Path formulation for multiparameter đ”» 3 -equivariant bifurcation problems
Annales de l'Institut Fourier, Volume 60 (2010) no. 4, pp. 1363-1400.

We implement a singularity theory approach, the path formulation, to classify đ”» 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 đ”» 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.

Nous utilisons une approche de la thĂ©orie des singularitĂ©s pour classifier des problĂšmes de bifurcation đ”» 3 -Ă©quivariants de corang 2, avec un ou deux paramĂštres de bifurcation distinguĂ©s, et leurs perturbations. Les diagrammes de bifurcation sont identifiĂ©s avec des sections sur des chemins dans l’espace des paramĂštres d’un dĂ©ployement miniversel đ”» 3 -Ă©quivariant F 0 de leur noyau. Les Ă©quivalences entre les chemins sont donnĂ©es par des diffĂ©omorphismes qui se relĂšvent le long de la projection de l’ensemble des zĂ©ros de F 0 dans l’espace de ses paramĂštres. Nos rĂ©sultats sont appliquĂ©s aux bifurcations dĂ©gĂ©nĂ©rĂ©es de solutions sous-harmoniques de pĂ©riode 3 dans des systĂšmes dynamiques rĂ©versibles, en particulier dans la rĂ©sonance 1 :1.

DOI: 10.5802/aif.2558
Classification: 37G40, 58K70, 58K40, 34F10, 34F15
Keywords: Equivariant bifurcation, degenerate bifurcation, path formulation, singularity theory, 1:1-resonance, reversible systems, subharmonic bifurcation
Furter, Jacques-Élie 1; Sitta, Angela Maria 2

1 Brunel University Department of Mathematical Sciences Uxbridge UB8 3PH (United Kingdom)
2 Universidade Estadual Paulista - UNESP Departamento de Matemåtica - IBILCE Campus de São José do Rio Preto - SP (Brazil)
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Furter, Jacques-Élie; Sitta, Angela Maria. Path formulation for multiparameter $\mathbb{D}_3$-equivariant bifurcation problems. Annales de l'Institut Fourier, Volume 60 (2010) no. 4, pp. 1363-1400. doi : 10.5802/aif.2558. https://aif.centre-mersenne.org/articles/10.5802/aif.2558/

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