Generalized Hopf bifurcation in a class of planar switched systems

نویسندگان

  • Song-Mei Huan
  • Xiao-Song Yang
چکیده

This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. 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. 1. Introduction Piecewise smooth dynamical systems are an important class of ordinary differential equations that arise in scientific problems and engineering applications. One of the most interesting class of piecewise smooth systems are the switched systems. By a switched system we mean a family of continuous time dynamical systems and a rule that determines at each time which system is responsible for the time evolution. Until now, much attention has been paid to stability, chaos and control of switched systems [1–4]. Here we are more concerned with the periodic orbits bifurcation in a class of planar switched systems with each smooth subsystem having a fixed point at the origin, whose associated eigenvalues are complex, and around which the direction of rotation is the same. Many new bifurcation phenomena caused by a discontinuity of dynamical system are unique to piecewise smooth systems, and have been widely studied in the literature [5,6]. The term 'C-bifurcation' was referred to these discontinuity-induced bifurcation (DIB) phenomena by Feigin [7] when he studied the doubling of the oscillation period in piecewise continuous systems. To give a flavour of the DIB-type Hopf bifurcation, let us first recall the classical Hopf bifurcation, …

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تاریخ انتشار 2012