Merging Bifurcation in a Pulse-Width Modulated Control System

Abstract

The paper describes bifurcation of a chaotic attractor in a pulse control system. Systems of this type are modelled by continuous piecewise smooth maps or differential equations with discontinuous right-hand sides. In addition to the classical bifurcations that occur in smooth systems, piecewise smooth systems exhibit a wide range of nonlinear phenomena associated with the so-called border collisions bifurcations. Border collision bifurcations arise when an invariant set, for example, a fixed point collides with a switching manifold. For many applications of nonlinear science, the study of bifurcations of chaotic attractors is one of the most important tasks. We focus on the bifurcation associated with merging of chaotic attractors known as "merging crisis". In particular, the paper examines a sequence of merging transitions in which a four-band chaotic attractor transforms into a single - band one in homoclinic bifurcations of unstable periodic orbits with negative multipliers.