THE CALCULATION OF THE HOPF/HOPF MODE INTERACTION POINT IN PROBLEMS WITH Z₂-SYMMETRY

Abstract

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.

Discretizing dynamical systems with Hopf-Hopf bifurcations

“…Note that it combines (2.3) twice in order to define the double Hopf condition. As already mentioned in the Introduction, the analysis presented in [1] constitutes the cornerstone of the present article, since there the genericity of an HH point is associated to the regularity of the real form of system (2.5). However, it is important to point out that in [1] the author considers dynamical systems with Z 2symmetry.…”

“…is regular at the HH point, see [1,Theorem 3.1]. Few remarks about the above presented system are in order.…”

“…In particular, the work of [1] constitutes the cornerstone of the present article. There, the genericity of a double Hopf point is associated to the regularity of a defining system at the point.…”

“…There, suitably modified defining systems of the underlying bifurcations are manipulated in order to obtain existence and closeness results. Such defining systems for the computation of Hopf points are studied in [2,3,10,12,24,27], and for Hopf-Hopf bifurcations in [1,9,29]. A broad overview of defining systems for various types of bifurcations can be found in [8].…”

Hopf Bifurcation in a Chaotic Associative Memory

“…The center manifold reduction and the normal forms to describe Hopf Bifurcation, based on Crawford's view [33], were used by many authors [35,36,37,38,39]. We present below, these two methodologies for discrete systems.…”