5 years ago

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

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

262 Fix(D:) - { z c

262 Fix(D:) - { z c Fix(~-)l z 1" Za" 0}. In each of these spaces we have to give the representative of one of the spaces Fix(O-) and Fix(Da). In Fix(D3 =) we find Fix(O-)- } Fix(D~ )[ z o = 0}. In Fix(D~) we the corresponding fixed point spaces are given by Fix(O-)- {z, Fix(D#) I z o-o } Fix(D#- {z ~Fix(m~) I Zo- lO~07E z 2 }. Filling in the stationary points and their local stable and unstable manifolds (which can be computed using the terms in the appendix) gives the following picture: (4.4) Region 2 + (x > O) d'< \ t z Fix(O-) "4/-+- Fix(D,') • Fix(O(2)-) x(D~ Fix(O-) In a similar fashion these diagrams can be drawn for all the regions. In order to get connections one just has to show, that the flow in the various sectors is directed in- ward, say for the critical value of the parameter x-0. Then a Poincare-Bendixson argument shows the existence of the connecting orbit. Using the form of the equivari- ants as displayed in the appendix this can be done easily for .all the sectors, where the picture indicates the existence of heteroclinic orbits. The computations are lengthy but straightforward. We will sketch these computations in one particular case following the statement of the theorem. It summarizes all the connections which can be obtained using our arguments. (4.5) Theorem: Assume (a,b)q Z. Then there exist heteroclinic solutions connecting a steady state with isotropy A to an equilibrium with isotropy z.. region z~ ~ Z r o(s) Dr, O(2)-. O- d D 6 O(2)-. O- o(2)- o- 2 ÷ O- 0(3) (subcriticaI) d o(3) o(2)-. D 6 D," o(2)- 3 ÷ O- o(2)- O(3) ] (subcritical) o(3), o- ] O(3) d D 6

263 4 ÷ D6 d O(3), O(2)-, O- O(2)- O(3), O- (subcritical) o- o(3) The results in the regions l- to 4- are obtained by changing the direction of time and exchanging subcritical and supercritical. Formally one could also exchange the groups O- and D~ and supercritical and subcritical! Proofi Let us do the computations for the region 2 ÷. The existence of the heteroclinic solutions joining the trivial solution and the solution corresponding to one of the maxi- mal isotropy subgroups is obvious. It is given by a trajectory in this one dimensional fixed point space. Therefore the only heteroclinic trajectory whose existence is not clear is the one connecting the dihedral solution with the axisymmetric solution. From the pictures in (4.4) it is clear that if we have to look in Fix(D~) for this solution. If we set ;~ =0 and compute the inner product between the vector (z0,z2) t and the vector field a¢~(v) ÷b¢2(v) for (a,b) in region 2" we find that it is negative if 4z=2,~3Zo 2. The region in Fix(D2D between the space with axisymmetnc symmetry and the space with dihedral symmetry satisfies 10 z22 0 there exists some e(r)>0 such that for x.~ e(r) the vector field of equation (2.2) restricted to Fix(D~) points inward (in the relevant region). Therefore a solution in the unstable manifold of the dihedral solution cannot leave the sphere of radius r and is confined to the region between Fix(O(2)-) and Fix(D6a). Therefore its u-limit set is nonempty. From lemma 42 and the Poincare-Bendixson theorem one finds that the only possible point in the e-limit set is the stationary solution with axisymmetric symmetry. This completes the proof of this particular case.

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