Nonlinearity 11 (1998) 14351455.
doi:10.1088/09517715/11/5/015
Bifurcations of periodic orbits with spatiotemporal 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 threedimensional
convection with imposed spatial periodicity, we analyse three examples of
bifurcations from a continuous group orbit of spatiotemporally 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, twodimensional pulsating waves (PW) and threedimensional
alternating pulsating waves (APW), have discrete spatiotemporal 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 twotorus, 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 spatiotemporal 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 steadystate bifurcations of periodic orbits with
discrete spatiotemporal symmetries, based on applying the equivariant
branching lemma to the irreducible representations of the spatiotemporal
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 patternforming systems.
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Movies relevant to this paper
Alternating rolls in threedimensional 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 spatiotemporal symmetry of advancing a quarter period in
time followed by a 90 degree rotation (counterclockwise 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.