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

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

172 a map Dq-equivariant

172 a map Dq-equivariant with respect to the following actions : 6 acts as (Sz, a, b), for ~ -- exp(2i~r p/q), a acts as (2,a,-b) on the source and as (-~,-a,b) on the target. Because (p, q) = 1 we can assume from now on that p = 1. The a-priori structure of ¢ is easily computed. We can decompose ¢ into its three components (¢1,¢2,¢3) and we get, respectively : (we denote u = zY., v = Re z q and w = Im z q) ¢1 =h+ig:E~C, h and g are Dq-equivariant functions (with respect to the standard action on C, ie. of the form p(u,v)z + r(u,v)~ "q-l) such that h(z,a,-b,A) = -h(z,a,b,A) and g(z,a,-b,A) = g(z,a,b,A), ¢2 = ¢21 + w ¢22 : E .--, It", with ¢21(u, v, a, -b, A) = -¢21(u, v, a, b, A) and ¢22(u, v, a, -b, A) = ¢22(u, v, a, b, A), $z = ¢31 + w ~2 : E -* tt b, with ¢31(u,v,a,-b,A) = C31(u,v,a,b,A) and ¢32(u,v,a,-b,A) = -¢32(u,v,a,b,A). The solution set of 4) = 0 can be split into several parts depending on the symmetry of their elements. Let remind ourselves of the principal ideas. The isotropy subgroup E= of x is {7 E Dq ] 7 x = z} (depending on the action) and the fixed point subspace Fix(E) of an isotropy subgroup E C Dq is {x I 7z = z , V7 E E}. The fundamental observation is that ¢(Fix(~)) C Fiz(~) (in our case the two Fix(~)'s are not the same !) This allowes us to control the symmetry of the solutions (given by the isotropy subgroup) and to systematize the investigation of ¢ = 0. Moreover it is easy to check that the Lyapounov-Schmidt reduction preserves also that property. Hence we can read the type of the sohttion in Xq from its type in C x tt `'+b. We can find those data in Table 1. The isotropy subgroups of elements belonging to the same orbit axe conjugate, this allowes us to simplify the description of the set of isotropy subgroups using the conjugacy of subgroups as an equiv- alence relation. The subgroup Z(a) (generated by a) is the representative of reversible qT-periodic solutions and we get its conjugates by phase shifts multiples of T. When q is even there is a second family of subharmonics, reversible with a phase shift of {T. The representative isotropy subgroup is Z(~a). subgroup source target type of solution Dq (0, a, O) (0, O, b) reversible T-perlodic Zq (0, a, b) (0, a, b) T-periodic Z(a) (R, a, O) (/It, O, b) reversible qT-perlodic Z(~a)(qeven) (R{ei'P/q},a,O)(iR{ei'P/q),O,b) reversible qT-perlodic 1 (z, a, b) (z, a, b) qr-periodic Table I. Isotropy subgroups and flxed-polnt subspaces.

173 From the fixed-point subspaces in Table 1, we can deduce the branching equations for each type of solutions: Dq Zq z(~) Z(ga)(q even) ¢3d0,a,0,~) = 0 q~l(0,a, b, ~k) = 0 ¢31(0, a, b, A) = 0 g(s,a,O,~) = 0 ¢31(s2, sq,a,O,)t) --'-- 0 g(s{~-~/~), ~, 0, ~) = 0 ¢31(s 2, -sq, a, O, ~) = 0 , sE 1t. ,sER The linear part of the bifurcation equations satisfy the following a-priori conditions h,(0) = g,(0) = 0, ¢~1b(0) = 0 =d ¢31a(0) = 0. We end this section with a lemma relating the multipliers of the trivial solution to the bifurcation function ¢. Let suppose 7~ = {(0, a, 0, A) I a E A} is a solution manifold of ¢ = 0. These solutions are reversible. It means we can now consider ~, and so C, as being parametrised by T~, with the properties of Part 2. 1. Hence there is a multiplier exp(i ¢(a, $)) of the monodromy matrix corresponding to the eigenvector u(a, )t) (in the complex domain) such that ¢(0, 0) = 2r p/q and u(O, O) = nl. From < u;, nl >= 2 and = 0 it is clear that = -1. Lemma 2 Under the previous hypotheses and at the origin, the order of the first nonzero partial derivative in ~ (only) oft is equal to the order of the first nonzero partial derivative in ~ (only) of ¢1z. Moreover the same statement holds for the partial derivatives with respect to a, and for the total derivatives in a and $ of ¢ and the partial derivatives in a and )t of ¢1 z. Proof. Let consider the following linear map £: CxR a+b --, ~ ¢ i ~ (z, a, b) ~-* X(Z) + ~I a Jr g22 b where ~1,2- (1,0,...) = A1, B1. We can write ¢l(~,a,b,~) =< V~,F(~(z,~,b) + ~(~(...),~),~) >. From now on, unless otherwise specified, we are evaluating every function at (0, a, 0, A). We are interested in the a and $ derivatives of Now ~t define ¢~ =< U~,F~(~ + w(.~,~),~). (Xz + ~,(X,~)X~). (5) U(t, a, ~) = exp[-i(¢()Q - 2rp/q) T ] q,(t, a, )~) u(a, )t),

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