Normal Forms, Differentiable Conjugacies, and Elementary Bifurcations of Maps

Paul A. Glendinning, David J. W. Simpson

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Abstract

We strengthen the standard bifurcation theorems for saddle-node, transcritical, pitchfork, and period-doubling bifurcations of maps. Our new formulation involves adding one or two extra terms to the standard truncated normal forms with coefficients determined by algebraic equations. These extended normal forms are differentiably conjugate to the original maps on basins of attraction and repulsion of fixed points or periodic orbits. This reflects common assumptions about the additional information in normal forms despite standard bifurcation theorems being formulated only in terms of topological equivalence.
Original languageEnglish
Pages (from-to)816-836
JournalSIAM JOURNAL ON APPLIED MATHEMATICS
Volume83
Issue number2
DOIs
Publication statusPublished - 30 Apr 2023

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