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

Hobbs, Catherine; Osinga, Hinke

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. © 2008 World Scientific Publishing Company.

• Journal Article Type: Article
• Publication Date: Jan 1, 2008
• Deposit Date: Jan 12, 2012
• Publicly Available Date: Feb 9, 2016
• Journal: International Journal of Bifurcation and Chaos
• Print ISSN: 0218-1274
• Electronic ISSN: 1793-6551
• Publisher: World Scientific Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 18
• Issue: 8
• Pages: 2207-2222
• DOI: https://doi.org/10.1142/S021812740802166X
• Keywords: stable manifold, stable set, backward invariant, cusp singularity, critical locus, unfolding
• Public URL: https://uwe-repository.worktribe.com/output/1007336
• Publisher URL: http://dx.doi.org/10.1142/S021812740802166X
• Contract Date: Feb 9, 2016