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

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

3.2. Phase oortraits in

3.2. Phase oortraits in Fix(O(2)-) 48 There are no pure 1=1 equilibria, as is appearent from the isotropy lattice (figure 1). It will be important in the next section to understand the behavior of the flow in the mixed- mode plane P2=Fix(O(2)-)= {x0,Y0}. Here the analysis follows closely Armbruster et al. [1988] who obtained a similar 2-dim. system. We therefore don't go through detailed proofs. In the absence of additional degeneracies, the analysis of the local dynamics in P2 reduces to the study of the system dxo = x0(7~l+~02-yo+SY02) t °= y0(~.2+cy0+dy02) + x02(l+(f-f')y0) Note that P2 contains one flow-invariant axis {x0=0}, which corresponds to the fixed-point subspace L=Fix(O(2)~Z~). If either ~.1>0 or I~.11 small and k2-~.l(f-f)/~/> 0, there exists a heteroclinic connection in P2 between the two type 1 equilibria a and 13 on L. This can be shown rigorously by means of the boundedness of the trajectories and the Poincar6-Bendixon theorem, as in Armbmster et al. [1988]. Taking c close to 0 does not change the qualitative picture of the phase portrait. The bifurcation diagram in the half-plane ~.z>0 is pictured on figure 3. The scenario for fixed ~.2>0 is as follows. The shaded areas correspond to non-existence of type 2 solutions. We start from a point in the shaded area of the left-hand side region, in which ez is a globally attracting sink, and we increase ~-1- At (9) ~1=k1-= - c - ~ 2 the type 2 equilibria bifurcate from a. In region IV type 2 are sinks, until a Hopf bifurcation gives rise to attracting limit cycles (region III). The Hopf bifurcation takes place when (10) ~,1=~,1'=-c-~. The limit cycles persist until they meet the stable manifold of the origin. Then the connexions from c¢ to 13 are established while type 2 solutions still exist as spiral sources (region II) and finally die off at the origin when ~,1=0. When

49 (11) kl=Z,l+- "-c+ c2~2 2 another branch of type 2 equilibria bifurcates from 13 and the ct-]3 connexions don't exist anymore (region 13. ~2 0 Figure 3. Bifurcation diagram in (~-l,~'2)-P Iane for equation in P2 (for ~,2>0 and under H1-H3). The (z-13 connexions exist in region II and in the r.h.s, shaded area. In region III exist attracting limit cycles. In regions I and IV type 2 equilibria are sinks. In the 1.h.s. shaded area o~ is a globally attracting sink. 3.3. Dynamics in Fix(DZ2) We set S=Fix(D~). Note that S=PI~P 2. The system in S truncated at order 3 is {~~t ° 28')Y2r2] dt = x°[~l + ~x02- Yo + (8+3~')Y02 + (~- (12) = Yo[~,2 + d(yo2+2y2r2)] + c(yo2-2y2r 2) + x02[ 1 + (f-f')Yo] dY2r = Y2r[~,2 + d(y02+2Y2r 2) - 2cy0y 2 + (f+3f')x02]. Note that if no additional degeneracies are assumed and if c;~0, the bifurcating equilibria in S and their stability are determined by the equations (12). The type 3 solutions bifurcate from the type 2 ones very close to their branching point with the type 1 solutions: one sees from table 1 that o~42) changes sign at a point close to ~q'. Then they travel in S until they are absorbed by the type 1 equilibria [3' and ~", when (13) ~q=~,l"=c-~.

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