Bifurcation from O(2) symmetric heteroclinic cycles with three interacting modes

The authors study a set of three complex ordinary differential equations describing modal interactions in a system equivariant under the group O(2) of planar rotations and reflections and appropriate to the interaction among Fourier modes with spatial wavenumbers in the ratio 1:2:4. Such systems are known to possess structurally stable heteroclinic cycles and they focus on the bifurcations occurring near a degenerate point at which these cycles simultaneously change their stability type and become unstable to travelling waves. They find a subtle interaction between local and global dynamics and they show that multiple branches of modulated travelling waves emerge. Their methods include centre manifolds and normal forms coupled with estimates on the global return of solutions obtained with the aid of numerical simulations.