## 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

*J*_{0}where the Jacobian of the map is zero. The critical locus is the image of*J*_{0}, denoted*J*_{1}, which is often only piecewise smooth due to the presence of cusp points that are persistent under perturbation. We investigate what happens when the stable set*W*^{s}of a fixed point or periodic orbit interacts with*J*_{1}near such a cusp point*C*_{1}. Our approach is in the spirit of bifurcation theory and we classify the different unfoldings of the codimension-two singularity where the curve*W*^{s}is tangent to*J*_{1}exactly at*C*_{1}. 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 organised by five different global bifurcation diagrams.Original language | English |
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Publication status | Unpublished - Oct 2007 |