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Bifurcations for Turing instability without SO(2) symmetry

Toshiyuki Ogawa, Takashi Okuda (2007)


In this paper, we consider the Swift–Hohenberg equation with perturbed boundary conditions. We do not a priori know the eigenfunctions for the linearized problem since the SO ( 2 ) symmetry of the problem is broken by perturbation. We show that how the neutral stability curves change and, as a result, how the bifurcation diagrams change by the perturbation of the boundary conditions.

Codimension 4 singularities on reflectionally symmetryc planar vector fields.

Freddy Dumortier, Santiago Ibáñez (1999)

Publicacions Matemàtiques

The paper deals with the topological classification of singularities of vector fields on the plane which are invariant under reflection with respect to a line. As it has been proved in previous papers, such a classification is necessary to determine the different topological types of singularities of vector fiels on R3 whose linear part is invariant under rotations. To get the classification we use normal form theory and the the blowing-up method.

Generic bifurcation of reversible vector fields on a 2-bidimensional manifold.

Marco Antonio Teixeira (1997)

Publicacions Matemàtiques

In this paper we deal with reversible vector fields on a 2-dimensional manifold having a codimension one submanifold as its symmetry axis. We classify generically the one parameter families of such vector fields. As a matter of fact, aspects of structural stability and codimension one bifurcation are analysed.

Path formulation for multiparameter 𝔻 3 -equivariant bifurcation problems

Jacques-Élie Furter, Angela Maria Sitta (2010)

Annales de l’institut Fourier

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...

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