Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems

Xingwu Chen, Jaume Llibre, Weinian Zhang

Research output: Contribution to journalArticleResearchpeer-review

6 Citations (Scopus)

Abstract

It is well known that the cyclicity of a Hopf bifurcation in continuous quadratic polynomial differential systems in ℝ2 is 3. In contrast here we consider discontinuous differential systems in ℝ2 defined in two half-planes separated by a straight line. In one half plane we have a general linear center at the origin of ℝ2, and in the other a general quadratic polynomial differential system having a focus or a center at the origin of ℝ2. Using averaging theory, we prove that the cyclicity of a Hopf bifurcation for such discontinuous differential systems is at least 5. Our computations show that only one of the averaged functions of fifth order can produce 5 limit cycles and there are no more limit cycles up to sixth order averaged function.
Original languageEnglish
Pages (from-to)3953-3965
JournalDiscrete and Continuous Dynamical Systems - Series B
Volume22
Issue number10
DOIs
Publication statusPublished - 1 Dec 2017

Keywords

  • Cyclicity
  • Discontinuous differential system
  • Hopf bifurcation
  • Limit cycle

Fingerprint

Dive into the research topics of 'Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems'. Together they form a unique fingerprint.

Cite this