Dynamics in the 1:2 steady state mode interaction with broken O(2) symmetry


Jeff Porter

University of Leeds


The process of pattern selection in dissipative systems can be influenced in a profound way by the dynamics of simple mode interactions.  In particular, it is known that the normal form equations describing the interaction between wavenumbers of ratio 1:2 in O(2)-symmetric systems contain structurally stable (attracting) heteroclinic cycles.  In a neighboring region of parameter space, but slightly further from onset, one finds an intricate sequence of additional heteroclinic cycles and chaotic dynamics of Shilnikov type.  Because much of this picture depends on the presence of invariant subspaces imposed by symmetry it is of vital importance to understand the effects of small symmetry-breaking perturbations.  Following Chossat (Nonlinearity 6, 1993) and Ashwin et al. (J. of Comput. Appl. Math. 70, 1996) we consider the consequences of breaking reflection symmetry, i.e., reducing the symmetry group from O(2) to SO(2).  While the above authors demonstrated that one could expect quasiperiodic solutions upon perturbing one of the "old" structurally stable heteroclinic cycles we show how the former cycles persist along a curve of codimension-one in the unfolding plane, and uncover a number of new heteroclinic and homoclinic connections made possible by the symmetry-breaking.  These new connections are organized by codimension-two heteroclinic cycles (T-points).  We compare our results with the recent experiments of Nore et al. on von Karman swirling flow (preprint, 2002) where the variable rotation of the upper and lower plates can be used to investigate the breaking of O(2) symmetry in a controlled manner.