Bifurcations of the global stable set of a planar endomorphism near a cusp singularity

C. A. Hobbs, H. M. Osinga

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

The dynamics of a system defined by an endomorphism is essentially different from that of a system defined by a diffeomorphism due to interaction of invariant objects with the so-called critical locus. A planar endomorphism typically folds the phase space along curves J0 where the Jacobian of the map is singular. The critical locus, denoted J1, is the image of J0. It is often only piecewise smooth due to the presence of isolated cusp points that are persistent under perturbation. We investigate what happens when the stable set Ws of a fixed point or periodic orbit interacts with J1 near such a cusp point C1. Our approach is in the spirit of bifurcation theory, and we classify the different unfoldings of the codimension-two singularity where the curve Ws is tangent to J1 exactly at C1. The analysis uses a local normal-form setup that identifies the possible local phase portraits. These local phase portraits give rise to different global manifestations of the behavior as organized by five different global bifurcation diagrams.

Original languageEnglish
Pages (from-to)2207-2222
Number of pages16
JournalInternational Journal of Bifurcation and Chaos
Volume18
Issue number8
DOIs
Publication statusPublished - Aug 2008
Externally publishedYes

Keywords

  • Backward invariant
  • Critical locus
  • Cusp singularity
  • Stable manifold
  • Stable set
  • Unfolding

ASJC Scopus subject areas

  • Modelling and Simulation
  • Engineering (miscellaneous)
  • General
  • Applied Mathematics

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