Nonattracting chaotic sets

Abstract

We have already encountered situations where chaotic motion was non-attracting. For example, the map Eq. (3.3) had an invariant Cantor set in [0, 1], but all initial conditions except for a set of Lebesgue measure zero eventually leave the interval [0, 1] and then approach x = ±∞. Similarly, the horseshoe map has an invariant set in the square S (cf Figure 4.1), but again all initial conditions except for a set of Lebesgue measure zero eventually leave the square. The invariant sets for these two cases are examples of nonattracting chaotic sets. While it is clear that chaotic attractors have practically important observable consequences, it may not at this point be clear that nonattracting chaotic sets also have practically important observable consequences. Perhaps the three most prominent consequences of nonattracting chaotic sets are the phenomena of chaotic transients, fractal basin boundaries , and chaotic scattering . The term chaotic transient refers to the fact that an orbit can spend a long time in the vicinity of a nonattracting chaotic set before it leaves, possibly moving off to some nonchaotic attractor which governs its motion ever after. During the initial phase, when the orbit is in the vicinity of the nonattracting chaotic set, its motion can appear to be very irregular and is, for most purposes, indistinguishable from motion on a chaotic attractor. Say we sprinkle a large number of initial conditions with a uniform distribution in some phase space region W containing the nonattracting chaotic set.