MULTIPLE-ATTRACTOR BIFURCATIONS AND QUASIPERIODICITY IN PIECEWISE-SMOOTH MAPS

Abstract

It is known that border-collision bifurcations in piecewise-smooth maps can lead to situations where several attractors are created simultaneously in so-called "multiple-attractor" or "multiple-choice" bifurcations. It has been shown that such a situation leads to a fundamental source of uncertainty regarding which attractor the system will follow as a parameter is varied through the bifurcation point. Phenomena of this type have been observed in various physical and engineering systems. We have recently demonstrated that piecewise-smooth systems can exhibit a new type of border-collision bifurcation in which a stable invariant curve, associated with a quasiperiodic or a mode-locked periodic orbit, arises from a fixed point. In this paper we consider a particular variant of the multiple-attractor bifurcation in which a stable periodic orbit arises simultaneously with a closed invariant curve. We also show examples of simultaneously appearing stable periodic orbits and of the simultaneous generation of periodic and chaotic attractors.