ( 1 ). For ~ < ~c, the chaotic set in the invariant plane y = vy = 0 is not an attractor: typical initial condi t ions all go to lYl = oo. For ~ > Xc, the chaotic set in the invariant plane attracts a posit ive Lebesgue-measure set in the initial condi t ion space. Variation o f h± with ~ appears to be smooth through the transition.

in regions resulting from fattening of these three dimensional leaves in the fourth direction, with the fattening decreasing to zero as y = vy = 0 is approached (Fig. 7d). Hence we have points in the lYl = oo basin that occupy four dimensional "hyperwedges" (compare Fig. 1 ), where the knife edge of the wedge corresponds with the periodic orbit's one dimensional stable manifold curve for the y = Vy = 0 dynamical system (Eq. (2)).

Referring now to the regions of dense black dots in Figs. 6a and 6d, we thus identify the wispy structures that extend down toward y = 0 as arising from the two dimensional intersections of the four dimensional hyperwedges with the two dimensional Vx = Vy = 0 surface of the figure. Thus these structures are analogous to the two dimensional wedges of Fig. 1.

As the parameter ~ is decreased to about 1.7887, the attractor in the invariant hyperplane is destroyed, and typical initial conditions are all repelled to [Yl --- o~. This transition is reflected in the Lyapunov spectrum of Eq. (1). Fig. 8 shows the largest out-of-plane Lyapunov exponent, ha_, evaluated along typical orbits on the attractor in the invariant plane. This is done by integrating the equations for the tangent vector representing the infinitesimal variations from the invariant hyperplane y = vy = 0: d S v y / d t = -VSVy - 2(x + -g)Sy, d S y / d t = ~vy. The exponent appears to vary smoothly through the transition. The remainder of this section will address the applicability of Eqs. (15), (17), and (20) to the dynamics of Eq. (1), when ~ is near its critical value. [We did not investigate the applicability of Eq. (19) because of the prohibitive computational cost of numerically integrating the stochastic version of Eq. (I) , at an accuracy corresponding to the deterministic results. ]

5.1. Applicability o f Eq. (15)

We investigated the scaling of the fraction of a line segment (0 < x < 1, vx = Vy = 0, y = const.) that is in the basin of the attractor at infinity by randomly sampling initial conditions from the segment and noting whether or not they were attracted to the invariant plane. For small y and near the transition, the probability of being attracted to infinity exhibited clear power-law scaling in y. The scaling exponent i/ of Eq. (15) was estimated empirically as the slope of a log-log plot


. . . . . . . . i . . . . . . . . i . . . . . . . .

Y-=1.8500 [ o data (with sampling error)[ , ¢ /

_ _ _-- . .

0.1 8 7

6

5

4

3

2

0.01

0.001 ~ ' ~' 6'~'.'01 ~ ' ~' ~'~''0.1 2 ' ~' ~'~' Y

Fig. 9

. . . . . . i

o data (with fit error) t 2 1 I - -

./J" 0.1 ~ ~ ~ ~ ~ ÷ ~

0.01 0.1 X - - X c

Fig. 10

Fig. 9. Typical scaling of the probability P. that an initial condition [x(0) ,y(0) ,0 ,0] is in the basin of infinity, as a function of y. Power-law scaling conforms to prediction of Eq. (15), with empirical scaling exponent r/nt given by slope of best least-squares fit to log-log plot.Error bars show uncertainty in proportions tiP. = ~ / P . (1 - P, ) /N .

Fig. 10. Dependence of empirically determined scaling exponent r/fi t of Eq. (15) with parameter excess x - Xc. Linear dependence predicted by Eq. (21) shown by straight line. Error bars show uncertainty of least-squares-estimated slopes in plots like Fig. 9.

of P* vs. y; a typical result is shown in Fig. 9. Eq. (21) predicts that the scaling exponent r/scales linearly with the parameter excess x - Xo Numerical results supporting this conjecture are shown in Fig. 10, based on the empirically determined scaling exponents r/fit, determined at different values of the parameter ~.

Stronger support for the universality of the random-walk results was obtained by computing the scaling parameter r/ from the fluctuations in the finite-time estimates of the Lyapunov exponent hi . We estimated the diffusion coefficient D by following a randomly chosen ensemble of orbits in the y = Vy = 0 attractor, along with their associated tangent vectors. The diffusion coefficient D was estimated as the slope of a graph of (1/2)nE((Sh±)2), as a function of the number of forcing periods n over which Eq. (1) was integrated [see Eq. (13a)]. The limit-based definition of Eq. (13b) was not used in estimating D to avoid problems inherent in point estimates. A typical result showing the linear increase of the quantity (1/2)nE((Sh±) 2) with n is shown in Fig. l l. Estimates of D at several values of the parameter ~ were used to predict the scaling exponents r~ (h±, D) = v / D = [h±l/D, for comparison to the empirically determined scaling exponents r/fit. As shown in Fig. 12, the drift-diffusion results are supported by the dynamics of Eq. (1).

5.2. Applicability o f Eq. (I 7)

We measured the fine-scaled geometry of the basin structure of Eq. ( 1 ) by determining the fat fractal dimension of the basin of the y = vy = 0 attractor. As discussed in Section 4, we randomly sampled initial conditions on a line segment (0 < x _< 1, Vx = Vy = 0, y = const.). For each initial condition, we determined whether a second initial condition on the same line randomly

E . O t t , e t al . / P h y s i c a D 76 ( 1 9 9 4 ) 3 8 4 - 4 1 0 399

2000 , , , ,

T=1.8500 ] o data (with sampling error)[ . ,

[ ~ - . .

1500

..~ 1000

{~ I I I I I I 0.2 0.4 0.6 0.8 1.0xl05

n

Fig. 11. Dependence of (scaled) variance in finite-time Lyapunov exponents as function of number of map iterations over which exponents are estimated. Straight line shows best least-squares fit, giving diffusion coefficient of Eq. (13a). Error bars show (scaled) 67% confidence intervals for estimated sample variances.

1.4

1.2

~. 1.0

.~ 0 . 8

'~ 0.6

8 0.4

0.2

0.0

i i i

[] ~(h,D) (with estimated error) (horizontal offset for clarity) *

o¢

1.80 1.85 1.90

Y Fig. 12. Comparison of Eq. (15) scaling exponents derived from empirical fits of type shown in Fig. 9 (circles) and exponents derived from estimates of normal Lyapunov exponents and diffusion coefficients (squares), as function of critical parameter ~. For circles, error bars show uncertainties in least-squares-estimated slopes in plots like Fig. 9; for squares, error bars show uncertainties estimated by propagation of errors in formula for rl, Eq. (15).

chosen within a distance e of the first initial condition went to the same or the other attractor. We then varied e and determined the corresponding variation in the probability (p) that pairs of initial conditions in the same e-interval go to different attractors. The results, for ~ = 1.9, are shown in Fig. 13. Note that the uncertainty exponent 4, given by the slope of the log(p) vs. loge graph, is extremely small. This is consistent with our expectations based on Eq. (17): ~ depends quadratically on h~_ which, by definition, is itself small near the transition to a riddled basin. The prediction of Eq. (17) for ~ is also shown in Fig. 13. Although the predicted value is of the correct order of magnitude, it is not in exact agreement with the empirical results. This is possibly because the comparison was made at ~ = 1.9, which, according to Fig. 8 is rather far from the transition at Xc ~ 1.7887. However, for ~ nearer the transition, the magnitude of h L, and especially of 4, become so small that determining ~ as the slope of a graph like Fig. 13 becomes statistically untenable. We

400

..~ 0.1

E. Ott, et al. / Physica D 76 (1994) 384-410

i i i i i

~-=1.9 I data (with sampling error) I fit [#~,=0.0175+_0.0038] I theory [~(h,D)=0.0093_+0.0012] I

0.01 / 10 -10 10 .8 10 .6 10 -4 10 -2

E

Fig. 13. Variation in proportion of initial condition pairs { [x(0) ,y (0) ,0 ,0 ] , [x(0) + ~,y(0) ,0 ,0]} (t~ random and - e < ~ < e ) that go to different attractors, as a function of initial condition separation e. Solid line gives best least-squares fit of log-log plot of data. Dashed line gives prediction of Eq. (17), based on estimate of normal Lyapunov exponent and diffusion coefficient. Error bars show uncertainty in proportions ~(p) = ~/(p)(1 - (p))/N.

are therefore forced to rely on the order-of-magnitude agreement farther from the transition. It is worth noting, for its own sake, the extreme insensitivity of the uncertainty probability (p)

to changes in the measurement uncertainty c of initial condition placement. Fig. 13 shows that an eight order of magnitude improvement in measurement accuracy results in only a few percent decrease in the uncertainty over which attractor will capture an initial condition. This example emphasizes the serious practical consequences of encountering a system with riddled basins: even with small initial condition uncertainty, there would not even be reproducibility of apparently identical experiments.

5.3. Applicability of Eq. (20)

Our final investigation of the dynamics of Eq. ( 1 ) was performed for ~ < Xc, i.e., such that the chaotic set in y = vy = 0 is not an attractor. Under these conditions, typical initial conditions near the invariant plane experience a chaotic transient before moving off toward lYl -- o0. We investigated the statistical distribution of the transient at ~ = 1.75 by starting an ensemble of initial conditions near the invariant plane, and noting the number of map iterations required for them to permanently leave the region y < 1. Only transients longer than 300 iterations were considered, eliminating about 1000 out of 4096 initial conditions that made very rapid escapes from the region of the invariant plane. The remaining tail of the distribution, shown in Fig. 14, has an approximately exponential form, with a characteristic decay time of ( T ) f i t = 1993 + 37 iterations. This is reasonably close to the prediction of Eq. (20), (T)(h±,D) = 2400 + 730 (the large confidence interval for the theoretical estimate results from the h72 dependence of (T) and the high relative error estimate in h± ). We also note that the prediction of Eq. (20) is asymptotically valid for large decay times. The distribution in Fig. 14 shows a tendency to shallower slope (longer characteristic time) for larger decay times.

In summary, we have numerically verified that the main deterministic results of Section 4


1 0 ° , , , ,

\ I" d~t~ I ] . data._+error I

i i0.i ~ exp(-n/(T)) ((T)= 1993:L'37) I

10.2

lo-3

10"4 0 I I I "s I 5000 10000 15000 20000

n

Fig. 14. Tail of distribution of escape times n for initial conditions randomly chosen near the invariant plane y = vy = 0, for X < ~c. Heavy dots give individual escape times; small dots show uncertainty in fraction df = v/f ( I - f)/N. Straight line shows exponential fail-off of fraction remaining near invariant plane, with characteristic time given by average of data.

are consistent with the dynamics of a high-dimensional, continuous-time dynamical system. This consistency supports the universality of the conjectures based on the simple random walk model.

6. Derivation of diffusion approximation results

6.1. Measure of the y > 1 basin on a horizontal line (Eq. (15))

The solution to Eq. (14) subject to the boundary condition P ( 0 , y l , s ) = 0 is

Y(y,y,,s) = { BA(exp[-XaY]exp[-Xay], - exp[xby] ), Y0 >< yyl, < Yl,

where

+ V2 1 ,

( i 4 D s ) Xb=~ 1 + --~-- + 1 ,

A = exp[--XbYl] (Xa + Xb)D'

(24a)

(24b)

(24c)

(24d)

B= e x p [ - x b Y l ] - e x p [ x a y l ] (1¢ a + K b )D

+ Xb ) = rill1 + 4Ds/v 2, (Xa

(24e)

402

r 1 = v / D .

Thus

O0

f P ( Y , Yl

0

E. Ott, et al. / Physica D 76 (1994) 384-410

0 ~

= A[(1 - e x p [ - l C a y l ] ) / X a - (exp[xbYl ] - 1 ) / / ¢ b ] -b Bexp[-XaYl ]/r,.a.

For small s we obtain Xb ~- v / D , Xa ~ s / v , A ~ - ( 1 / v ) e x p [ - q y l ] , B ~- ( 1 - e x p [ - q y l ] ) / v where q = v / D . Hence,

OO

1 - P, = l i m s l - P d y = 1 - exp [ - ? / y l ], s--~0 J

0

which yields Eq. ( 15 ).

6.2. Validity o f the di f fusion approx ima t ion

If we take 7 = 1/~ > l, then the step sizes in y to the right and left are equal (~ = ~). In this case, the random walk can be solved exactly [9]. It is instructive to compare the resulting exact solution to that obtained in Section 6.1 using the diffusion approximation. Let z = Yl/~ be an integer, and let • (z) denote the probability that a walker starting at z ultimately gets absorbed at z = 0 (i.e., is in the y > 1 basin). Then ~ ( z ) obeys the equation,

q~( z ) = a ~ ( z + 1) + f l ~ ( z - 1),

whose general solution is

q~( z ) = A + B ( ~ / p ) z

Applying the boundary conditions, ~ ( + c ~ ) = O, ~ ( 0 ) = 1, we have A = 0 and B = 1. Thus

q ~ ( z ) = ( a / f l ) z = y~, (25a)

= [ l n ( f l / a ) ] / 7 . (25b)

We now compare the exact result for the exponent (25b) with the diffusion approximation result (15). Setting y = 1/~, we have from Eq. (15)

fl - a q = v / D - 2d-fl~" (26)

Now let a = (1 - ~ ) / 2 , fl = (1 + ~)/2, where 0 is a measure of the deviation from the critical point (h± = ( f l - a ) ~ = 0 when ~ = 0). Eqs. (25a) and (26) then yield

? ] - ~ -- ~t~ 2 -~ O(04 ) . (27)

Thus the diffusion approximation and the exact result agree near the transition (0 ~ 0) with an error that is quadratic in t~. This is consistent with the requirement Eq. (12).


To understand further the source of the error (27) and the necessity of (12) for validity of the diffusion approximation, we can adopt the following picture. As time proceeds, the initial delta function distribution P spreads and translates; its width is ~ and its peak is centered at Yl + vn. At any time n, the number W of widths in the y distance from the absorbing boundary to the peak of P is

W = (Yl + v n ) / ~ .

This number is minimum at the time n = nm = Yl/v. It is roughly at this time that most of the loss to the absorbing boundary occurs. Also at n = nm, the number of widths is W = ~ / D , which can be large since yl is large. Thus, for the validity of the diffusion approximation, it is crucial that, at the time nm = yl/V, the diffusion approximation be accurate far enough out in the tail of the distribution function. The diffusion approximation is clearly not valid, if one goes too far out in the tail: If we, for the moment, remove the absorbing wall at y = 0, then the exact distribution must be zero for y < Yl - n~ (y = Yl - ny corresponds to all steps being leftward) and Y > Yl + n~ (y = Yl + n~ corresponds to all steps being rightward); but the spreading Gauss±an of the diffusion approximation says that the distribution is nonzero everywhere in - ~ < y < c¢. Analysis shows that the diffusion approximation is valid for points y satisfying

lY- (Yl + vn)[ << nT, n~.

Inserting n = r/m and y = 0 (appropriate to getting the total absorption at y = 0 fight), then yields Eq. (12).

Having shown the source of the error in the diffusion approximation for values of g away from the transition, we now ask how the general theory can be improved to yield q away from the transition. This can be done adapting ideas from the paper of Pikovsky and Grassberger [5 ]. Define finite time Lyapunov exponents h± (t, r0) for the stretching ratio of differential vectors transverse to the invariant hypersurface. Here r0 denotes an initial condition on the chaotic attractor and 0 to t is the time interval used in the calculation of h±. Say we look at many randomly chosen points r0 distributed according to the natural measure on the attractor. We then form a histogram of hA to produce a probability distribution function F (hA, t). For large t, the distribution takes the general large deviation form

F (-hA, t) ~ t -1/2 e x p ( - t ¢ (hA)).

Here the scaling function ¢(hA) behaves quadratically around its minimum value attained at hA = hA where ¢(hA) = 0. Thus, in the limit t --* c¢, F(hA, t) = g(hA--hA), corresponding to the known result that almost every initial point in the basin of the attractor yields the sameLyapunov exponents. For finite t, we see that there are fluctuations of hA about hA given by F(hA , t). For hA near ha_, we have that ¢ (hA) ~ ( 1/2) ¢" (h±) (hA - hA )2, and F (hA, t) is thus Gauss±an in that approximation. If the Gauss±an approximation holds, then it will result that our diffusion treatment is valid. Again using the quantity • (z) to denote the probability of a walker initiated at point z ever being absorbed at z = 0, taking z = In(1/y) , and noting that qb (z) is independent of time, we have

(z) = / d ~ ± ~ (z - "~±t)F(~., t),


where the right hand side represents the evolution of • (z) over a time interval of duration t. Assuming @ ,,~ y~ or (from z = I n ( l / y ) ) @(z) ,,~ exp(-~/z) we have

1 ,~ f dh± exp It (r/hi - ~b(h±)) ]

which for large t implies the saddle point condition

= =

Thus we have two equations for the location of the saddle Lyapunov exponent h±s and the exponent ~/. This is our desired general result for ~/, and it is valid away from as well as near the transition. Note that in the diffusion limit it is sufficient to know onl)~ the two constants D and h± to obtain t/ (via r / = Ih±l/D), while now knowledge of a function, O(h±), is required. If we use the quadratic approximation for ~b(h±), the saddle point condition yields h±s = - h ± and

= 2[h±l~b".

Recalling our definition of D (see Eq. (13)), and comparing with the large deviation form of F(-h±,t), we have 2~b" = 1/D, yielding our previous result from the diffusion approximation,

~1 = Ih±l/O.

6. 3. Uncertainty exponent (Eq. (17))

Consider a horizontal interval of length e initially on the line y = Yl and centered at x = x~. Depending on x~, there are many possible walks that originate from Yl at time n = 0. Let ~(y~ ,y, e ) denote the number of iterates for the particular walk from Yl to y for which an initial horizontal

interval of the line y = Yt expands to a horizontal length on one. For this particular walk, let r denote the number of rightward steps and ~ the number of leftward steps. Referring to Fig. 3, we have

Yl + r ~ - gy = y, (28a)

r + ~ = ~(Yl,Y,e). (28b)

The condition that the interval expand horizontally to a length of one is

e(1/a)r(1/ f l ) t = 1, (28c)

where we have noted that a rightward step corresponds to x falling in the interval (a, 1 ) while a leftward step corresponds to x falling in the interval (0 ,a) . Combining Eqs. (28a), (28b) and (28c) we have

~ ( y l , y , e ) = a l n ( 1 / e ) + ( Y - Y l ) b , (29)

where

a = ( y - 3 ) / [ y l n ( l / f l ) - ~ l n ( 1 / a ) ]

b = [ ln(1/a) - ln(1/ f l )] /[y ln(1/ f l ) - ~ ln (1 / a ) ] .


Since we are going to use the diffusion approximation, we can approximate for small h± = a ~ - f l ~ , in which case

a ~- 1/hll. (30)

We now ask, what is the probability that two points chosen at random in the same e interval of the horizontal line segment, y = Yl, 0 < x _< 1, are in different basins? We denote this probability p (e, x~, y~ ) where x~ denotes the center of the e interval. Let P0 be the probability that a randomly chosen point in the e interval is in the basin of the y = 0 attractor, and let Pl = (1 - P0) be the probability that it is in the basin of the y > 1 attractor. We then have p = 2popl. Since we consider Yl to be small (see Eq. (11 )), Pl is also small, so that

p ~ 2p~. (31)

To find Pl, we consider e intervals that, under repeated iterates, expand to occupy 0 < x < 1 before they ever reach y _> 1. We apply the map ~ times (given by Eq. (29)) so that the e length interval centered about Xl maps to some y value and covers the x interval (0, 1 ). In this case the mapped interval has a total length y~ ~ exp(- r /y) in the y > 1 basin. Hence, this is also the fraction of the original e interval in the y > 1 basin,

P l "~ exp [-r /y] .

Putting these results together, and averaging over xl, we have o o

w> ~ f 2 = exp[-r/y]P(y, y l , ~ ( y l , y , e ) ) d y . (32)

0

Introducing P, the Laplace transform of P, Eq. (32) becomes

oo + i o o + a

/p)~2fexp[-r/7]~-7~ / f -Fexp[s~]dsdy. (33) 0 - - ioo+a

Interchanging the orders of integration we have

+ioo +t7

(P) ~- trc'l f exp[saln(1/e)]E(s)ds, (34) - ioo+0"

where o o

£(s ) = / exp[- r /y] exp[ ( y - Yl )b]-Pdy. (35) , /

0

In the above, a is chosen so that all singularities are to the left of the integration path in s and so that the integral in y converges. Using Eq. (24), the integral (35) is easily evaluated. Examination of the resulting expression for E (s) reveals that the only singularity of E (s) is a branch point at s. = - v2 /4D coming from the square root x/1 + 4Ds/v 2 appearing in Eqs. (24c,d). Hence for small e (note that E(s) is independent of e ), Eq. (34) yields

(p) ~ exp[(s .aln(1/e))] = e -as" (36)


Comparing with Eq. (16) and utilizing Eq. (30), we obtain the uncertainty exponent, ~b = v2/(4Dhll), as stated in Eq. (17).

6.4. Noise (Eq. (19))

m

We solve for the Laplace transformed P subject to the boundary conditions (10) and (18), and for simplicity, we take Yl = Y~ - 0+. The Laplace transform N(s) of N(n) [where N(n) is the fraction of particles sprinkled in the vicinity of y = 0 that are still in the region y < 1 at time n] is

- - f 1 Xal(1 - exp[ -xay~] ) +x~l(1--exp[xbyv]) N(s) = J -PaY = - B ~bex----P[--xaY----------~] +--~aex{~[-Xb--~] (37)

0

Changing the sign of the square root x/1 + 4Ds/v z does the following:

ICb --+ --Ka, Ka ~ --Nb.

Since the expression (37) is invariant to this change, it is an even function of the square root, and there is thus no branch point singularity of N(s ) . The only possible singularity of N(s ) is a pole at the value of s satisfying

Xb exp[--tCbYv ] + /Ca exp[KbYv ] = 0. (38)

We look for a root of the transcendental equation (38) for which 4Ds/v 2 is small. In this case (24c) and (24d) yield Xb -~ 7, r.a -~ S/V. Thus Eq. (38) becomes

S _ (--~y~) + (s/v) exp(r/y,) = 0, ~/exp

or s = -v~/exp [ - ( r /+ (s/v) )y~ ] = - v rl~ ~ exp ( - s y , / v ) . For sy~/v << 1, this gives the location of the pole as

S = Sp ~ - v 2 v g / D . (39)

Since N(n) .,~ exp(spn) for large time, this yields the decay time given by Eq. (19).

6.5. Chaotic transient decay time (Eq. (20))

We now consider the case where h± is small and positive. Since h± > 0 (v < 0), y = 0 is no longer an attractor. Nevertheless, as discussed in Section 4, the chaotic invariant set at y = 0 still has an observable consequence in the form of a chaotic transient, and this transient has a well defined characteristic decay time which we wish to evaluate. To do this we evaluate the Laplace transform -p(y, Yl,S) (note that Eq. (24) does not apply for v < 0 so the calculation must be redone) and take its integral from y = 0 to y = c~. This yields

( 3 O

f - p d y 1 - exp[-xdYt ] (40) OlCclC d

0

where Xc,d = ~ 2 [~/1 + ~ 4-1]. The only singularity is the branch point at s = -v2/4D, which

yields Eq. (20).


7. Conclusions

The main results of this paper are as follows.

( l ) A simple model exhibiting a riddled basin has been shown to be analyzable in terms of a random walk.

(2) Results from the model have been calculated near the critical point for (i) the scaling of the measure of the riddling basin near the attractor whose basin is riddled (Eq. (15)), (ii) the uncertainty exponent characterization of the arbitrarily fine scaled basin structure (Eq. (17)), (iii) the effect of noise (Eq. (19)), and (iv) the average chaotic transient decay time just past the critical point (Eq. 20)).

(3) Based on our understanding of these results as arising from fluctuations of the perpendicular Lyapunov exponent h±, we have conjectured that the results are general for systems near the riddling transition, and we have tested this conjecture with numerical experiments.

Acknowledgements

This work was supported by the Office of Naval Research (Physics), by the Department of the Navy Space and Naval Warfare Systems Command, by the Air Force Office of Scientific Research, and by the National Science Foundation.

Appendix. Map with intermingled basins for which there is a random walk model

In all of the preceding considerations of this paper we have dealt with a situation in which there is one basin such that pieces of a second basin were arbitrarily near any point in the first basin (the first basin was riddled by the second basin). The second basin, however, was not riddled: there were open sets completely contained in the second basin. In this appendix we consider a two dimensional map for which there are two basins each one of which is riddled by the other (an arbitrarily small circle about any point in either of the two basins contains pieces of the other basin). In this situation the two basins are said to be intermingled [1]. The map we consider is similar to the map of Section 2 in that it admits a random walk model, as in Section 3, and can be analyzed in a manner similar to that of Section 6. (Actually there is a class of maps with the above properties, but we shall only consider one example to illustrate this class.)

The map we consider is

xn/a(yn) , forxn'< a(yn), Xn+l = (xn - a(yn)) / f l (yn) , forxn > a(Yn), (A.1)

{ ( l - y ) + (~ 7)Yn Y~+, (1 ~ 1 - ~ ) ) ~ f o r x n < a ( y n ) ,

= (7 + 1 )Yn, (A.2) (7 + 1) + - - - - 1-~n forxn > a(yn),

where f l (y) = 1 - a ( y ) , and we assume that }, > 1, and restrict our attention to the region 0 < x < 1, -1 _< y _< 1. From Eq. (A.2) we see that the lines y = 1 and y = -1 are both invariant


Y

i -1

(a)

0t_ ~_

-x=~(y)

1 ~ X

Yn+l i S lope = 1/7

1 . . . . . . " ,o,

. . / S,ope= , x<cqy) I / / / " '

- . \ • It/...,.- ! ', , " ~ " : ~ I 1 Yn I I / /

Slope = Y~l l / / / / ~ - - x > a ( y )

& Z . I Slope = 1/T I

Fig. A.1. The function x = a ( y ) , and the function Yn+l versus yn.

under the map. Eq. (A.1) is the same as Eq. (3a) (see Fig. 2) except that now a depends on y. The functions x = a(y) and Yn+l versus Yn are illustrated in Fig. A.1. Expanding Eq. (A.2) near y = - 1 , we have

(1 4- Yn)/7 forxn < a(Yn), 1 + Yn+l ~ 7(1 +Yn) forxn > a(Yn). (A.3)

Thus (7 > 1 ), for y near y = - 1, we see that there is attraction toward y = - 1 b y the factor 1 / 7

in x < a ( y ) a n d r e p u l s i o n f r o m y = - 1 b y t h e fac tor 7 in x > a ( y ) . H e n c e y = - 1 is a n a t tractor

if a ( - 1 ) - a _ > ½ and has y Lyapunov exponent

h~ -) = - ( a _ - f l - ) l n T , (A.4)

where fl_ = 1 - ct_. Similarly, expanding Eq. (A.2) near y = 1, we have

{ 7 ( 1 - y n ) fOrXn < ~(yn) 1 -Yn+l = (1 -Yn) /7 fOrXn > a(yn). (A.5)

Hence y = 1 is an attractor is a ( 1 ) = a+ < 1/2, and its y Lyapunov exponent is

h(j_ +) = - ( f l + - a + ) l n T , (A.6)

where fl+ - 1 - a +. We now make the change of variables,

( 1 + y'~ (A.7) y = In \-]--Z'-~) •

This reduces the y map to

= ~ Y n + Y i f xn>~(Yn) , Yn+ Yn - Y ifxn < ~(Yn), (A.8) 1 (

E, Ott, et al. / Physica D 76 (1994) 384-410 409

(a) probability ~(Yn)\ /probabil i ty o~(yn)

y - _ ) ~ Yn-7 Yn Yn+7 ~-.>~ (y = -1) (y = +1)

, a ( y )

(X . . . . . . . . . . . . . . . . . . . . . . . . . .

(b)

Fig. A.2. Illustration of the random walk model (fl(y) = 1 - ~(~)).

with y = lny and ~ ( y ) = a ( y ( y ) ) , where, from Eq. (A.7), we have

y ( y ) = tanh(½Y).

Note that Eq. (A.7) maps y = - 1 to y = - ~ and y = + 1 and y = + ~ . Thus, for xl randomly chosen, we have a random walk as illustrated in Fig. A.2 (analogous to Fig. 3).

The proof of intermingling for this example proceeds from the following two considerations:

(i) For any horizontal line y = Yt, - 1 < y~ < 1, and almost every Xl in (0, 1) with respect to Lebesgue measure, either l imn-~ Yn = + ~ (i.e., Yn ~ + 1 ), or l i m n ~ Yn = - ~ ( i . e . , Y , ~ - 1 ), and both of these outcomes have positive probability for a randomly chosen xl. (This follows from the properties of a random walk.)

(ii) An arbitrarily small horizontal line segment, when iterate forward in time, expands and eventually spans (0,1) in x. Since the expanded segment has pieces of both basins (by (i) above), the original small segment must also (compare Section 6.3). Hence there is intermingling.

Using the random walk picture (Fig. A.2) we can proceed with analyses as in the main body of the paper. For example, if we assume that ~ (y ) becomes constant for sufficiently large finite [y[ (i.e., ~ (y ) = a+ for y > y , > 0 and ~ (y ) = a_ for y < -[Y,I, 0 _< y , < e~), then the random walk problem can be solved exactly by following the method in Section 6.2: Let z = y/y, and consider z > z+ + 1 - y , / y . Also let ~+ (z) denote the probability that the walker goes to z = -oo (i.e., belongs to the basin of the y = - 1 attractor). Then ~+ (z) satisfies

¢)+(z) = a + ~ + ( z + 1) + f l + ~ + ( z - 1), (A.9)

whose general solution is ~+ (z) = A + B(a+/fl+) z. Since ~ + ( + c ~ ) = 0, we have A = 0 (the value of B depends on the form of ~ (y ) in ]Yl < Y,), so that

q,+ ( z ) = B( ,~+l~+)zy~ + (A.10)

~+ = [ l n ( f l + / e ~ + ) ] / 7 , ( A . 1 1 )

which is analogous to the expression for ~ in Eq. (25b). Using the diffusion approximation, we can also obtain results analogous to those of Section 4.

R e f e r e n c e s

[1] J.C. Alexander, J.A. Yorke, Z. You and I. Kan, Int. J. Bifur. Chaos 2 (1992) 795 ; I. Kan, preprint (1992); Bull. Am. Math. Soc., to be published; E. Ott, J.C. Sommerer, J.C. Alexander, I. Kan and J.A. Yorke, Phys. Rev. Lett. 71 (1993) 4134.


[2] J.C. Sommerer and E. Ott, Nature 365 (1993) 136. [3] S.W. McDonald, C. Grebogi, E. Ott and J.A. Yorke, Physica D 17 (1985) 125. [4] J. Milnor, Commun. Math. Phys. 99 (1985) 177. [ 5 ] Similar diffusion models arise in the dynamics problems treated by

A.S. Pikovsky and P. Grassberger, J. Phys. A 24 (1991) 4587; L. Yu, E. Ott and Q. Chen, Phys. Rev. Lett. 65 (1990) 2935; Physica D 53 (1992) 102; A.S. Pikovsky, Phys. Lett. A 165 (1992) 33 ; H. Fujisaka et al., Prog. Theor. Phys. 76 (1986) 1198; J. Heagy, N. Platt and S. Hammel, Phys. Rev. E, to be published.

[6] J.D. Farmer, Phys. Rev. Lett. 55 (1985) 351. [7] C. Grebogi, S.W. McDonald, E. Ott and J.A. Yorke, Phys. Lett. A l l 0 (1985) 1. [8] C. Grebogi, E. Ott, F.J. Romeiras and J.A. Yorke, Phys. Rev. A 36 (1987) 5336;

C. Grebogi, E. Ott and J.A. Yorke, Phys. Rev. Lett. 57 (1986) 1284. [9] W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1966)

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