Generalized Hopf bifurcations: a homological analysis and obstructions to their existence

Abstract

A generalized Hopf bifurcation takes place when varying a parameter of a (semi)dynamical system causes an attractor $K$ to become a repeller. In doing so, it expels a (parameter-dependent) family of attractors. In this paper we concentrate on two questions: (1) what do the expelled attractors look like?, and (2) can any attractor $K$ undergo a generalized Hopf bifurcation? Regarding (1), we compute the cohomology of the expelled attractors in terms of the cohomology of $K$; in the case of continuous-time dynamics this can be improved to a computation of their Borsuk shape. As for (2), we show that certain compacta cannot undergo generalized Hopf bifurcations and, when the ambient space has dimension $ \leq 3$, give a complete topological characterization of which can.