Bifurcations of Periodic Solutions

Abstract

In this chapter we will study bifurcations of periodic solutions in discontinuous systems of Filippov-type. The Poincaré map, which is introduced in Section 9.1, converts the continuous-time system to a discrete map. Fixed points of piecewise linear maps, which represent periodic solutions of Filippov systems, are studied in Section 9.2. Bifurcations of periodic solutions in smooth systems are briefly addressed in Section 9.3. Section 9.4 explains how a discontinuous bifurcation of a periodic solution can be created when a periodic solution touches a non-smooth switching boundary. The relation with discontinuous bifurcations of equilibria in non-smooth continuous systems is discussed. Bifurcations of fixed points in planar maps are discussed in Sections 9.5 and 9.6. Fundamental questions about discontinuous bifurcations of periodic solutions are raised in Section 9.7. Sections 9.8 and 9.9 treat a number of numerical examples which show discontinuous bifurcations. Section 9.10 draws conclusions from the numerical examples and tries to give answers to the questions of Section 9.7.KeywordsPeriodic SolutionUnit CircleBifurcation PointFloquet MultiplierStable Periodic SolutionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.