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

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

q=3 q=4 q_>5 182 0o b =

q=3 q=4 q_>5 182 0o b = --~orsin(30) + h.o.t, A = Ar 2 + h.o.t,/~ = Br 2 + h.o.t where A and B are constants, generically nonzero. b = -gv-~°r2sin(40) + h.o.t, A = --e4gl r2 + h.o.t, /~ = -elezr 2 + h.o.t no b - O(r q-2) , A = -e4glr 2 + h.o.t, # = -E1ezr 2 + h.o.t We should point out that the variables are r and 8, but only r is important for the bifurcation from the origin. Moreover we can see a striking similarity with the two pa- rameter study of subharmonics in systems without symmetry (Arnold [1]). For q > 5 the partial destruction of reversibility brings in tongues licking the branches of reversible sub- harmonics. As in the classical theory, in the case of strong resonances, q = 3, 4, the tongues are open subset such that a one dimensional path can generically emerge through them. We now show that indeed if q = 3, 4 there is generically a bifurcation of nonreversible subharmonics in a one parameter situation. Let consider the system (13-15) when A and/t are the same parameter, let say A. When q = 3 we can solve (13) to get and solve (15) to get The remaining equation (14) now looks like b= Oo sin(30) + h.o.t -he r = -~3(e~r 2 + e2b2(r, 0)) + h.o.t. rH(r, 0) = ~lr 2 + ~2r2sin2(3O) + ~or cos(30) +...= 0, (16) where gi,~2 are some complicated coefficients. Dividing by r we can calculate that H(O, Oj) = 0 where 0j = ~ + j~, 0 < j < 5. To use the implicit function theorem we assume (generically satisfied) that ~0 - ~0 0 # s,(0,0~) = o, - ,~,3,, + ~[ho + ~(o2 - ,2,3,,)]. Hence there are 6 branches of solutions of (16) given by rj(O) such that rj(Oj) = 0, 0 < j < 5. Obviously these branches are Dz-orbits of re. When q = 4 a similar analysis can be carried out. This time the bifurcation equations are functions of rZ,O,b,#. It is easy to solve (13) and (15) to get b = -Or~°r2sin(4#) +h.o.t, he ----- --ele3r 2 + h.o.t.

183 Puting them back into (14) and dividing by r 2 we get, as before, an equation R'(r2,0) = 0. This time there is a condition on the coefficients coming from o = H(O,O) = g~ - ~3~4 + ~ocos(40). Hence if ] gl - elea~4 ] ] ~o l-1_ < 1 there are 8 roots 0j, 0 < j < 8, to (17) and assuming that DIH(O, Oj) ~ 0 (complicated coefficient) we can solve (14) using the implicit function theorem, to get 8 branches of subharmonics, D4 related, bifurcating from the trivial solution. 4 Classification of Dn-equivariant bifurcation problems on R 3 when dimA -- 1. We identify R s with C x R and work with one bifurcation parameter. Let first remind ourselves of the actions of Dn. On the source the action, denoted by 7, is simply the direct product of the standard one on C and the identity on R. On the image the action "~ is different because the symmetry operator acts as (z,a) ~ (-2,-a). We can use the standard actions, p = 1, because a simple group homomorphism will give us any action with a different p. We are following the classical ideas and methods (cf. Damon [4], Melbourne [16]), with the necessary changes to adapt to our (-~, 7)-equivariance. Let E denote the set (f : R a+l --, It 3 [ f(7(x, A)) = ~/f(z, A)} of Dn-equivariant bifurcation functions. We want to classify the elements of E and their perturbations with respect to changes of coordinates preserving the equivariant stucture. Let start by some definitions we need to make all this precise. We axe working exclusively with germs of functions around the origin. We define the following sets (with a natural algebraic stucture) : El = {h : R a+l --* R ] h is 7- invariant} (local ring with maximM ideal m) E = {X: R a+l --* l~ a [ X is 7- equivariant} (a module over El) A4 = {S :R TM --* GL(3, R) [ S(7(x,A)) 7 = ~ S(x,A)} (a module over El) A4o - (S • .M [ S(0) belongs to the connected component of the identity} EA denotes the ring of the real functions in A of maximal ideal rnA. The changes of coordinates belong to the group = {(s,x,h) I s e ~o, x e E, x~(o) e ~o, x(o) = o, h E m~, h(o) > o}. 2) acts on E in the obvious way : (s, x, h)(f) = s f(x, h). The recognition problem deals with the identification of the equivalence classes of E/:D. An other aspect examined is the classification of all possible perturbations of a given germ : the unfolding problem. An unfolding of a given f E E is a (~, 7)-equivariant function F : R 3+1+t --* R 3 such that F(z, A, O) = f(x, A), l E N being the number of parameters. Let consider two unfoldlngs F and G of f, with maybe a different number (17)

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