Volume 99A, number 9 LETTERS 26 December 1983

FINAL STATE SENSITIVITY: AN OBSTRUCTION TO

Celso GREBOGI 1 Steven W. McDONALD 1 , 1'2 and James A. YORKE 3 University of Maryland, College Park, MD 20742, USA

Received 21 October 1983

It is shown that nonlinear systems with multiple commonly require very accurate initial conditions for the reliable prediction of final states. A scaling exponent for the final-state-uncertain volume dependence on un- certainty in initial conditions is defined and related to the fractai dimension of basin boundaries.

Typical nonlinear dynamical systems may have more is the "basin bounctary". It the initial conaitions are than one possible time-asymptotic final state. In such uncertain by an amount e, then (cf. fig. 1), for those cases the final state that is eventually reached depends initial conditions within e of the boundary, we cannot on the initial state of the system. In this letter we say a priori to which the eventually wish to consider the extent to which uncertainty in tends. For example, in fig. 1, 1 and 2 represent two initial conditions leads to uncertainty in the final state. initial conditions with an uncertainty e. The orbit To orient the discussion, consider the simple two- generated by initial condition 1 is attracted to attrac- dimensional phase space diagram schematically de- tor B. Initial condition 2, however, is uncertain in the picted in fig. 1. There are two possible final states sense that the orbit generated by 2 may be attracted ("attractors") denoted A and B. Initial conditions on either to A or to B. In particular, consider the fraction one side of the boundary, Z, eventually asymptote to of the uncertain phase space volume within the rec- B, while those on the other side of Z eventually go to tangle shown and denote this fraction f. For the case A. The region to the left (right) of Z is the "basin of shown in fig. 1, we clearly have f ~ e. It is one of the attraction" for attractor A (or B, respectively) and main points of this letter that, from the point of view of prediction, much worse scalings of f with e fre- quently occur in nonlinear dynamics. In particular, if 1 Laboratory for Plasma and Fusion Energy Studies, and De- partment of Physics and Astronomy. f ~e ~ , (1) 2 Also, Department of Electrical Engineering. 3 Institute for Physical Science and Technology, and Depart- with c~ < 1, we shall say that there is final state sensi- ment of Mathematics. tivity. In fact, a substantially less than one is, we be- lieve, fairly common. In such a case, a substantial im- provement in the initial condition uncertainty, e, yields only a relatively small decrease in the uncertainty of the final state as measured by f. While ¢x is equal to one for simple basin boundaries, such as that depicted in fig. 1, highly convoluted boundaries with noninteger () dimension also occur. We use here the capacity definition of dimen- sion [ 1 ], Fig. 1. A region of phase space divided by the basin boundary Y. into basins of attraction for the two attractors A and B. 1 In N(8) and 2 represent two initial conditions with uncertainty e. d = lim (2) 6--,0 ln(1/8) '

0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland 415 Volume 99A, number 9 PHYSICS LETTERS 26 December 1983

where, if the dimensionality of the relevant phase space is D, then N(6) is the minimum number of D ( dimensional cubes of side 6 needed to cover the basin boundary. In general, since the basin boundary di- c vides the phase space, its dimension d must satisfy d ~> D-1. It can be proven ,1 [2] that the following c relation between the index c~ and the basin boundary dimension holds -(

c~ = O - d. (3) -( For a simple boundary, such as that depicted in fig. 1, we have d = D-l, and eq. (3) then gives c~ = 1, as ex- O rr/2 Jr pected. For a fractal basin boundary, d > D-l, and Fig. 2. Basins of attraction for eqs. (4). Arrows denote the eq. (3) gives a < 1 (i.e., there is final state sensitivity). two attractors. To see heuristically how (3) comes about from (2), set the cube edge 6 equal to the initial condition un- certainty e. The volume of the uncertain region of these parameters we find numerically that the only phase space will be of the order of the total volume attractors are the fixed points (0, -0.3) and (Tr, 0.3). of all the N(e) D-dimensional cubes of side e needed Fig. 2 shows a computer generated picture of the basins to cover the basin boundary. Since the volume of one of attraction for the two fixed point attractors. This of these D-dimensional cubes is e D, the uncertain vol- figure is constructed using a 256 X 256 grid of initial ume of phase space is of the order of eDN(e). Noting conditions. Each initial condition is iterated until it is that N(e) ~ e -d satisfies (2), we use this to estimate close to one of the two attractors (100 iterates of the the uncertain phase space volume as eDN(e) ~ e D-d, map is always sufficient to accomplish this). If an orbit which is the result predicted by eqs. (1) and (3). goes to the attractor at 0 = 0, a black dot is plotted at We now illustrate the above with a concrete exam- the corresponding initial condition. If the orbit goes ple. Subsequently, we argue that the demonstrated to the other attractor, no dot is plotted. Thus the phenomena occur commonly in nonlinear dynamics black and blank regions are essentially pictures of the (e.g., in the [3] and in experiments on basins of attractions for the two attractors to the ac- fluids such as those of Berg6 and Dubois [4]). curacy of the grid used and of the computer plotter. We consider the two-dimensional map (Due to the symmetry of the map we have only shown 0 ~< 0 ~ rr in fig. 2). Fine scale structure in the basins On+ 1 = On + a sin 20 n - b sin 40 n - x n sin On , (4a) of attraction is evident. This is a consequence of the Cantor set nature of the basin boundary * 2. In fact, Xn+l = -Jo cos On , (4b) magnifications of the basin boundary show that, as where 0 and 0 + 27r are identified as equivalent. This we examine it on a smaller and smaller scale, it con- map has two fixed points (0, x) = (0, -J0), and (Tr, J0), tinues to have structure. which are attracting for I1 + 2a - 4bl < 1. Numerical experiments on eqs. (4) with different sets of parame- ters have been performed. Here, as an example, we re- ~:2 The Cantor set structure of the basin boundary for eqs. (4) is due to the presence of a horseshoe in the dynamics port results forJ 0 = 0.3, a = 1.32, and b = 0.90. For [5]. In fact, eqs. (4) were chosen because they are a par- ticularly straightforward example of this. (This point will *1 For the purposes of the proof of eq. (3) and for those of be extensively discussed in ref. [9]). Cantor set basin bound- the present letter, eq. (1) should be regarded as shorthand aries due to horseshoes axe not curves (i.e., they cannot be for a = lime~olnf/ln e. In addition, we note that, if the represented as x = x(u), 0 = O(u) with x(u) and O(u) con- basins are unbounded regions, then we restrict attention tinuous functions of u), and have been known for a long to a bounded region of phase space (e.g., the rectangular time (see, for example ref. [6]). Fractal basin boundaries region of fig. 1) for the purposes of calculating f in (1) which are curves (unlike that for fig. 2) have been dis- and N(6) in (2). cussed by Grebogi et al. [7].

416 Volume 99A, number 9 PHYSICS LETTERS 26 December 1983

We now wish to explore the consequences for pre- for e = 0.125, 59% of the initial conditions are uncer- diction of this infinitely fine scaled structure. To do tain; for e = 0.002, 26%; for e = 3 × 10 -5, 12%. Thus this, consider an initial condition (00, x0). We ask, even apparently small uncertainty in the initial con- what is the effect of a small change e in the x-coordi- ditions yieMs substantial fractions of the phase space nate? Thus we iterate the initial conditions (00, x0), which are uncertain as to which final state is eventually (00, x 0 + e), and (00, x 0 - e) until they approach one attained. The implication is that extraordinarily high of the attractors. If either or both of the perturbed accuracy of initial conditions may sometimes be nec- initial conditions yield orbits which do not approach essary for the reliable prediction of the eventual final the same attractor as the unperturbed initial condi- state. tion, we say that (00, x0) is uncertain. Now say that Furthermore, the example just discussed is by no we randomly choose a large number of initial condi- means extreme or unusual. As evidence for this we tions in the rectangle shown in fig. 2, and let fdenote cite two examples: the fraction of these which we find to be uncertain. (1) The Lorenz system. In his pioneering study, From the definitions off and f (f is the fraction of Lorenz [3] examined a system of three first-order or- uncertain phase space volume) we expect that f is ap- dinary differential equations modeling the B6nard in- proximately proportional to f, and hence [from (1) stability. In this model a chaotic attractor occurs when and (3)] f~ e D-d. Fig. 3 shows results from a set of the Rayleigh number r exceeds a critical value r c ~ 24.06. numerical experiments on the scaling of f with e. In For 1 < r < r c there is no chaotic attractor, but there generating this figure, 8192 randomly chosen initial remain two nonchaotic attractors, one representing conditions were used for each value of e. The statisti- steady counterclockwise convective flow and the other cal error in the number N' of uncertain initial condi- representing steady clockwise convective flow. Based tions at each e was estimated to be x,~-', and this was on the results of ref. [8], it follows that [9], for F used in determining the error bars shown in the figure. r e > r > r e ~- 13.93, the Lorenz system has final state Linear dependence of log fwith log e is evident, thus sensitivity (i.e., a < 1) with respect to the two steady indicating an approximate power law dependence, eq. convective attractors. In addition, the sensitivity can (1). We find from fig. 3 that c~ ~ 0.2. Thus, from (3), be much more severe than in our numerical example, the dimension of the basin boundary is d ~ 1.8. Note, figs. 2 and 3. For example, using ref. [10] we can in particular, the numerical values that result (cf. fig. 3): crudely estimate [9] ct "- 0.1 at r c- r = 4, ct ~ 10 -2 at r c - r = 1.6, and c~ "" 10 -3 at r c - r = 0.8. with ct-+0asr-~r c. 1.00 , , I I I I (2) The experiments of Berg@ and Dubois [ 4]. These authors have performed experiments on the B6nard 0.50 :\. instability in a low aspect ratio rectangular cell for high Rayleigh number and high Prandtl number. They \. observe that the system can have multiple attractors, \, and that a rather long chaotic transient [ 10,11] exists \ before the system settles into one of the attractors. 0. I0 Since the system evolution during the transient phase depends strongly on initial conditions, it is to be ex- \, pected that the final state will also, and that t~ < 1 in 0.05 eq. (1) will apply. At somewhat lower Rayleigh num- \\\ ber (450 >~ r > 200) different stable attractors still simultaneously coexist but long chaotic transients do \ not occur. Even in this range it is probable that final state sensitivity will occur (e.g., in our example, fig. 2, 0.01 L l h I I I I I i 10-2 t0-4 I0-6 iO-e io-IO the average decay time to the vicinity of one of the E attractors is only about five iterates). Fig. 3. Log-log plot ofTversus e. In conclusion, the notion of final state sensitivity

417 Volume 99A, number 9 PHYSICS LETTERS 26 December 1983

has been introduced, and its implications for predic- [2] E, Ott, E.D. Yorke and J.A. Yorke, to be published. tion have been illustrated by a numerical experiment [3] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. and by suitable interpretation of previous theoretical [4] P. Berg6 and M. Dubois, Phys. Lett. 93A (1983) 365; and private communication. [8] and experimental [4] work. [5] S. Smale, Bull. Am. Math. Soc. 73 (1967) 747. [6] M. Levy, Mem. Amer. Math. Soc. 32 (1981) 244, and We thank J. Guckenheimer for valuable comments references therein. and P. Berg6 and M. Dubois for discussions concerning [7] C. Grebogi, E. Ott and J.A. Yorke, Phys. Rev. Lett. 50 their experiments. This work was supported by the (1983) 935; and erratum 51 (1983) 942. [8] J.L. Kaplan and J.A. Yorke, Commun. Math. Phys. 67 Department of Energy (Office of Basic Energy Sciences) (1979) 93. and by the Air Force Office of Scientific Research. [9] S.W. McDonald, C. Grebogi, E. Ott and J.A. Yorke, to be published. References [10] J.A. Yorke and E.D. Yorke, J. Stat. Phys. 21 (1979) 263. [11] C. Grebogi, E. Ott and J.A. Yorke, Physica 7D (1983) [1] J.D. Farmer, E. Ott and J.A. Yorke, Physica 7D (1983) 181. 153.

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