Tracking unstable orbits in chaos using dissipative feedback control

Abstract

Using a form of linear feedback we call dissipative feedback control, we show how to use external forcing to control a chaotic dynamical system to a fixed point or an unstable periodic orbit when the location of the fixed point or unstable periodic orbit may change slowly with time. The ability to follow a desired state of the system by an external control even when that state is slowly varying in time we call tracking. This slow ‘‘drift’’ of states is the usual situation in actual experimental realizations of chaotic systems in nonlinear circuits and other physical manifestations, and this drift can be accounted for by providing a slow dynamics for the location of the fixed point or periodic orbit. We discuss the theoretical aspects of this idea and show its feasibility in some experiments with nonlinear circuits with chaotic behavior.