Bistable boundary conditions implying codimension 2 bifurcations

Abstract

We consider generic families $X_\param$ of smooth dynamical systems depending on parameters $\param\in P$ where $P$ is a 2-dimensional simply connected domain and assume that each $X_\param$ only has a finite number of restpoints and periodic orbits. We prove that if over the boundary of $P$ there is a S or Z shaped bifurcation graph containing two opposing fold bifurcation points while over the rest of the boundary there are no other bifurcation points, then, if there is no fold-Hopf bifurcation in $P$ then there is at least one cusp in the interior of $P$.