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

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

elliptic s~-Ix-al q=3

elliptic s~-Ix-al q=3 178 Figure 2: Sheets of reversible subharmonics for the case II. Case II : Here the quadratic behaviour of ~b with respect to a, and ~x(0, 0) # 0, folds the path of classical generic branches. The normal form depends on q : (/~a 2 + el A, e, 0) (for q = 3), (hu + ~a 2 + elA, 6,0) (for q = 4) and (e2u + ~a 2 + ~lA, e,0) (for q > 5). The bifurcation equations are ha 2 +Sa 2 +q~es q-2 =O (+onlyifqisodd). h being 0, if q = 3, or e2, if q _> 5. The q bifurcating manifolds cross T¢ along the parabola 5a 2 + elA = 0, s = 0. The exact shape of the manifolds depends on q, a cylinder if q = 3 or an hyperbolic or elliptic paxaboloid if q _> 4 (Figure 2). Note that in this case we do not have A-independent bifurcations. This is because of the quadratic behaviour of g,,, a is not allowed anymore to fully compensate for 3~. X X

q=3 S 179 s=lk-aa q ~ - - q>5 ~ a ~ even ~ ~, odd Figure 3: Sheets of reversible subharmonics for the case VI. Case VI : For all q the normal form is (~a, elA,~). This is again a situation where !ha(O, O) ~ 0 and so the bifurcation occur from the branch (0, ~, O) (intersection of the manifolds of reversible subharmonics). Indeed the equations are 6a + elAS q-2 = 0 (+ only if q is odd). The bifurcation manifolds axe foliated into classical generic branches in (z, a), whose opening depend on A and is reversed as A changes sign (Figure 3). .o Z, $

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