a Department of Computer Science, Kursk State Technical University, 50 Year of October street 94, 305040 Kursk, Russia
b Center for Chaos and Turbulence Studies, Department of Physics, Technical University of Denmark, 2800 Lyngby, Denmark
Abstract:
This paper studies resonance phenomena in a piecewise-smooth dynamical system with external periodic action and examines transitions to chaos via border-collision bifurcations of cycles on a two-dimensional torus. As an example we consider a control system with pulse-width modulation described by a three-dimensional set of piecewise-linear non-autonomous equations. It is shown that the domains of synchronization of quasiperiodic oscillations for piecewise-smooth dynamical systems differ in an essential way 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 which contains regions of bistability. The structure of border-collision bifurcation boundaries of synchronization tongues and transitions to chaos via border-collision bifurcations of cycles on a two-dimensional torus are described in detail.