Geodesic
Regular polyhedra and bifurcations of symmetric equilibria of ordinary differential equations
Sbornik. Mathematics, Tome 191 (2000) no. 8, pp. 1243-1258 Cet article a éte moissonné depuis la source Math-Net.Ru

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All local 1-parameter bifurcations of symmetric equilibrium states corresponding to triple eigenvalue 0 are considered. In each case the corresponding “bifurcation group” the restriction of the full symmetry group of the differential equations to the centre manifold, is associated with symmetries of a regular (3-dimensional) polyhedron. It is shown that in all cases but one the bifurcation event is just a version of equilibrium branching. The proofs are based on the existence of functions (similar to Lyapunov functions) whose derivative by virtue of the equations has constant sign. These functions do not depend on the bifurcation parameter and are homogeneous of degree zero. 
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È. È. Shnol'. Regular polyhedra and bifurcations of symmetric equilibria of ordinary differential equations. Sbornik. Mathematics, Tome 191 (2000) no. 8, pp. 1243-1258. http://geodesic.mathdoc.fr/item/SM_2000_191_8_a6/

[1] Shnol E. E., Nikolaev E. V., “O bifurkatsiyakh simmetrichnykh polozhenii ravnovesiya, otvechayuschikh dvukratnym sobstvennym znacheniyam”, Matem. sb., 190:9 (1999), 127–150 | MR | Zbl

[2] Arnold V. I., Afraimovich V. S., Ilyashenko Yu. S., Shilnikov L. P., “Teoriya bifurkatsii”, Dinamicheskie sistemy, Itogi nauki i tekhniki. Sovr. probl. matem. Fundam. napr., 5, VINITI, M., 1986, 5–218 | MR

[3] Arnold V. I., Dopolnitelnye glavy teorii obyknovennykh differentsialnykh uravnenii, Nauka, M., 1978 | MR

[4] Swift J., Barany E., “Chaos in the Hopf bifurcation with tetrahedral symmetry: convection in a rotating fluid low Prandtl number”, European J. Mech. B Fluids, 10:2, suppl. (1991), 99–104 | MR | Zbl

[5] Guckenheimer J., Worfolk P., “Instant chaos”, Nonlinearity, 5 (1992), 1211–1222 | DOI | MR | Zbl

[6] Golubitsky M., Stewart I., Schaeffer D., Singularities and groups in bifurcation theory, V. II, Springer-Verlag, New York, 1988 | MR | Zbl

[7] Shafarevich I. R., “Osnovnye ponyatiya algebry”, Sovr. probl. matem. Fundam. napr., 11, VINITI, M., 1986

[8] Landau L. D., Lifshits E. M., Kvantovaya mekhanika. Nerelyativistskaya teoriya, Nauka, M., 1989

[9] Ruelle D., “Bifurcations in the presence of a symmetry group”, Arch. Rational Mech. Anal., 51 (1973), 136–152 | DOI | MR | Zbl

[10] Shafarevich I. R., Osnovy algebraicheskoi geometrii, V. 1, Nauka, M., 1988 | MR

[11] Guckenheimer J., Holmes Ph., “Structurally stable heteroclinic cycles”, Math. Proc. Cambridge Philos. Soc., 103 (1988), 189–192 | DOI | MR | Zbl

[12] Solomon L., “Invariants of finite reflection groups”, Nagoya Math. J., 22 (1963), 57–64 | MR | Zbl

[13] Burbaki N., Gruppy i algebry Li, Gl. IV–VI, Mir, M., 1972 | MR | Zbl