Narrow Results

A direct method for finding the mode interaction point of a symmetry breaking Hopf bifurcation and a symmetry preserving Hopf bifurcation in problems with ℤ₂-symmetry is developed. It has been shown that the mode interaction point corresponds to an isolated solution of an extended system. The existence of this solution relies on the occurrence of the mode interaction point and this is interpreted in the context of the mode interaction, using centre manifold reduction. A numerical example with the symmetry group O(2), which has a branch of ℤ₂-symmetric nontrivial steady state solutions, is considered to provide clarification of the method.

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 ℤ₂-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 ℤ₂-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 Poincaré 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.

This paper is concerned with the strong resonance bifurcations with a reflection symmetry i.e. ${\mathbb{Z}}_2$-symmetry in maps. We compute the normal form of $1:2$ resonance and $1:3$ resonance bifurcations with ${\mathbb{Z}}_2$-symmetry. We use standard normal form techniques in order to obtain the reduced map. Then, we will obtain explicit formulae for normal form coefficients of bifurcations with ${\mathbb{Z}}_2$-symmetry. By using critical coefficients, we avoid the computation of the center manifold and the transformation of the linear part of the map into Jordan form. So this method can be used in the study of bifurcations with ${\mathbb{Z}}_2$-symmetry in general problems. To illustrate our results, we will analyze local bifurcations of the strong resonance bifurcations with ${\mathbb{Z}}_2$-symmetry numerically and then we will present some applications from economics and neural networks.

We provide the normal forms, the bifurcation diagrams and the global phase portraits on the Poincaré disk of all planar Kukles systems of degree $3$ with ${\mathbb{Z}}_2$-symmetries.