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

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

2.1 Linearisation 170 We

2.1 Linearisation 170 We know that F is Fredholm of index 0. Let us look more carefully at the linear part. L = F=(0, 0) is defined by Lv = -i~ + A(t)v, where A(t) = fx(t, O, O) is T-periodic. Prom the Floquet theory we can find the transition matrix ~(t) such that ~(0) = I. ~ defines the monodromy matrix : C = ~(T). Well-known results about • and C axe : 1. c -~ = RCIZ, ~(t + T) = ~(t) C, detC = 1, 2. if p is an eigenvalue of C with eigenvector ~ then p-lalso belongs to a(C) with eigenvector R~, 3. KerL = {Re(@(t)u) l u • ger(Cq- I)}, gerL" = {Re((~'(~))-'u) I u • ge~(C'~ - Z)}. The eigenvalues of C are called the multipliers of L. From 2. we know they come as : p = 1, or conjugate pairs on the unit circle, or quadruples (#,p,#-~,p-x) outside of the unit circle. In this work we assume that Ker(Cq - I) = V1 (D Vo , where V1 = {v ICv = exp(2iTrp/q)v, q > 3, 1 < p < q, (p,q) = 1}, dimVl = 2 , Vo = {v t Cv= v} = A @ B , A = {v • go I 2Zv = ~}, B = {v • Vol R. = -,}. As pointed out in 3. the corresponding elements of KerL, L* are given by applying iI), ~* on the vectors of V1 @ V0. We denote by V1, V0 the corresponding subspaces of Yq. Our first lemma relates Vo and R. More precisely, let suppose R reverses k and fixes (n - k) coordinates then Lemma 1 Under the previous assumptions, dimA > (n - 2k) and dimB > (2k - n), in particular dimVo > I n - 2k I. Proof. Let C1 = ~(-T/2) and C2 = ~(T/2), and so C = C~1C2 and C1 = RC2R. Let denoteM=FixRandN= {v•R" IRv=-v}. We claim that A=MfqC~lMand B = N f3 C~IN. Let z • M f3 C~IM, there is y G M such that y = Clx. Then Cz = C{IC2z = C~IRClz = C{11~y = z, hence x E A. Now consider z E A, by definition Clx = C2z and Rx = z. Let y = Clz, to see that y E M we compute Ry = RClz = C2Rx = Cux = Clz = y. Using a similar argument we get B = N fl C{IN. Because C1 is a diffeomorphism dim(C~lM) = dimM and we can conclude with the classical formula for the minimal dimension of art intersection of subspaces. 1:3 As a corollary dimVo is nontrivial when n is odd, as a particular example of n ~ 2k. There is an important distinction between A and B. If A represent the reversible part

171 of KerL, the elements of B are antireversible corresponding to a symmetry breaking. We can generically consider two cases. If dimA > dimB there is a trivial manifold of reversible solutions with a pair of multipliers locked on S x. This allowed Vanderbauwhede [23] to treat the problem when V0 is trivial as a one bifurcation parameter equation, instead of the two needed in the classical study of the problem in nonreversible systems (Arnold [1]). Moreover in that situation we also expect the reversible subharmonics to appear in (dimA-dimB)-families of branches. By contrast if dimA < climb the reversible solutions are isolated, but it is still possible to have branches of simple T-periodic solutions. In general subharmonics do need additional parameters, in particular if we are looking for reversible ones. As we shall see, when n = 3 the situation is very much llke the nonreversible situation, with strong (q = 3,4) or weak (q >_ 5) resonances. In both cases the parameters corresponding to multipliers whose eigenspaces contribute to V0 but which are not fixed at 1 (we can call them "detuning parameters") can be considered ,as unfolding parameters. 2.2 Bifurcation Equations We can introduce the Dq-invariant splittings Xq = KerL (~ N, Yq = ImL @ M with the projection P : Yq --* M. Our equation (2) is equivalent to the system : PF(u + w,A) = O, u E KerL, wEN, (3) (I-P)F(u+w,A) = 0. (4) Using the implicit function theorem we can solve (4) near (0,0) and end up with the reduced bifurcation equation : ](u, A) = PF(u + w(u, A), A) = 0. We then introduce some coordinates. Let ul , f**, at ... an, bl ... bb be abase for 'I/I ~ A~ B = Ker(C q - I). A base of KerL is constructed by applying ~I! on the previous vectors, we denote u1 = u2 = ... & = Bk = ¢(t)bk . . . . We can proceed similarly for the adjoint L*. We denote the respective vectors of an associated base by adding a star superscript, ie. U~ = ,~*(t)u~ .... We can identify and C via the isomorphism X : C -, V1, z ~ Re(zU1) and A @/} trivially to tt a+b. Let consider the following scalar product on Yq (real or complex) 1 jO qT < X,Y >= -~ X(s) .~'(s) ds . After scalings, from now on we assume that < u~,ul >= 2, P is given by the three projections : PI : Yq-* V1, X v-*< U~,X > P2 :Yq-'* A, X ~-, (...,< A~,X >,...) P3 : Yq"* B, X ~ (...,< B~,X >,...) In that coordinate system ] is represented by ¢ : E = C X R a+b × R !---'¢ C X I{. a+b,

Reading grade 6 2.A.5.b - mdk12
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