Local analysis of C-bifurcations in n-dimensional piecewise-smooth dynamical systems

This paper is concerned with the analysis of non-standard bifurcations in p iecewise smooth (PWS) dynamical systems. These systems are particularly rel evant in many areas of engineering and applied science and have been shown to exhibit a large variety of nonlinear phenomena including chaos. While th ere is a complete understanding of local bifurcations for smooth dynamical systems, there is an urgent need for a complete theory regarding bifurcatio ns in PWS systems. Although it is often claimed in the Western literature t hat no such theory exists, an analytical Framework to describe these bifurc ations appeared in the Russian literature in the early Seventies, when Mark Feigin published his pioneering work on the analysis of C-bifurcations (al so known as border-collision bifurcations) in n-dimensional PWS systems. Ou r aim is to bring his results in a more complete Form to a wider audience w hile putting them in the context of modern bifurcation analysis. First, a t ypical C-bifurcation scenario is described. Then, an appropriate local map is derived and used to derive a set of elementary conditions describing the possible consequences of a C-bifurcation. This set of conditions is finall y used to classify all the possible codimension one C-bifurcations in a gen eral class of PWS systems. The method presented is then applied to the case of a two-dimensional map and used to obtain a complete mapping of its para meter space. The possibility of a sudden jump to a chaotic attractor at a C -bifurcation is also illustrated in the case of a one-dimensional map. Fina lly. the method is applied to a set of first-order ordinary differential eq uations, and the results compared with numerical simulations, which graphic ally illustrate the wide range of possible behaviours in PWS systems. (C) 1 999 Elsevier Science Ltd. Ail rights reserved.