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

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

Appendix 264 Let z_ 3

Appendix 264 Let z_ 3 ..... z 3 denote the coordinates in V. Then the two independent cubic order maps ck' k-l,2 on V have the form: (ek) J- X a= z = , where = ranges over all multiindices of length 3 and (¢k)j denotes the j - th component of Ck" The following table lists the coef- ficient a= of the term z= in (ek) J. Set rl- -/016. j oc k-I k-2 3 (-3,3,3) 3~1 -2 (-2,2,3) -3'~ 2 (-1,1.3) ~ -2 (- 1.2.2) I 0 (o.o.3) o 1 (0,I,2) --/2 rl 0 (l,l,l) 0.4 0 2 (-3,2,3) 3rj -2 (-2.1,3) -2 0 (-2,2,2) -~n 2 (-] ,0.3) 1/~ ~ 0 (- 1, ! ,2) ~1 -2 (o,o,2) -2,~ l (o.].I) ~ o L I j = k-1 k-2 1 (-3,1,3) n -2 (-3.2.2) 1 0 (- 2,0,3) ~ rl 0 (-2,1.2) -~ 2 (- l,- 1.3) -~ o (-I.o.2) o (-I,1,1) ~,~ -2 (0,0. l ) -~,~ l 0 (-3,0,3) 0 -2 (-3.1.2) ~"~1 0 C-2,-1,3) 1/~ ~ 0 C-2,0.2) -4~1 2 (-2,1,1) ~2- 0 (- l,- 1.2) & ~ 0 (- l,O, l) l-~ n -2 (o,o,o) -~,~ [ These terms may be obtained solving a linear system, see Sattinger [1979]. Chossat and Lauterbach [1989]. Chossat, Lauterbach and Melbourne [t989] for more details.

References 265 Chossat, P. [1Q70]: Bifurcation and stability of convective flows in a rotating or nonrotating spherical shell. SIAM J. Appl. Math. 37, 624-547 Chossat, P. & Lauterbach. R. []989].- The instabifity of axisymmetric solutions in problems with spherical symmetry, SIAM J. Appl. Anal. 20(I), 31 -38 Chossat, P. & Lauterbach. R. & Melbourne, [. [1QQO]= Steady sate bifurcation with 0(3) symmetry, Arch. Rat. Mech. and Anal. (in press) Chow. S.-N. & Lauterbach, R. [1988], A bifurcation theorem for critical points of variational prob]ems, Nonl. AnaL, TMA 12, 51-6l Cicogna, G. [IQ8I]: Symmetry breakdown from bifurcation, Lettere el Nuovo Cimente, 3l. 600-602 Fiedler. B. & Mischaikov, K. [108O]= Dynamics of bifurcations £o[ variational problems with 0(3) equivariance: a Conley index approach, Preprint SFB 123, No. 536 Golubitsky, M.. Stewart, [. & Schaeffer, D.G. [lC)88]. - Sing~']arities and groups in bifurcation theory, VoL lI, Springer Vertag, Heidelberg New York Henry, D. [1981], Geometric theory of semilinear parabofic equations, Springer Lecture Notes 840, Springer Verlag, New York - Heidelberg Ihrig, E. & Golubitsky, M. [1984]: Pattern selection with 0(3) symmetry, Physica 13D, 1-33 Kato, T. [I97b]: Perturbation theory for linear operators, Springer Verlag, New York-Heidelberg Knightly, G.H. and Sather, D. [1980]: Buckled states o1' a spherical shell under uniform external pressure, Arch. Rat. Mech. and Anal. 72, 315-380 Lauterbach, R. [1988]: Problems with spherical symmetries: studies on bifurcations and dynamics for O(3)-equivariant equations. Habilitationsschrift, Univ. Augsburg Lauterbach, R. [198g]= Maxima] isot[opy subgroups and bifurcation: an example, Preprint Sattinger, D.H. rIOT(:;)]: Group theoretic methods in bifurcation theory, Springer Lecture Notes 752, Springer Verlag, New York- Heidelberg Vanderbauwhede. A. [I982]: Local bifurcation and symmetry, Research Notes in Mathematics 75, Pitman, Boston

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