Computing Hopf Bifurcations I

Abstract

This paper addresses the problems of detecting Hopf bifurcations in systems of ordinary differential equations and following curves of Hopf points in two-parameter families of vector fields. The established approach to this problem relies upon augmenting the equilibrium condition so that a Hopf bifurcation occurs at an isolated, regular point of the extended system. We propose two new methods of this type based on classical algebraic results regarding the roots of polynomial equations and properties of Kronecker products for matrices. In addition to their utility as augmented systems for use with standard Newton-type continuation methods, they are also particularly well adapted for solution by computer algebra techniques for vector fields of small or moderate dimension.