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.

Keywords:

• mode interaction
• steady-state bifurcation
• symmetry
• robust heteroclinic network

Acknowledgements

S.B.S.D. Castro and I.S. Labouriau acknowledge the support from Centro de Matemática da Universidade do Porto, financed by Fundação para a Ciência e a Tecnologia through the programmes POCTI and POSI. O. Podvigina was financed by the grants ANR-07-BLAN-0235 OTARIE from Agence Nationale de la Recherche (France) and 07-01-92217-CNRSL_a from the Russian Foundation for Basic Research.

Notes

Note

1. We are using Samovol's theorem instead of the better known theorem by Sternberg 26 because the resonance conditions are tighter and easier to check.