Path formulation for multiparameter đ”» 3 -equivariant bifurcation problems
Annales de l'Institut Fourier, Volume 60 (2010) no. 4, p. 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 : https://doi.org/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
@article{AIF_2010__60_4_1363_0,
     author = {Furter, Jacques-\'Elie and Sitta, Angela Maria},
     title = {Path formulation for multiparameter $\mathbb{D}\_3$-equivariant bifurcation problems},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {4},
     year = {2010},
     pages = {1363-1400},
     doi = {10.5802/aif.2558},
     zbl = {1204.37054},
     mrnumber = {2722245},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2010__60_4_1363_0}
}
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/item/AIF_2010__60_4_1363_0/

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