Nonlinearity 11 (1998) 1435-1455. doi:10.1088/0951-7715/11/5/015

## Bifurcations of periodic orbits with spatio-temporal symmetries

A.M.Rucklidge(1) and M. Silber(2).

(1) Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge, CB3 9EW, UK

(2) Department of Engineering Sciences and Applied Mathematics,
Northwestern University, Evanston, IL 60208, USA

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 pulsating waves (PW) and three-dimensional alternating pulsating waves (APW), have discrete spatio-temporal symmetries characterized by the cyclic groups Z_n, n=2 (PW) and n=4 (APW). These symmetries force the Poincare' return map M to be the nth iterate of a map G: M=G^n. 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 pulsating waves can lead to solutions that drift along the group orbit, while bifurcations with Floquet multiplier +1 of alternating pulsating waves do not lead to drifting solutions. The third example we consider, alternating rolls, has the spatio-temporal symmetry of alternating pulsating waves as well as being invariant under reflections in two vertical planes. This leads to the possibility of a doubling of the marginal Floquet multiplier 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 (1996). 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.

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### Movies relevant to this paper

Alternating rolls in three-dimensional compressible magnetoconvection in a periodic 2x2x1 box, starting with parameter values from Matthews et al (1995). The movies shows contours of the vertical velocity in a horizontal plane in the middle of the layer: solid lines denote fluid travelling upwards, dashed lines denote fluid travelling downwards, and the dotted line denotes zero vertical velocity.

• Alternating rolls (0.46MB) The spatial symmetries (reflections in planes at x=1/4 and at y=1/4) are manifest, as is the spatio-temporal symmetry of advancing a quarter period in time followed by a 90 degree rotation (counter-clockwise about the centre of the box). The movie shows one period of the oscillation.
• Travelling alternating rolls (drifting slowly to the left) (0.81MB) After a bifurcation of type B- (increasing the temperature difference across the layer), the alternating rolls begin to drift slowly to the left. The movie shows two periods of the oscillation; the only remaining spatial symmetry is the reflection in y=1/4.