On Hopf bifurcation in non-smooth planar systems

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:248

Article

Han, Maoan¹  Zhang, Weinian² 

[1] Shanghai Normal Univ, Dept Math, Shanghai 200234, Peoples R China

[2] Sichuan Univ, Dept Math, Chengdu 610064, Peoples R China

关键词: Hopf bifurcation;    Non-smooth dynamical system;    Limit cycle;    Piecewise linear system;   

DOI  :  10.1016/j.jde.2009.10.002

来源: Elsevier

【 摘 要 】

As we know, for non-smooth planar systems there are foci of three different types, called focus-focus (FF), focus-parabolic (FP) and parabolic-parabolic (PP) type respectively. The Poincare map with its analytical property and the problem of Hopf bifurcation have been Studied in Coll et al. (2001) [3] and Filippov (1988) [6] for general systems and in Zou et al. (2006) [13] for piecewise linear systems. In this paper we also study the problem of Hopf bifurcation for non-smooth planar systems, obtaining new results. More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order I for foci of FF or FP type and order 2 for foci of PP type) (Theorem 2.3), different from the case of smooth systems. For piecewise linear systems we prove that 2 limit cycles call appear near a focus Of either FF, FP or PP type (Theorem 3.3). (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

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