CONTROLLING CHAOS IN HIGHLY DISSIPATIVE SYSTEMS - A SIMPLE RECURSIVE ALGORITHM

We present a recursive proportional-feedback (RPF) algorithm for contr olling deterministic chaos. The algorithm is an adaptation of the meth od of Dressler and Nitsche [Phys. Rev. Lett. 68, 1 (1992)] to highly d issipative systems with a dynamics that shows a nearly one-dimensional return map of a single variable X measured at each Poincare cycle. Th e result extends the usefulness of simple proportional-feedback contro l algorithms. The change in control parameter prescribed for the nth P oincare cycle by the RPF algorithm is given by deltap(n) = K(X(n) - X( F)) + Rdeltap(n-1), where X(F) is the fixed point of the target orbit, and K and R are proportionality constants. The recursive term is show n to arise fundamentally because, in general, the Poincare section of the attractor near X(F) will change position in phase space as small c hanges are made in the control parameter. We show how to obtain K and R from simple measurements of the return map without any prior knowled ge of the system dynamics and report the successful application of the RPF algorithm to model systems from chemistry and biology where the r ecursive term is necessary to achieve control.