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Publication status | Unpublished - Oct 2007 |
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Sponsorship: The research of HMO was supported by an Advanced Research Fellowship grant of the Engineering and Physical Sciences Research Council (EPSRC)