Bifurcations of periodic orbits with spatio-temporal symmetries

Abstract

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 threedimensional 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 Poincaré return map M to be the n th 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. When the bifurcation breaks these reflections, the map G has a "two-symmetry," as analysed by Lamb (1996). This leads to a doubling of the marginal Floquet multiplier and the possibility of bifurcation to two distinct types of drifting solutions.

Amplitude equations close to a triple-(+1) bifurcation point of D4-symmetric periodic orbits in O(2)-equivariant systems

“…This completes the derivation of the coefñcients of the amplitude equations (20)- (22) or, invoking (23), the coefñcients of (12)- (16). These are as given below, in (31).…”

“…Other authors adopt a more formal approach and study the problem in the context of equivariant bifurcation theory [10,16,11]. Probably, the method used in [2, p.135, VI.1] for ribbon solutions in the Taylor-Couette problem is the most closely related to the method developed here.…”

“…FIGURE 16. From left to right, (a), (b) instantaneous streamlines and temperature perturbation, respectively, for U*(0), (c), (d) for U¡(T/2), (e), (f) for £/ 2 *(0), and (g), (h) for E/ 2 *(T/2).…”

“…The adjoint operator £ T is obtained imposing that {ui, Cu2) = (£ T MI,M2) for all wi,«2 in the range of Q. Using these, the deñnition (40), and that Pj = Pg, Pgf = f, and (1 -Pg)ip = ip, we obtain upon repeated integration by parts invoking the boundary conditions (6) (16)(17)(18)(19). They are ordered to be in correspondence to £/?.…”

Secondary Instabilities of Hexagons: A Bifurcation Analysis of Experimentally Observed Faraday Wave Patterns

“…(a) (b) (c) Figure 4. Reconstructed patterns from the two solutions that arise in representation 7, using the Fourier functions (12)(13) added to a function of the form of (11). (a) has the spatial symmetries of pattern (a) and no spatio-temporal symmetries (cf.…”