Implementation of Hopf and double-Hopf continuation using bordering methods

Abstract

We discuss the computational study of curves of Hopf and double-Hopf points in the software package CONTENT developed at CWI, Amsterdam. These are important points in the numerical study of dynamical systems characterized by the occurrence of one or two conjugate pairs of pure imaginary eigenvalues in the spectrum of the Jacobian matrix. The bialternate product of matrices is extensively used in three codes for the numerical continuation of curves of Hopf points and in one for the continuation of curves of double-Hopf points. In the double-Hopf and two of the single-Hopf cases this is combined with a bordered matrix method. We use this software to find special points on a Hopf curve in a model of chemical oscillations and by computing a Hopf and a double-Hopf curve in a realistic model of a neuron.