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5 years ago

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

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

50 It can be shown that

50 It can be shown that the type 3 equilibria are sinks in S (and in V) when they bifurcate from type 2. Then it is numerically observed that they undergo a Hopf bifurcation (with a stable limit cycle). This limit cycle disappears in a heteroclinization connecting 13' and 13" with type 2. This process is accompanied by the formation of a seemingly strange attractor. Figures 6 and 7 (at the end of the paper) show a numerical simulation of the limit cycle and the "strange attractor" in S. Beyond this point the unstable manifold of the type 2 equilibria connects to 0t' and ~", which are sinks in S. This connexion, which is observed numerically, can be proved rigorously when c is close to 0 thanks to the following lemma. Lemma 2 If Ill-H2 hold, c = 0 and the following conditions are satisfied: { 8+38'+f-f'

51 When c=0, the eigenvalue in the direction x 0 is equal to ~,I-Y0 for any point on the orbit 0 of equilibria defined by ~,2+d(y02+2y~2)=0 in PI. We can restrict ourselves to the half-space y2r>0. If (17) ~1 > 1 =--2 ~ ~' this eigenvalue is positive from ~x to points which are beyond 9'- Hence the unstable manifold of the type 2 equilibria must connect to points on C lying between 9' and a". Now suppose that c=o(1). The portion of ~ connecting 13' to cx" becomes a connecting trajectory 13'---~cx". If the conditions of lemma 2 and condition (17) are satisfied the unstable manifold of the type 2 solutions enters a tubular neighborhood of 0 on the boundary of which the vector field points inwards. This unstable manifold is therefore driven to cx" and connects to it. Remark. Under the same conditions, when a limit cycle is present in Pz it must also connect to o~' and oC'. This observation is supported by the numerical simulations. 4. Symmetry-induced heteroclinic cycles 4.1. Existence 1 The lattice of isotropy types (figure 1) indicates where one should look for heteroclinic cycles. It follows from the discussion of the previous section that the part of the lattice involving T 1=D2~ Z~, T2=O(2)- and Z1=O(2)~ ;7~ can satisfy conditions (i)- (iii) of proposition 1. This is enough to insure the existence of the heteroclinic cycle: Proposition 3. Assume that hypotheses H1-H3 are satisfied. Then a heteroclinic cycle connecting type 1 equilibria of type ct and 13 exists for every (~q,~,2) such that 0

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