Brian Hunt

Optimal Periodic Orbits of Chaotic Systems

The presence of chaos in physical systems has been extensively demonstrated and is very common. In practice, however, it is often desired that chaos be avoided and/or that the system performance be improved or changed in some way. Given a system with a chaotic attractor, one approach might be to make some large and possibly costly alteration in the system that completely changes its dynamics in such a way as to achieve the desired behavior. Here we assume that this avenue is not available. Thus we address the question of how one can obtain improved performance in a chaotic system by making only small time-dependent perturbations. A key observation is that a chaotic attractor typically has embedded within it an infinite number of unstable periodic orbits. Since we wish to make only small perturbations to the system, we do not envision creating new orbits with very different properties from the existing ones. Thus we seek to exploit the already existing unstable periodic orbits. Once such an orbit is discovered, the control method proposed by Ott, Grebogi, and Yorke can be used to determine small time-dependent parameter perturbations that stabilize this orbit. A question that naturally arises is which periodic orbit to use. Ideally one would like to have information on all the periodic orbits embedded in an attractor, and to choose the best one. In practice, however, one can only expect to determine and control unstable periodic orbits with sufficiently low period, and to choose the best performing of these orbits. We claim that this approach typically yields optimal or near-optimal performance; that is, one could not do much better even if one could control higher period (or nonperiodic) orbits. We assume that the performance of a orbit can be quantified as the time average of a given smooth function of state space over the orbit. Given ergodicity, typical (uncontrolled) orbits on the chaotic attractor will yield a common value of the performance average. However, unstable periodic orbits and other unstable motions embedded in the attractor will in general yield different values of the performance average. We consider the question of which orbit(s) on the attractor yield the largest (optimal) performance average. We present numerical evidence and analysis that indicate that optimal performance is typically achieved by an unstable periodic orbit with low period.