Controlling chaos in highly dissipative systems: A simple recursive algorithm

Abstract

We present a recursive proportional-feedback (RPF) algorithm for controlling deterministic chaos. The algorithm is an adaptation of the method of Dressler and Nitsche [Phys. Rev. Lett. 68, 1 (1992)] to highly dissipative systems with a dynamics that shows a nearly one-dimensional return map of a single variable X measured at each Poincaré cycle. The result extends the usefulness of simple proportional-feedback control algorithms. The change in control parameter prescribed for the nth Poincaré cycle by the RPF algorithm is given by δ${\mathit{p}}_{\mathit{n}}$=K(${\mathit{X}}_{\mathit{n}}$-${\mathit{X}}_{\mathit{F}}$)+Rδ${\mathit{p}}_{\mathit{n}\mathrm{-}1}$, where ${\mathit{X}}_{\mathit{F}}$ is the fixed point of the target orbit, and K and R are proportionality constants. The recursive term is shown to arise fundamentally because, in general, the Poincaré section of the attractor near ${\mathit{X}}_{\mathit{F}}$ will change position in phase space as small changes are made in the control parameter. We show how to obtain K and R from simple measurements of the return map without any prior knowledge of the system dynamics and report the successful application of the RPF algorithm to model systems from chemistry and biology where the recursive term is necessary to achieve control.