## 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.