Analysis of the T-point-Hopf bifurcation with Z2-symmetry : application to chua's equation

Abstract

The aim of this work is twofold - on the one hand, to perform a theoretical analysis of the global behavior organized by a T-point-Hopf in Z(2)-symmetric systems; on the other hand, to apply the obtained results for a numerical study of Chua's equation, where for the first time this bifurcation is considered. In a parameterized three-dimensional system of autonomous differential equations, a T-point is a point of the parameter space where a special kind of codimension-two heteroclinic cycle occurs. A more degenerate scenario appears when one of the equilibria involved in such a cycle undergoes a Hopf bifurcation. This degeneration, which corresponds to a codimension-three bifurcation, is called T-point-Hopf and has been recently studied for a generic system. However, the presence of Z(2)-symmetry may lead to the existence of a double T-point-Hopf heteroclinic cycle, which is responsible for the appearance of interesting global behavior that we will study in this paper. The theoretical models proposed for two different situations are based on the construction of a Poincare map. The existence of certain kinds of homoclinic and heteroclinic connections between equilibria and/or periodic orbits is proved and their organization close to the T-point-Hopf bifurcation is described. The numerical phenomena found in Chua's equation strongly agree with the results deduced from the models