Bifurcations of the Global Stable Set of a Planar Endomorphism Near a Cusp Singularity

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 zero. The critical locus is the image of J0, denoted J1, 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 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 W s 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 organised by five different global bifurcation diagrams.