Finding optimal orbits of chaotic systems
Abstract. Chaotic dynamical systems can exhibit a wide variety of motions, including periodic orbits of arbitrarily large period. We consider the question of which motion is optimal, in the sense that it maximizes the average over time of some given scalar ``performance function". Past work has shown that optimal motions tend to be periodic orbits with low period but does not describe, beyond a brute force approach, how to determine which orbit is optimal in a particular scenario. For one-dimensional expanding maps, we have developed a constructive method for computing the optimal average and corresponding periodic orbit. We demonstrate that this method works quite well in practice and discuss progress toward a method for higher dimensional systems.