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

This paper is concerned with the analysis of non standard bifurcations in piecewise smooth PWS dynamical systems. These systems are particularly relevant in many areas of engineering and applied science and have been shown to exhibit a large variety of nonlinear phenomena including chaos. While there is a complete understanding of lo cal bifurcations for smooth dynamical systems, there is an urgent need for a complete theory regarding bifurcations in PWS systems. Although it is often claimed in the Western literature that no such theory exists, an analytical framework to describe these bifurcations appeared in the Russian literature in the early Seventies when Mark Feigin published his pioneering work on the analysis of C-bifurcations also known as border-collision bifurcations in n-dimensional PWS systems. Our aim is to bring his results in a more complete form to a wider audience while putting them in the context of modern bifurcation analysis. First, a typical 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 finally used to classify all the possible codimension one C-bifurcations in a general 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 parameter 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. Finally, the method is applied to a set of first-order ordinary differential equations and the results compared with numerical simulations which graphically illustrate the wide range of possible behaviours in PWS systems.