Heteroclinic cycles and periodic orbits for the O(2)-equivariant 0:1:2 mode interaction

We study the quadratic normal form describing the generic interaction of Fourier modes of wavenumbers 0, 1 and 2 under the symmetry group O(2) of rotations and reflections, in the case that the homogeneous quadratic terms preserve ‘energy’: the sum of the squares of absolute values of the (complex) variables. This system is a generalization of the 1:2 mode interaction studied by Dangelmayr [G. Dangelmayr, Steady-state mode interactions in the presence of O(2)-symmetry, Dyn. Stab. Syst. 1 (2) (1986) 159–185], Armbruster et al. [D. Armbruster, J. Guckenheimer, P. Holmes, Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry, Physica D 29 (1988) 257–282] and others, and its restriction to the 1:2 subspace is a degenerate case of that system. It displays all the classes of fixed points, periodic orbits (standing and travelling waves), invariant tori (modulated travelling waves) and heteroclinic cycles found in the 1:2 interaction, as well as new heteroclinic cycles connecting pure and mixed modes, chaotic cycles, and ‘strange’ periodic orbits. We describe the key dynamical features, show that the degenerate 1:2 case possesses a second organizing center at which bifurcation curves coalesce, provide representative bifurcation sets and diagrams for the 1:2 and 0:1:2 systems, and use a conservative limit to understand the periodic orbits in the latter system.