Optimal periodic orbits of chaotic systems

Invariant sets embedded in a chaotic attractor can generate time averages that differ from the average generated by typical orbits on the attractor. Motivated by two different 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.