Bistable boundary conditions implying codimension 2 bifurcations

Abstract We consider generic families X θ of smooth dynamical systems depending on parameters <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> where P is a 2-dimensional simply connected domain and assume that each X θ 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 .