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

Hobbs, C. and Osinga, H. (2008) Bifurcations of the global stable set of a planar endomorphism near a cusp singularity. International Journal of Bifurcation and Chaos, 18 (8). pp. 2207-2222. ISSN 0218-1274

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Publisher's URL: http://dx.doi.org/10.1142/S021812740802166X

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 behaviour as organised by five different global bifurcation diagrams.

Item Type:Article
Uncontrolled Keywords:stable manifold, stable set, backward invariant, cusp singularity, critical locus, unfolding
Faculty/Department:Faculty of Environment and Technology > Department of Engineering Design and Mathematics
ID Code:16367
Deposited By: Dr C. Hobbs
Deposited On:12 Jan 2012 11:36
Last Modified:10 Sep 2014 18:19

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