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

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

and and so 174 = ~ +

and and so 174 = ~ + ~(~, A). ~/ is qT-periodic,/)(., 0, 0) = ul and it is easy to verify that i Fr(~, A) U = ~(¢(a,A) - 27rp/q)~], 1 Fz(:~, X) Re ~] = -#( ¢(a, X) - 2~r p/ q) Im ~]. (6) (6) will help us to evaluate (5). Let us remark that :~ = w~ and :~z = Re U~. We shall illustrate the proof on the first $ derivative. For the general case a proof by induction using the chain rule (Faa di Bruno's formula) get us the results. Let's take the $ derivative of (5) and evaluate it at the origin, we find Now let turn to (6) and we find replacing into (7) : ¢iz~ =< V~, F~, • w~2z + F,~. ~ + L. w,~. 2z > • F~2 • f~ARe U1 + FxA. Re U1 + L . Re U1 = -~ ¢A(0, 0) Im U1 , i lzA(0 .... ,0) = igzA(O ,. ..,0) = ~ CA(0,0). I-1 3 Analysis of the bifurcation equations Let start by looking at the set of solution T of period T. The equations axe given by the system ¢21(0, a, b, A) = ¢3,(0,a,b,A) = 0, Z2-equivariant with a acting as (a,-b) on the source and as (-a, b) on the target. The subset T~ of reversible solutions satisfy the equation ¢31(0, a,O,A) = O, A priori we have a nice set-up for the existence of branches. But the eqlfivarlant structure can force 7" to have in 7"£ a submanifold of dimension larger than 1. As an example let consider the generic situation (modulo some coordinates changes) when diraA -- 2 and dimB = 1 : b(el al -{-...) = 0 b(e2a2 +...) = 0 2 #la21 + #2a~ + #3b 2 - e3A + .... 0 , e i --1,i=1,2,3. And so T~ is given by the surface : (7)

175 and the following curve is the remaining part of T : A = e3#3b 2 + h.o.t, al = O(b2), a2 = O(b2). Concerning the generic bifurcation of proper subharmonics the symmetry does not prevent the simple dimension count showing that a > b is a necessary condition to get curves of reversible subharmonics. As a consequence 7"6 will be nontrivial in this case. The situation is different for nonreversible subharmonics where no bifurcation generically occurs with the exception of some cases for q = 3 or 4. A two bifurcation parameters setting can help to get at least some nonsymmetric subharmonics, but we need (dimB-dimA + 1) parameters to get reversible solutions. At this stage we should mention the work of Hummel [12] on the bifurcations of periodic points of general maps. Using the flow of (1) at T the map formulation of our problem in Fix(R) is rather classical, subharmonics corresponding to periodic points of reversible maps. As we shall see, reversibility is often an importaalt factor, for instance when dimB = 0. On the contrary, when dimA = 0 and dimB = 1 we find the same generic situations as described in [12]. It is an other illustration of the property of the problem in that situation of not being very sensitive to reversibility. We turn now our attention to particular examples more thorougly investigated to illustrate the properties of the bifurcation equations. 3.1 dimB=0 In this situation, because ¢31 do not exist and ~b21(0, a, 0, A) = 0, 7~ equa~ T and is parametrized by (0, a, 0, A). We then get the following bifurcation result. Theorem 3 Under the previous notations, suppose that dimB = 0 and that CA(O, O) # 0 then there are 2q-families of branches of reversible qT-periodic solutions of (2) branching off the trivial manifold T~. These families are parametrised by A. Proof. When dimB = 0, the branching equation for Z(a) is simply g(s, a, 0, A) = 0. g is Dq-equivariant and so has the form g = gl(u, v, a, A)z + g2(...)yq-1. From Lemma 2 we know that if CA(0, 0) # 0 then gzx(O) = glA(0) # 0. Dividing by s and applying the implicit function theorem we simply get the existence of a unique function A(s, a) satisfying the branching equation, leading to a A-family of 2 branches of reversible q~e-periodic solutions of (2). If q is odd there are q isotropy subgroups conjugate to Z(a), if q is even there are only q/2 of them but in that case the other family conjugate to Z($a) satisfy a similar equation, o 3.1.1 dimA=0 When dimA = 0 the problem reduces to Dq acting on C. The generic situation has been studied by Vanderbauwhede [23]. There are 2q branches of reversible solutions bifurcating out of the trivial branch. The singularity theory approach to this analysis can be found in Gervais [9] building on the partial classification of Buzano-Geymonat-Poston [3]. The complete classification up to topological codimension 2 is to be found in Furter [7],[8].

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