Generalized Hopf bifurcation in a class of planar switched systems

Abstract

This article presents an analysis on a generalized Hopf bifurcation in a class of planar switched systems with the phase portraits of the subsystems being locally those of similarly oriented foci at the origin, where the appearance or disappearance of a periodic orbit is not due to the crossing of complex conjugate eigenvalues of the linearization of smooth subsystems through the imaginary axis, but due to the switching law between these smooth subsystems. The mechanism of the generalized Hopf bifurcation dealt with in this article is of significance from switched systems point of view.

Generalized Hopf bifurcation emerged from a corner in general planar piecewise smooth systems

“…When n = 4 and the lines of discontinuity coincide with the axes in system (1.1), the GHB has been studied in great detail in [24]. Here we review these results as follows.…”

“…The existing results on GHB emerged from a corner in planar PWSs with the Jacobian matrix of each smooth subsystem having a pair of complex conjugate eigenvalues can be found in [22][23][24][25]. Under the assumption that the Jacobian matrix of each smooth subsystem is in the Jordan normal form, Akhmet [22,23] and Zou and Küpper [25] studied, respectively, the cases when the discontinuity boundaries are given by finitely many nonlinear curves and several straight lines emanating from the origin.…”

“…Under the assumption that the Jacobian matrix of each smooth subsystem is in the Jordan normal form, Akhmet [22,23] and Zou and Küpper [25] studied, respectively, the cases when the discontinuity boundaries are given by finitely many nonlinear curves and several straight lines emanating from the origin. Huan and Yang [24] studied the case when the Jacobian matrix of each smooth subsystem is of the most general form and the discontinuity boundaries are given by several radials emanating from the origin.…”