Bistable boundary conditions implying codimension 2 bifurcations

Abstract

We consider generic families Xθ of smooth dynamical systems depending on parameters θ ∈ P where P is a 2-dimensional simply connected domain and assume that each Xθ 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, there is a set of bifurcation curves in P that contain an odd number of cusps. In particular, there is at least one codimension 2 bifurcation point in the interior of P.