Border-Collision Bifurcations on a Two-Dimensional Torus and Transitions to Chaos in a Control System with Pulse-Width Modulation

Abstract

Resonance phenomena in a piecewise-smooth dynamical system with external periodic action and transitions to chaos via border-collision bifurcations of cycles on a two-dimensional torus are studied in this work. A control system with pulse-width modulation which is described by a three-dimensional system of piecewise-linear non-autonomous equations is considered as an example.It is shown that domains of synchronization of quasi periodic oscillations for piecewise-smooth dynamical systems essentially differ from the classical Arnol'd tongues. The difference lies in the inner structure and bifurcational transitions. There are two different kinds of synchronization domains, one of them consists of domains that contain regions of bistability. The structure of border-collision bifurcation boundaries of synchronization tongues and transitions to chaos via border-collision bifurcation of cycles on a two-dimensional torus is studied in detail.