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

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

258 that C().)=c(),)I.

258 that C().)=c(),)I. where I stands for the identity on V. Since we assumed that there is a change of stability of the equilibrium u- 0 in X '~ at X- 0 we may assume that (Ill) c(x) 0. In this case we have generically (IV) c'(0) ~ 0. In the sequel we will study equation (2.2) with assumptions (I) to (IV). Let us re- strict to the case d-7. Fiedler and Mischaikov discuss also the cases d-5 and d-Q. From our point of view these cases are relatively simple. Due to the fact that quadratic terms play a crucial role in the cases where d- 2[. 1, [ even. the computations in our approach become much easier than in the case d- 7. The first step towards an understanding of the dynamics is to determine the stationary solutions and their stabilities. We recall the equivariant branching lemma, tt characterizes a class of subgroups which are (for a given representation) always isotropy subgroups of equilibria. It is due to Vanderbauwhede [1982] and Cicogna [iQ8I]. The version stated here is based on lhrig and Golubitsky [1984]. see also Golubitsky. Stewart and Schaeffer [1988]. (2.3) Theorem (Equivariant Branching Lemma): Let F be a compact Lie group acting ab- solutely irreducibly on a linear space V. Assume that g:VxlR--V is equivariant (in its first variable) with respect to the action of F, that (2.3.1) g(v,x)- c(),)v + o(llvl12), (2.3.2) c(O) - O, c'(O) = O, (2.3.3) AcF is a maximal isotropy subgroup with dim(Fix(A))-l. Then each neighborhood of (O,O) cFix(A) × R contains solutions (v,),) ~ Fix(A) x ~, v =O. Locally these solutions form a single branch 4-{ (v(s).X(s))l s~ [-,.,]}. For a proof, see lhrig and Golubitsky [1Q84] or Golubitsky, Schaeffer and Stewart [1~8]. (2.4) Remark: The assumption dim (Fix(A))-l implies that A is a maximal isotropy sub- group. The converse is not true, see Ihrig and Golubitsky [1984]. Degree theory implies that the conclusion remains true if we just assume that dim (Fix(a)) is odd. In this con- text there are two open problems: (i) the question of bifurcation if z~ is maximal, but dim (Fix(a)) is not odd (for an example see Lauterbach [198Q]), (ii) what can be said about the nonexistence of solutions having nonmaximal isotropy. (2.5) Remark: The equivariant branching lemma gives no stability information. m) The Symmetry Group 0(3): Bifurcation and Stability In view of the equivafiant branching temma the poset (partial ordered set) of isotropy subgroups of 0(3) is of great interest. Obviously this poset depends on the representa- tion. Let us just recall a few facts on the irreducible representations of O(3). For more details see Chossat, Lauterbach and Melbourne [IQSQ]. Every absolutely irreducible representation is (up to equivalence) characterized by the dimension d of the underly- ing space. The number d is always odd, usually one writes d-2[. 1, and [ is used to characterize the representation. The isotropy subgroups for all representations of 0(3) were computed in lhfig and Golubitsky [1Q84], however there are a few minor errors in

259 their lists. The correct lists are contained in Chossat. Lauterbach and Melbourne [t990] or Golubitsky. Schaeffer and Stewart [1988]. (3.1) The poser of isotropy subgroups in 0(3) for I-3 is given below. A line connecting two groups in the diagram indicates that the bigger group contains a subgroup conjugate to the smaller one. It is important to note that lines do not mean inclusion but subconjugacy. Again this is discussed in Chossat, Lauterbach and Melbourne [1989]. Let us briefly recall the notation for subgroups of 0(3): any subgroup L of OC3) is either a subgroup of SO(3) or it corresponds in a unique way to a pair (H,K) of subgroups of SO(3) such that K c H and indCH:K)-1 or 2, namely H-~rCL), where ~:O(3)-.,. SO(3) is the canonical surjection and K-L CISO(3). In the first case L can be an isotropy subgroup only if -I acts as plus identity. In this case it is more natural to consider the action of SO(3). In the second case any such pair defines a subgroup L of 0(3). We just list the notation for the pairs which are of interest in the case ~-3: 0(2)--(0(2),SO(2)), D z n "(Dn'~n)' Dd2n "(D2n'Dn )" O--(O,T) and T.2~. ~ - (7.2n,Z n). Here O denotes the group of rigid motions of the regular octahedron, T the group of rigid motions of the tetrahedron. With this notation the poset of isotropy subgroups for [-3 has the following form: dim (FixCA)) OC3) 0 /{\ //S 2 Z 2-\ / 4 1 7 (3.2) An immediate consequence of the equivariant branching lemma is the existence of bifurcating solutions with symmetry D6 a. O-, O(2)-. Next we want to adress the que- stions of stability and of branching with nonmaximal isotropy. In Chossat, Lauterbach and Melbourne [1989] it is shown that for generic problems the stability assignments for the solutions with maximal isotropy depend on the third order terms and there are no solutions with nonmaximal isotropy. Here we just need the fact that there are no soluti- ons in FixCD~) for k- 2,3. This follows if we show that there are no solutions other than those with maximal isotropy in Fix(~2). In order to be able to prove this and to state the stability properties we need the precise form of an equivariant vector field for [-3. From a general result in Chossat and Lauterbach [IO89] we conclude that there are two independent equivariant maps of cubic order, say el(v) and ¢2(v). The algebraic form of these maps is displayed in the appendix. Based on this information, we determine the stabilities of the solutions with maximal isotropy and exclude the existence of solu- tions in Fix(z2-) other than those with maximal isotropy. (3.3) Let us write the vector field g(v,),)- :k~'÷aelCV)+be2(v).r(v.),). Replacing cO,) by ~, is just a scaling argument. Set n" ~/3/5 . Let Z c ~2 be the union of the four lines

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