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

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

66 (b) Next we discuss

66 (b) Next we discuss the expected type of bifurcation in the PBC case. When m > 0, the effective action of 0(2) on the bifurcation is by O(2)/E' = O(2)/Z m ~- 0(2). Hence, genetically we expect a pitchfork of revolution. When m = 0 and Z'= O(2), the group O(2)/Z' is trivial and genetically we expect a limit point bifurcation (in the branch of constant solutions). On the other hand when Z' = SO(2) we have O(2)/Z' ~ Z 2, and generically we expect a pitchfork bifurcation. The solutions to P(u) --- 0 satisfying NBC are found in FiX(BN) = {z ~ ker d\$ > : Rz = z}. Thus, when m > 0, NBC picks out those solutions in the pitchfork of revolution that are invariant under the reflection Rx = -x, rather than a general reflection x ~ x0-x. These solutions form a pitchfork, for the following reason. Let T(x) = x + 2rc/2m. (1.7) The translation T lies in the normalizer of the isotropy subgroup D m =

U T(u) 67 m=2 -~ ................ 0 ........ ,,°,'~"JJJ" -:¢ 0 Fig. 2 As for Fig. 1 but with m even. Now only T acts by -I.. Assume that 9(u) depends on two parameters and that when these parameters are set to zero the linearized NBC problem has a double zero eigenvalue and no other eigenvalues on the imaginary axis. Then 0 lies in the intersection of two curves in parameter space along which Lu = 0 has a simple eigenvalue. Let m and n be the mode numbers along these curves, guaranteed by Proposition 1.2. Corollary 1.3 Suppose that m ~ n. Then dim ker L = 2. Remark Without the effect of 0(2) symmetry, we might have expected a Takens-Bogdanov type singularity at ~. = 0. The distinct mode numbers rule this out. Proof Extend to PBC and let = Mz + ... (1.8) be the O(2)-equivariant vector field on IR 4 -~ IR2xlR 2 obtained by centre manifold reduction. Since m ¢ n, 0(2) acts by distinct irreducible (indeed absolutely irreducible) representations on each copy of gt 2. Hence (,,° 1 M = 0 c2I 2 " The assumption of a double zero eigenvalue in NBC implies that c 1 = c 2 = 0. Thus M - 0. Restricting (1.8) to NBC proves the result. []

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