IngentaConnect Heteroclinic cycles and wreath product symmetries

Authors: Dias A.P. S.¹; Dionne B.²; Stewart I.³

Source: Dynamics and Stability of Systems, Volume 15, Number 4, 1 December 2000 , pp. 353-385(33)

Abstract:

We consider the existence and stability of heteroclinic cycles arising by local bifurcation in dynamical systems with wreath product symmetry = Z₂ G, where Z₂ acts by ± 1 on R and G is a transitive subgroup of the permutation group S_N (thus G has degree N). The group acts absolutely irreducibly on R_N. We consider primary (codimension one) bifurcations from an equilibrium to heteroclinic cycles as real eigenvalues pass through zero. We relate the possibility of such cycles to the existence of non-gradient equivariant vector fields of cubic order. Using Hilbert series and the software package MAGMA we show that apart from the cyclic groups G (previously studied by other authors) only five groups G of degree

7 are candidates for the existence of heteroclinic cycles. We establish the existence of certain types of heteroclinic cycle in these cases by making use of the concept of a subcycle. We also discusss edge cycles, and a generalization of heteroclinic cycles which we call a heteroclinic web. We apply our method to three examples.

Language: English

Document Type: Research article

Affiliations: 1: Universidade do Porto 2: University of Ottawa 3: University of Warwick

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