Near-Hamiltonian Hopf Bifurcation
by
William Langford
University of Guelph

The classical Hopf Bifurcation Theorem describes the generic mechanism for the birth or death of a periodic solution, near an equilibrium point of a dynamical system. However, it does not apply to the important case of Hamiltonian systems, where the generic bifurcation of periodic solutions is described by what has been called the Hamiltonian-Hopf bifurcation theorem.

This talk will explore the interface between the classical Hopf Bifurcation Theorem and the Hamiltonian case; that is, the general setting of systems that are nearly Hamiltonian (also called weakly dissipative). The Hamiltonian limit leads to an interesting singularity known as Whitney's umbrella, which will be described. In fact, the Hamiltonian case lies on the "handle" of Whitney's umbrella. There exist limit cycles that persist in the near-Hamiltonian case (just off the handle of the umbrella), that correspond to the two families of centers that exist in the Hamiltonian case.

Date received: October 27, 2003