Bistable boundary conditions implying codimension 2 bifurcations

TL;DR

This paper proves that under certain boundary bifurcation conditions in a 2-parameter family of smooth dynamical systems, a cusp bifurcation must exist inside the parameter domain, revealing a topological constraint on bifurcation structures.

Contribution

It establishes a new link between boundary bifurcation configurations and the existence of cusp bifurcations in parameter space for generic dynamical systems.

Findings

Boundary conditions with S or Z shaped bifurcation graphs imply interior cusps.

No fold-Hopf bifurcation in the domain guarantees at least one cusp.

The result applies to generic families with finitely many restpoints and periodic orbits.

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 $P$.