Generalized Hopf bifurcation emanated from a corner for piecewise smooth planar systems

Abstract

In this paper, we study the existence of periodic orbits bifurcating from a corner in a piecewise smooth planar dynamical system. This phenomenon is interpreted as generalized Hopf bifurcation (Philos. Trans. R. Soc. London Ser. A 359 (1789) (2001) 2483–2496; “Hopf bifurcation” for nonsmooth planar systems, Universität zu Köln, 2000; Northeast. Math. J. 17(4) (2001) 383–386; Northeast. Math. J. 17(3) (2001) 261–264; Generalized Hopf bifurcation for non-smooth planar dynamical systems, University Cologne, 2003). In the case of smoothness, Hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This characterization does not apply to a piecewise smooth system due to the lack of linearization. In fact, the generalized Hopf bifurcation is determined by interactions between the geometrical structure of the corner and the eigenstructure of each smooth 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 a return map.