APPROACHING NONLINEAR DYNAMICS BY STUDYING THE MOTION OF A PENDULUM III: PREDICTABILITY AND CONTROL OF CHAOTIC MOTION

Abstract

We apply the concepts of predictability and control of chaotic motion to the driven damped pendulum. A physical measure of predictability is defined and determined from experimental data as well as from the equations of motion. The results are presented in predictability portraits which constitute an intrinsic pattern of zones of varying predictability. The origin of these patterns is related to the unstable periodic orbits and their invariant manifolds within the attractor. In order to control the chaotic motion of the pendulum we implement an extension of the OGY feedback control method, which we call “local control method.” With this control scheme any motion of the pendulum which is a solution of the systems equations of motion can be stabilized. We apply the control formalism in order to stabilize experimentally unstable periodic orbits as well as arbitrarily chosen chaotic trajectories.