Generalized Hopf bifurcation of a non-smooth railway wheelset system

Abstract

In this paper, we investigate the generalized Hopf bifurcation of a non-smooth railway wheelset system. It is to note that the system is a four-dimensional non-smooth differential equation. First, we show how to overcome the non-smoothness and reduce the four-dimensional system to a two-dimensional non-smooth system by the center manifold theorem. Since the two-dimensional central manifold is still non-smooth, we cannot apply the classical Hopf bifurcation theorem. Hence, we need to construct and analyze a Poincare map so that a criterion for determining the generalized Hopf bifurcation occurring in the system is given. Finally, to demonstrate our theoretical results, we also give some numerical simulations which are presented to exhibit the corresponding bifurcation diagrams.