Nous utilisons une approche de la théorie des singularités pour classifier des problèmes de bifurcation -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 -équivariant 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 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 -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 -miniversal unfolding of their cores. Equivalence between paths is given by diffeomorphisms liftable over the projection from the zero-set of onto its unfolding parameter space. We apply our results to degenerate bifurcation of period- subharmonics in reversible systems, in particular in the 1:1-resonance.
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
@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}, pages = {1363--1400}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {60}, number = {4}, year = {2010}, doi = {10.5802/aif.2558}, zbl = {1204.37054}, mrnumber = {2722245}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2558/} }
TY - JOUR AU - Furter, Jacques-Élie AU - Sitta, Angela Maria TI - Path formulation for multiparameter $\mathbb{D}_3$-equivariant bifurcation problems JO - Annales de l'Institut Fourier PY - 2010 SP - 1363 EP - 1400 VL - 60 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2558/ DO - 10.5802/aif.2558 LA - en ID - AIF_2010__60_4_1363_0 ER -
%0 Journal Article %A Furter, Jacques-Élie %A Sitta, Angela Maria %T Path formulation for multiparameter $\mathbb{D}_3$-equivariant bifurcation problems %J Annales de l'Institut Fourier %D 2010 %P 1363-1400 %V 60 %N 4 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2558/ %R 10.5802/aif.2558 %G en %F 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, Tome 60 (2010) no. 4, pp. 1363-1400. doi : 10.5802/aif.2558. http://www.numdam.org/articles/10.5802/aif.2558/
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