scholarly journals Path formulation for multiparameter \mathbb{D}_3-equivariant bifurcation problems

2010 ◽  
Vol 60 (4) ◽  
pp. 1363-1400
Author(s):  
Jacques-Élie Furter ◽  
Angela Sitta
Keyword(s):  
Equivariant Bifurcation ◽  
Bifurcation Problems

Bifurcations, symmetries and the notion of fixed subspace

2021 ◽  
pp. 2130006
Author(s):  
Giampaolo Cicogna
Keyword(s):  
Topological Index ◽  
Ad Hoc ◽  
Index Theory ◽  
Typical Case ◽  
One Dimensional ◽  
Dimension Formula ◽  
Symmetry Properties ◽  
Equivariant Bifurcation ◽  
Stationary Bifurcation ◽  
Bifurcation Problems

In the context of stationary bifurcation problems admitting a symmetry, this paper is focused on the key notion of Fixed Subspace (FS), and provides a review of some applications aimed at detecting bifurcating solutions in various situations. We start recalling, in its commonly used simplified version, the old Equivariant Bifurcation Lemma (EBL), where the FS is one-dimensional; then we provide a first generalization in a typical case of non-semisimple critical eigenvalues, where the presence of the symmetry produces a non-trivial situation. Next, we consider the case of FSs of dimension [Formula: see text] in very different contexts. First, relying on the topological index theory and in particular on the Krasnosel’skii theorem, we provide a largely applicable statement of an extension of the EBL. Second, we propose a completely different and new application which combines symmetry properties with the notion of stability of bifurcating solutions. We also provide some simple examples, constructed ad hoc to illustrate the various situations.

On the Stability of Equivariant Bifurcation Problems and Their Unfoldings

1992 ◽  
Vol 35 (2) ◽  
pp. 237-246 ◽  
Author(s):  
Ali Lari-Lavassani ◽  
Yung-Chen Lu
Keyword(s):  
Bifurcation Theory ◽  
Local Level ◽  
Equivariant Bifurcation ◽  
Bifurcation Problems ◽  
The Stability ◽  
Equivariant Version

AbstractIn their book Singularities and Groups in Bifurcation Theory M. Golubitsky, I. Stewart and D. Schaeffer have introduced an equivariant version of Martinet's notion of V (for variety)-equivalence with parameter. In this paper we give a unified proof that, in this context, infinitesimal stability is equivalent to stability at the local level of germs and that stability in the unfolding category is equivalent to versality.

scholarly journals Algebraic path formulation for equivariant bifurcation problems

1998 ◽  
Vol 124 (2) ◽  
pp. 275-304 ◽  
Author(s):  
JACQUES-ELIE FURTER ◽  
ANGELA MARIA SITTA ◽  
IAN STEWART
Keyword(s):  
Equivariant Bifurcation ◽  
Bifurcation Problems

scholarly journals Origin Preserving Path Formulation for Multiparameter ℤ2-Equivariant Corank 2 Bifurcation Problems

2020 ◽  
Vol 30 (09) ◽  
pp. 2050140
Author(s):  
Jacques-Elie Furter
Keyword(s):  
Singularity Theory ◽  
Cylindrical Panel ◽  
Qualitative Behavior ◽  
Partial Classification ◽  
Equivariant Bifurcation ◽  
Bifurcation Problems ◽  
Multiparameter Bifurcation ◽  
Generic Bifurcation ◽  
Minimal Modification

A singularity theory, in the form of path formulation, is developed to analyze and organize the qualitative behavior of multiparameter [Formula: see text]-equivariant bifurcation problems of corank 2 and their deformations when the trivial solution is preserved as parameters vary. Path formulation allows for an efficient discussion of different parameter structures with a minimal modification of the algebra between cases. We give a partial classification of one-parameter problems. With a couple of parameter hierarchies, we show that the generic bifurcation problems are 2-determined and of topological codimension-0. We also show that the preservation of the trivial solutions is an important hypotheses for multiparameter bifurcation problems. We apply our results to the bifurcation of a cylindrical panel under axial compression.

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