Optimal periodic orbits of chaotic systems occur at low period

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 that indicate that the optimal average is typically achieved by a low-period unstable periodic orbit embedded in the chaotic attractor. In particular, our results indicate that, if we consider that the function to be optimized depends on a parameter gamma, then the Lebesgue measure in gamma corresponding to optimal periodic orbits of period p or greater decreases exponentially with increasing p. Furthermore, the set of parameter values for which optimal orbits are nonperiodic typically has zero Lebesgue measure.