Steady-state mode interactions for D3 and D4-symmetric systems

Abstract

This paper investigates generic bifurcations from a D-m-invariant equilibrium of a D-m-symmetric dynamical system, for m = 3 or 4, near points of codimension-2 steady-state mode interactions. The center manifold is isomorphic to R-3 or R-4 and is non-irreducible. Depending on the group representation, in the unfolding of the linearization we find: symmetry-breaking bifurcations to primary branches, secondary steady-state and Hopf bifurcations; three symmetry-breaking bifurcations of the Bogdanov-Takens type (one transcritical for D-3 and two Z(2)-symmetric for D-4); and a structurally stable heteroclinic cycle in a three dimensional representation of D-4.