Stabilization Control of Quasi-periodic Orbits

Abstract

A quasi-periodic orbit possesses the properties of both a periodic orbit and a chaotic orbit, which are almost periodic and aperiodic, respectively. Whereas the stabilization of a periodic orbit is widely discussed, that of a quasi-periodic orbit remains an enigma because of its aperiodicity. We focus on a quasi-periodic orbit on an invariant closed curve in discrete-time systems, which is the simplest case of dynamics on a high-dimensional invariant torus. In this case, a quasi-periodic orbit can be characterized by its rotation number, which is reflected in the design of control methods. We apply three control methods, the external force control, the delayed feedback control, and the pole assignment method, to stabilize an unstable quasi-periodic orbit. Although these control methods have been used to stabilize unstable periodic orbits, we show that they are also applicable to unstable quasi-periodic orbits.