Nonlinear feedback for control of chaos

Abstract

We generalize a method of control of chaos which uses delayed feedback at the period of an unstable orbit to stabilize that orbit. The generalization consists of substituting some portion of the nonlinear dynamical system with a delayed dynamics, rather than using a linear delay function for control. A further generalization, in which the control function retains memory of all previous periods, allows the region of the parameter space over which control can be achieved to be extended, but at the price of losing the ability to achieve superstability. Nonliner feedback results in a larger basin of attraction to the stabilized orbit than linear feedback. For a simple test mapping studied (the logistic map) the dimension of the system increases from one to two by introducing control. We show in the case involving memory, for a particular choice of the relationship between the control parameters, that the superstable orbit can be recovered without reducing the parameter space that can be controlled. This particular solution, in addition to having the largest basin of attraction of the methods considered, retains the dimension of the uncontrolled system.