OPTIMAL PERIODIC-ORBITS OF CHAOTIC SYSTEMS OCCUR AT LOW PERIOD

Invariant sets embedded in a chaotic attractor can generate time avera ges that differ from the average generated by typical orbits on the at tractor. Motivated by two different topics (namely, controlling chaos and riddled basins of attraction), we consider the question of which i nvariant set yields the largest (optimal) value of an average of a giv en smooth function of the system state. We present numerical evidence and analysis that indicate that the optimal average is typically achie ved by a low-period unstable periodic orbit embedded in the chaotic at tractor. In particular, our results indicate that, if we consider that the function to be optimized depends on a parameter gamma, then the L ebesgue measure in gamma corresponding to optimal periodic orbits of p eriod p or greater decreases exponentially with increasing p. Furtherm ore, the set of parameter values for which optimal orbits are nonperio dic typically has zero Lebesgue measure.