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

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

3 46 ~.2Yo+bxo2+Cyo+dyo

3 46 ~.2Yo+bxo2+Cyo+dyo 3=0 o'(12)=~.2+3/2~q, { xo,yo : 0 2). 2)=-2b 702, o~22)+o~32)=~,z+2C~0+ 3d~ 0 2, { Y2,Y-2 } : O~42)=~'2-2cY0+dy02' D~ .1+~Y0+Yx02+(28--8')y22=0 {x0,Y0,Y2,Y_2}: 0 and eigenvalues of a ~.2Yo+bx02+c(y0Z-2y22) 3×3 mau'ix M, +dy0(y02+2y22)=0 {Xl,X_l,Yl,y_ 1}: 0 (double) and ~.2-2cyo+d(yo2+2y22)=O 0"(13 ). 0"~3 ) = 3~-~ ~02 o(13)+o~3)=-313~o The matrix M which appears in the "eigenvalues" column for type 3 equilibria has no special structure and one needs in general the help of a computer to compute its eigenvalues. Its entries can be found in Moutrane [1988]. From table 1 we can find conditions for the stability of these equilibria. Here of course we mean orbital stability since these solutions form continuous group orbits. 3. The local dynamics and connexions in the invariant subspaces We do not attempt to give a complete description of the dynamics in a neighborhood of the bifurcation point. Since we are interested in the existence of structurally stable heteroclinic cycles we rather try to isolate those situations which may lead to such heteroclinic cycles (next section). We now make the following (physically relevant) assumptions: (H2) d

3.1. Phase portrait in FixfD2~.Z ~ 47 When x=0 the problem is reduced to the pure mode 1=2 bifurcation. The dynamics is then completely described by the equation restricted to the plane Fix(D2~7~)= {y0,Y2+Y2} which we denote by P1 (Golubitsky, Schaeffer [1982]). Because we assume (H1), the bifurcation of type 1 solutions (pure modes) is slightly transcritical (unstable) with a turning point. We denote by ct and 13 resp. the negative and positive bifurcating equilibria on the invariant line L=Fix(O(2)~ Z~). Moreover P1 contains 3 copies of L: L itself and two conjugate lines L' and L", obtained by rotations of 120 ° and 240 ° in P1- The hypothesis d0,d

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