Hopf bifurcation in higher dimensional differential systems via the averaging method

Jaume Llibre, Xiang Zhang

Research output: Contribution to journalArticleResearchpeer-review

21 Citations (Scopus)

Abstract

We study the Hopf bifurcation of C{script}3 differential systems in R{double-struck}n showing that l limit cycles can bifurcate from one singularity with eigenvalues ±bi and n - 2 zeros with l ∈ {0, 1, ..., 2n-3}. As far as we know this is the first time that it is proved that the number of limit cycles that can bifurcate in a Hopf bifurcation increases exponentially with the dimension of the space. To prove this result, we use first-order averaging theory. Further, in dimension 4 we characterize the shape and the kind of stability of the bifurcated limit cycles. We apply our results to certain fourth-order differential equations and then to a simplified Marchuk model that describes immune response.
Original languageEnglish
Pages (from-to)321-341
JournalPacific Journal of Mathematics
Volume240
DOIs
Publication statusPublished - 1 Apr 2009

Keywords

  • Averaging theory
  • Generalized Hopf bifurcation
  • Limit cycles

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