__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.