On the bifurcations of equilibria corresponding to double eigenvalues

È. È. Shnol'   ; E. V. Nikolaev

Sbornik. Mathematics, Tome 190 (1999) no. 9, pp. 1353-1376

Résumé

Systems of ordinary differential equations having a finite symmetry group are considered. One-parameter local bifurcations of symmetric equilibria corresponding to a double pair of purely imaginary eigenvalues are studied. It is shown that in one case a two-dimensional torus is generated from the equilibrium. The torus contains limit cycles; their number does not depend on the values of the parameter. The trajectories of the system that do not leave a certain fixed domain may only tend to the equilibrium under study or to the 2-dimensional torus or to one of two (disjoint) limit cycles. In all the other cases an invariant surface is generated from the equilibrium which is diffeomorphic to the three-dimensional sphere. The behaviour of the trajectories on this surface depends on the symmetry group and is not studied in this paper. In the appendix we provide information on codimension 1 bifurcations corresponding to double zero eigenvalues.

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