Bifurcation of switched nonlinear dynamical systems

Abstract

This paper proposes a method to trace bifurcation sets for a piecewise-defined differential equation. In this system, the trajectory is continuous, but it is not differentiable at break points of the characteristics. We define the Poincare mapping by suitable local sections and local mappings, and thereby it is possible to calculate bifurcation parameter values. As an illustrated example, we analyze the behavior of a two-dimensional nonlinear autonomous system whose state space is constrained on two half planes concerned with state dependent switching characteristics. From investigation of bifurcation diagrams, we conclude that the tangent and global bifurcations play an important role for generating various periodic solutions and chaos. Some theoretical results are confirmed by laboratory experiments.