Heteroclinic cycles and wreath product symmetries

Abstract

We consider the existence and stability of heteroclinic cycles arising by local bifurcation in dynamical systems with wreath product symmetry = Z 2 G, where Z 2 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 h 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.