Global stabilization of periodic orbits in chaotic systems by using symbolic dynamics

Abstract

In this report, a control method for the stabilization of periodic orbits for a class of discrete-time systems that are topologically conjugate to symbolic dynamics is proposed and applied to a population model in an ecosystem and the Smale horseshoe map. A periodic orbit is assigned as a target by giving a sequence in which symbols have periodicity. As a consequence, it is shown that any periodic orbits can be globally stabilized by using arbitrarily small control inputs. This work is the first attempt to systematically design a control system based on symbolic dynamics in the sense that one estimates the magnitude of control inputs and analyzes the Lyapunov stability.