Generalized Hopf bifurcation for non-smooth planar dynamical systems

In this paper, we study the existence of periodic orbits bifurcating from stationary solutions in a non-smooth planar dynamical system. This phenomenon is interpreted as generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the pure imaginary axis. This method does not apply to a non-smooth system due to the lack of linearization. In fact, the generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigenstructure of each subsystem. We combine a geometrical method and an analytical method to investigate the generalized Hopf bifurcation. The bifurcating periodic orbits are obtained by studying the fixed points of return maps.