Path formulation for multiparameter ${𝔻}_3$-equivariant bifurcation problems

Furter, Jacques-Élie; Sitta, Angela Maria

Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1363-1400.

Nous utilisons une approche de la théorie des singularités pour classifier des problèmes de bifurcation ${𝔻}_3$-équivariants de corang 2, avec un ou deux paramètres de bifurcation distingués, et leurs perturbations. Les diagrammes de bifurcation sont identifiés avec des sections sur des chemins dans l’espace des paramètres d’un déployement miniversel ${𝔻}_3$-équivariant $F_0$ de leur noyau. Les équivalences entre les chemins sont données par des difféomorphismes qui se relèvent le long de la projection de l’ensemble des zéros de $F_0$ dans l’espace de ses paramètres. Nos résultats sont appliqués aux bifurcations dégénérées de solutions sous-harmoniques de période 3 dans des systèmes dynamiques réversibles, en particulier dans la résonance 1 :1.

We implement a singularity theory approach, the path formulation, to classify ${𝔻}_3$-equivariant bifurcation problems of corank 2, with one or two distinguished parameters, and their perturbations. The bifurcation diagrams are identified with sections over paths in the parameter space of a ${𝔻}_3$-miniversal unfolding $F_0$ of their cores. Equivalence between paths is given by diffeomorphisms liftable over the projection from the zero-set of $F_0$ onto its unfolding parameter space. We apply our results to degenerate bifurcation of period-$3$ subharmonics in reversible systems, in particular in the 1:1-resonance.

Classification : 37G40, 58K70, 58K40, 34F10, 34F15
Keywords: Equivariant bifurcation, degenerate bifurcation, path formulation, singularity theory, 1:1-resonance, reversible systems, subharmonic bifurcation
Mot clés : bifurcation équivariante, bifurcation dégénérée, formulation des chemins, théorie des singularités, résonance 1 :1, systèmes dynamiques réversibles, bifurcation sous-harmonique

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