5 years ago

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

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

3.1.2 dimA=l 176 We axe

3.1.2 dimA=l 176 We axe now looking at the simplest example when dimA > dimB. In the last section we classify the bifurcation diagrams up to topological codimension 1. Here we are going to use and to comment on those results. We know that T~ is a plane (0, a, A). There is a point in choosing both a and A as parameters, it is indeed the way we described Theorem 3. This will provide an example where a group acts also on the parameters. But here we axe not going to follow this line, to preserve the special external role for A. From that choice, the changes of coordinates preserve only the sequence in A of slices (z, a) of the bifurcation diagram. A maybe more acurate description would split z and a as space variables. Before we continue, to illustrate the description, let consider the following model equa- tion ~" + A~ + f(t,x,~,~) = 0, (8) where f is 1-periodic in t, /(t, O, 0, 0) = 0, Vt, and /(-t, z, -y, z) = -f(t, z, y, z) for reversibility. We look for the bifurcations of periodic solutions from the trivial branch at A = (2r/q) 2. The kernel of the lineaxisation is generated by eos(2~t), sin(TM) and 1, and q q our previous theory applies. We can therefore think of the variable a as being the mean of the solutions of (8). From the lists of next section we are going to look particularly at the cases of zero codimension (I, II, VI), where there axe only reversible subhaxmonics appearing (forming q conjugate manifolds), and the first case with a bifurcation of nonreversible subharmonics (IX). Lemma 2 related the (a, A)-derivatives of g with those of the multiplier ~b. From the flowcharts (Figure 5,6 and 7) we can see that these derivatives are heavily involved in the values of important coefficients. The normal forms are given by three functions (Pl,/>2, i03) of u = z$., v = Re z q , a (the coordinate on A) and A. Let us denote .rm z q by w. We can recover & as ( i(plz + p2zq-1), wp3 ). The greek letters/5 and ei's represent 4-1. Using the branching equations, we get :

q even q=3 a g 177 s~llx-ag odd Figure 1: Sheets of reversible subharmonics for the case I. Case I : This is the case where ~a(0,0) ~ 0. From the proof of Theorem 3 we see we can similarly always solve the bifurcation equations in function of a. This is expressed by the normal form (~a, ~, 0) (for all q's), where we see the rote of A being trivial. There are q cylinders (in the A direction) of reversible subharmonics, from the curve 5a 4- e s q-2 = 0, A = 0 (+ only if q is odd). (9) In Figure 1 we have drawn, for each q, one of those conjugate cylinder. As usual we get the others by successive rotations of ~ around the R-axis (in the space C x R2). Instead q of branches of qT-periodic solutions bifurcating only at A = 0, we get bifurcations all the way. Because A is the distinguished bifurcation parameter, our bifurcation manifolds are foliated in constant A. / f a $

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