Numerical normal forms for codim 2 bifurcations of fixed points with at most two critical eigenvalues

Yu A. Kuznetsov*, H.G.E. Meijer

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

41 Citations (Scopus)
21 Downloads (Pure)

Abstract

In this paper we derive explicit formulas for the normal form coefficients to verify the nondegeneracy of eight codimension two bifurcations of fixed points with one or two critical eigenvalues. These include all strong resonances, as well as the degenerate flip and Neimark-Sacker bifurcations. Applying our results to n-dimensional maps, one avoids the computation of the center manifold, but one can directly evaluate the critical normal form coefficients in the original basis. The formulas remain valid also for n = 2 and allow one to avoid the transformation of the linear part of the map into Jordan form. The developed techniques are tested on two examples: (1) a three-dimensional map appearing in adaptive control; (2) a periodically forced epidemic model. We compute numerically the critical normal form coefficients for several codim 2 bifurcations occurring in these models. To compute the required derivatives of the Poincaré map for the epidemic model, the automatic differentiation package ADOL-C is used.

Original languageEnglish
Pages (from-to)1932-1954
Number of pages23
JournalSIAM journal on scientific computing
Volume26
Issue number6
DOIs
Publication statusPublished - 25 Nov 2005
Externally publishedYes

Keywords

  • Automatic differentiation
  • Bifurcations of fixed points
  • Normal forms

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