VOLUME 78, NUMBER 24 PHYSICAL REVIEW LETTERS 16 JUNE 1997 From High Dimensional Chaos to Stable Periodic Orbits: The Structure of Parameter Space Ernest Barreto,* , ² Brian R. Hunt, ‡ Celso Grebogi, ² , ‡, § and James A. Yorke ‡, § University of Maryland, College Park, Maryland 20742 (Received 10 October 1996) Regions in the parameter space of chaotic systems that correspond to stable behavior are often referred to as windows. In this Letter, we elucidate the occurrence of such regions in higher dimensional chaotic systems. We describe the fundamental structure of these windows, and also indicate under what circumstances one can expect to find them. These results are applicable to systems that exhibit several positive Lyapunov exponents, and are of importance to both the theoretical and the experimental understanding of dynamical systems. [S0031-9007(97)03367-X] PACS numbers: 05.45.+b A characteristic feature of one dimensional chaotic dynamical systems is the appearance of stable behavior as system parameters traverse chaotic regions. For example, in the bifurcation diagram of the quadratic map x ! x 2 2 a, large areas of chaotic behavior are visible, but are punctuated by parameter intervals in which stable periodic behavior is observed. These intervals, commonly called windows, have long been believed to be present arbitrarily close to every parameter value that leads to chaos. Only recently has this been proven to be true [1]. In this Letter, we address the fundamental problem of the occurrence of stable periodic behavior amid high di- mensional chaos. We propose a conjecture that describes the nature of parameter space for chaotic maps, and, fur- thermore, indicates under what circumstances one may rea- sonably expect to have numerous parameter space regions that lead to stable periodic behavior (i.e., windows). This conjecture can be of considerable practical importance for experimentalists, since it is often desirable to establish non- chaotic behavior in the vicinity of parameter values that give rise to chaos. We begin by describing the content of our conjecture in practical terms. We then motivate the work, and conclude with a precise mathematical statement of our result. Most chaotic systems discussed in the scientific literature are almost certainly “fragile” in the sense that a slight alteration of a large number N of parameters will destroy the chaos and replace it by a stable periodic orbit. Let k be the number of positive Lyapunov exponents of a chaotic attractor, but suppose that only n , N parameters can be varied in an experiment. We conjecture that if n $ k, then typically a slight change applied to these n parameters can destroy the chaos. If, however, n , k, then the chaos typically cannot be so destroyed. In this case, we expect that for an experimentally significant parameter space region near the original setting, the chaotic attractor will persist. For example, if k  1, then as one parameter is slightly varied, numerous stable regions will be observed. If k  2, then slight changes to a single parameter will typically not destroy the chaos. However, if two parameters are available, then the parameter space can be systematically searched in two dimensions, and many windows can be located. Knowledge of these windows may be helpful in controlling the system, even in the presence of noise. Alternatively, if the location of a desired window is to be calculated, our conjecture indicates that one must typically solve for at least n  k parameters. We now motivate the work. Our conjecture is based on the idea that a window is constructed around a spine locus. For simplicity, we consider maps that contain critical points [2]. For one dimensional maps, the spine locus corresponds to parameter values that give rise to superstable orbits. To illustrate, consider a map x ! Fx; a, where a is a scalar parameter. The stability of a period p orbit is governed by m  d dx F p x  d dx Fx p  d dx Fx p21  ··· d dx Fx 1 , where the derivatives are evaluated at each point in the orbit. The orbit is asymptotically stable if jmj , 1, and an orbit that contains a critical point of F, where dFdx  0, has m  0 and is called a superstable orbit. As the parameter varies in the vicinity of the spine, m sweeps through the interval 21, 1. In this way, the extent of the window is delineated. For the quadratic family x ! x 2 2 a, the windows are intervals in the (one dimensional) parameter space built around isolated spine points. For maps with more parameters, bifurcation diagrams are usually drawn entirely in parameter space, with points shaded differently to represent the type of dynamics generated. In the case of the two parameter quadratic family x ! x 2 2 a 2 2 b, the spine locus consists of two parabolas; see Fig. 1. The black curves, defined by the condition m  0, are the spine locus; these clearly determine the shape of the window. Of importance for our purposes is the dimension of the spine locus. In particular, we note that the condition m  0 is a single constraint, and hence the spine locus is of codimension one in the parameter space (i.e., one less than the parameter space dimension). The dimension of the spine determines the geometry of the window in the following sense. If the spine is a 0031-9007 97 78(24) 4561(4)$10.00 © 1997 The American Physical Society 4561