A heteroclinic network in mode interaction with symmetry

Abstract

We study a robust heteroclinic network existing in generic mode interactions of symmetric dynamical systems. Each mode lies in C 3 and is equivariant under the action of D 6 ⋉ T 2 × Z 2. With this symmetry there are eight different types of non-trivial steady states. This work is motivated by Boussinesq convection on a plane layer with periodic boundary conditions on a hexagonal lattice. The mode interaction takes place in a centre eigenspace isomorphic to C 6 when the trivial steady state becomes unstable to two modes of the form of rolls with spatial periods in the ratio. Due to relations between the normal form coefficients, only four types of steady states can be involved in the network. We examine the normal form restricted to R 6, a flow-invariant subspace, then we describe the dynamics near the network and discuss subnetworks and switching near them.