Numerical periodic normalization for codim 2 bifurcations of limit cycles: computational formulas, numerical implementation, and examples

V. De Witte, F. Della Rossa, W. Govaerts, Yuri Kuznetsov

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

Explicit computational formulas for the coefficients of the periodic normal forms for codimension 2 (codim 2) bifurcations of limit cycles in generic autonomous ODEs are derived. All cases (except the weak resonances) with no more than three Floquet multipliers on the unit circle are covered. The resulting formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems on the interval [0, T], where T is the period of the critical cycle, as well as multilinear functions from the Taylor expansion of the ODE right-hand side near the cycle. The formulas allow one to distinguish between various bifurcation scenarios near codim 2 bifurcations of limit cycles. Our formulation makes it possible to use robust numerical boundaryvalue algorithms based on orthogonal collocation, rather than shooting techniques, which greatly expands its applicability. The implementation is described in detail with numerical examples, where numerous codim 2 bifurcations of limit cycles are analyzed for the first time.
Original languageEnglish
Pages (from-to)722-788
Number of pages67
JournalSIAM Journal on Applied Dynamical Systems
Volume12
Issue number2
DOIs
Publication statusPublished - 2013

Keywords

  • normal forms
  • limit cycles
  • bifurcations
  • codimension 2

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