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

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

52 equilibria. In P1 the

52 equilibria. In P1 the heteroclinic cycle connects the point 13 to the points o~' and 0t" (see figure 2). These points are obtained from o~ by suitable rotations R' and R". In terms of Euler angles (CPl,0,¢P2), we have R , =(0~-,~-) ~ ~x and R"=(0,2,0) (Chossat [1982]). From o~' and ¢t" the cycle goes respectively to 15' and 13" via the planes P2' and P2" obtained from P2 by the rotations R' and R". Then in P113' and 1~" are connected respectively to 0t and ~", 0t and 0t'. It follows that this heteroclinic cycle involves only the three pairs (ot,[5), (ct',13'), (o~",t]") (figure 4). Figure 4. the heteroclinic cycle involving ot and ~ equilibria viewed in P1 Let us now suppose that (~,1,~2) lies in the region IV of the parameter plane (figure 2). We know from lemma 2 that a connexion type 2---){tx',tx"} can exist for an open range of parameter values. This connexion is robust under small perturbations, therefore it allows a heteroclinic cycle involving type 2 equilibria to exist: Prot~osition 4 Assume that hypothesis H1-H3 and conditions (14) of lemma 2 are satisfied. Then a heteroclinic cycle connecting ct to type 2 equilibria exists for every (LI,~, 2) in region IV of fig.2 such that ~'1 > ~'1 *+E, where e>0 tends to 0 with c.

53 Remark 2. This heteroclinic cycle involves structurally stable connexions to equilibria with complex attracting eigenvalues, so that there is the possibility of "Sil'nikov-like" chaos. Proposition 5 Under the hypothesis of prop.4, a heteroclinic cycle connecting oc to limit cycles in P2 exists for every (X1,X 2) in region HI of fig.2 such that ~1 > Xl*+e, where e>0 tends to 0 with c. Remark 3. Numerical simulations show this heteroclinic cycle even when Icl is not very small (see figures 9 and 10). The heteroclinic cycles in propositions 4 and 5 are again simple to describe. Now the only type 1 equilibria which enter in a cycle are o~, o~' and co", and three pairs of (conjugate) type 2 equilibria (or limit cycles) are involved. We finally mention the possibility for another kind of heteroclinic cycle to bifurcate in this problem: we already noticed that under certain circumstances (sign of fourth order coefficient in the pure 1=2 part of the vector field), a secondary branch of non- axisymmetric equilibria can bifurcate from the type I. We called them type 4 equilibria. Because of the isotropy of type 1 solutions, this is a typical problem of bifurcation with O(2) symmetry. The critical eigendirections are Y2 and Y-2, hence the critical wave number (for the 0(2) action) is m=2. The bifurcation is a pitchfork and in the plane P1 it leads to three pairs of type 4 equilibria, each pair being exchanged by a rotation of angle cp =-2 around the axis of symmetry of the basic type 1 solution (figure 5),

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