Bifurcations of periodic orbits with spatio-temporal symmetries

Abstract

Motivated by recent analytical and numerical work on two- and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic solutions of partial differential equations. Our approach is based on centre manifold reduction for maps, and is in the spirit of earlier work by Iooss (1986) on bifurcations of group orbits of spatially symmetric equilibria. Two examples, two-dimensional PWs and three-dimensional alternating pulsating waves (APW), have discrete spatio-temporal symmetries characterized by the cyclic groups , n = 2 (PW) and n = 4 (APW). These symmetries force the Poincaré return map to be the nth iterate of a map : . The group orbits of PW and APW are generated by translations in the horizontal directions and correspond to a circle and a two-torus, respectively. An instability of PWs can lead to solutions that drift along the group orbit, while bifurcations with Floquet multiplier (FM) +1 of APWs do not lead to drifting solutions. The third example we consider, alternating rolls, has the spatio-temporal symmetry of APWs as well as being invariant under reflections in two vertical planes. This leads to the possibility of a doubling of the marginal FM and of bifurcation to two distinct types of drifting solutions. We conclude by proposing a systematic way of analysing steady-state bifurcations of periodic orbits with discrete spatio-temporal symmetries, based on applying the equivariant branching lemma to the irreducible representations of the spatio-temporal symmetry group of the periodic orbit, and on the normal form results of Lamb J S W (1996 Local bifurcations in k-symmetric dynamical systems Nonlinearity 9 537-57). This general approach is relevant to other pattern formation problems, and contributes to our understanding of the transition from ordered to disordered behaviour in pattern-forming systems.