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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

76 It has also been

76 It has also been suggested that contaminants on the surface of the fluid might lead to Robin boundary conditions, see Hocking [1987]. The extension to -~ _< x,y < r~ with PBC is straightforward for u. Let u = (u,v,w); then (u,v,w)(x,-y,z) = (u,-v,w)(x,y,z) 0 < x _< n, -~ < y < and (u,v,w)(-x,y,z) = (-u,v,w)(x,y,z) -n _< x _< n, -n < y < n. The extension for ~(x,y) depends as usual on whether NBC or DBC are selected: NBC: ~(x,-y) = ~(x,y) 0 < x < n, -n < y < n ~(-x,y) = ~(x,y) -n -< x < n, -n -< y -< n; DBC: ~(x,-y) = -~(x,y) 0 < x < n, -n < y < n ~(-x,y) = -~(x,y) -n < x < n, -n _< y < n. In either case the extended problem has D4+T 2 symmetry, and one naturally expects the eigenfunctions of (dS)F to have well-defined mode numbers, for example, ~ cos(mx)cos(ny) for NBC. By contrast a Robin boundary condition would force the eigenfunctions to be mixtures of these pure modes. The D4~-T2-action on -n -< x,y < n is generated by D4: nl: (x,y) ~ (-x,y) n2: (x,y) ~ (y,x) T2: (cpl A02): (x,y) ~, (x+q01,y+q~2). If, for given (re,n) with m ~ n, we choose ztei(mx+ny) + z2ei(mx-ny) + z3ei(nx+my) + z4ei(nx-my) as a basis for the eigenspace, then the coordinates (z 1, z2, z3, z4) E C 4 are transformed by ~:1 : (zl, z2, z3, z4) ~ (z2, Zl, z4, z3) 1~2 : (Zl' z2' z3' z4) ~ (z3, z4, Zl, z2) (q~l'q~2): (Zl, z2, z3, z4) ~ (ei(mg)l+nq)2)Zl ' ei(mq~l-nq~2)z2,ei(ng)l+mg~2) z3, ei(nq~l-m92)z4). In this case NBC are selected by invariance under the group generated by B N = {~Cl, ~c 3} ~Cl: (x,y) ~ (-x,y) and ~:3: (x,y) m (x,-y). Note that 3 = K2~ClK 2. For DBC we have the additional symmetry operation (c.~)(x,y,z) = -~(-x,y,z) which acts on the eigenspace by o: (z 1, z 2, z 3, z4) ~ (-z2,- Zl, -z4, -~3)" Now DBC are selected by invariance with respect to the subgroup B D = {or, K2ox:2}. The bifurcation problems relevant to the models of the Faraday experiment occur on FiX(BN) and Fix(BD). Interestingly, the effective symmetry group occuring in the bifurcation of mode (m,n) can be smaller than the D 4 symmetry suggested by the

77 experimental geometry. 'Upper bounds' on the resulting contraints can be determined by calculating the relevant normalizer in each case, but additional constraints may also arise, not inherited in this way from symmetries of the original problem, a point that we do not discuss further here. For NBC the normalizer constraints depend on the parities of the mode numbers as follows: ND4+T2(BN)/B N mode (m,n) D 4 m+n odd D 2 m+n even; m, n each odd Z 2 m+n even, m, n each even Simonelli and Gollub [1989] also demonstrate the existence of codimension two mode interactions between different standing waves, as had previously been obtained in the circular case. Although the dynamics near such points of multiple instability have not been studied in detail either experimentally or theoretically, it is clear that a wider variety of possibilities arises than in the circular problem. Most significant is the role of boundary conditions. For NBC or DBC the primary standing waves are pure modes, and the linearization (dS)0 of the centre manifold map is diagonal, that is, (dS) 0 = -I, as in the 0(2) mode interaction. If, however, the experimental conditions require Robin boundary conditions, then (dS) 0 should have a nilpotent part. The normal fores selected by these two linearizations involve quite different nonlinear terms, and presumably quite different dynamics. For example, a stability analysis of 2-cycle solutions indicates that in the nilpotent mode interaction the primary modes can undergo a secondary Hopf bifurcation, while in the diagonal interaction the first possibility for Hopf bifurcation occurs only along a secondary mixed mode branch. Also, preliminary numerical results suggest that chaotic behaviour is readily found for the nilpotent mode interaction, whereas for the diagonal case chaos is likely to occur only for parameters in a very thin region, as in the 0(2) mode interaction. In addition, when (dS) 0 = -I, the dependence of the normalizer on the mode numbers indicates that the 'parity' of the modes influences the dynamics. A detailed discussion of these issues will be given in Crawford, Golubitsky and Knobloch [1989]. Acknowledgements Much of this paper was written during the 1988-89 Warwick Symposium on Singularity Theory and its Applications, which was supported by a grant from SERC. The work of individual authors was partially supported by the following grants: (JDC) DARPA ACM Program; (MG) DARPA/NSF (DMS-8700897), NASA-Ames (2-432), and the Texas Advanced Research Program (ARP-1100); (MGMG) JNICT; (EK) DARPA/NSF (DMS-8814702); (INS) SERC.

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