Optimal periodic orbits of chaotic systems.

Invariant sets embedded in a chaotic attractor can generate time averages that diier from the average generated by typical orbits on the attractor. Motivated by two diierent topics (namely, controlling chaos and riddled basins of attraction), we consider the question of which invariant set yields the largest (optimal) value of an average of a given smooth function of the system state. We present numerical evidence and analysis which indicate that the optimal average is typically achieved by a low period unstable periodic orbit embedded in the chaotic attractor. 1 Many questions concerning dynamical behavior are addressed by consideration of the long-time average of a function F of the state vector x, hFi = lim t!1 1 t Z t 0 F (x(t 0))dt 0 ; (1a) hFi = lim t!1 1 t t X t 0 =1 F (x t 0); (1b) where t denotes time and is either continuous (Eq. (1a)) or discrete (Eq. (1b)). In this paper we consider systems, such that, for typical choices of the initial x, the trajectory generated by the dynamical system is chaotic, and has a well-deened long-time average (1). (Here \typical" is with respect to the Lebesgue measure of initial conditions in state space.) We note, however, that atypical initial conditions may generate orbits embedded in the chaotic attractor that have diierent values for hFi than typical orbits. For example, consider a chaotic attractor with a basin of attraction B. Even though there is a set of initial conditions in B all yielding the same value for hFi, and the state space volume (Lebesgue measure) of these initial conditions is equal to the entire volume of B, there is still a zero volume set of initial conditions (\atypical" initial conditions) whose orbits asymptote to sets within the chaotic attractor but for which hFi is diierent from the average attained by typical orbits. A familiar case where this happens is when the initial condition is placed exactly on an unstable periodic orbit embedded in a chaotic attractor (or on the stable manifold of the unstable periodic orbit). The question we address is the following. Which (atypical) orbit on the attractor yields the largest value of hFi? To our knowledge this question has not been previously addressed, yet it is fundamental to at least two important problem areas of current interest: a. Controlling chaos. In one often used method 1] for the control of …