Classification of D4, S1-equivariant bifurcation problems up to  topological codimension 2

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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Bifurcations, symmetries and the notion of fixed subspace

Reviews in Mathematical Physics ◽

10.1142/s0129055x21300065 ◽

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.

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On the Stability of Equivariant Bifurcation Problems and Their Unfoldings

Canadian Mathematical Bulletin ◽

10.4153/cmb-1992-034-x ◽

1992 ◽

Vol 35 (2) ◽

pp. 237-246 ◽

Cited By ~ 3

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.

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Unfolding of multiparameter equivariant bifurcation problems with two groups of state variables under left-right equivalent group

Applied Mathematics and Mechanics ◽

10.1007/bf02465393 ◽

2005 ◽

Vol 26 (4) ◽

pp. 530-538 ◽

Cited By ~ 1

Author(s):

Guo Rui-zhi ◽

Li Yang-cheng

Keyword(s):

State Variables ◽

Equivariant Bifurcation ◽

Bifurcation Problems

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NONDEGENERATE UMBILICS, THE PATH FORMULATION AND GRADIENT BIFURCATION PROBLEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s021812740902458x ◽

2009 ◽

Vol 19 (09) ◽

pp. 2965-2977 ◽

Cited By ~ 1

Author(s):

JACQUES-ELIE FURTER ◽

ANGELA MARIA SITTA

Keyword(s):

Cylindrical Panel ◽

Point Of View ◽

Singular Behavior ◽

Universal Unfolding ◽

Buckling Modes ◽

The Core ◽

Bifurcation Problems ◽

Multiparameter Bifurcation ◽

Special Parameter

Parametrized contact-equivalence is a successful theory for the understanding and classification of the qualitative local behavior of bifurcation diagrams and their perturbations. Path formulation is an alternative point of view making explicit the singular behavior due to the core of the bifurcation germ (when the parameters vanish) from the effects of the way parameters enter. We show how to use path formulation to classify and structure efficiently multiparameter bifurcation problems in corank 2 problems. In particular, the nondegenerate umbilics singularities are the generic cores in four situations: the general or gradient problems, with or without ℤ2 symmetry where ℤ2 acts on the second component of ℝ2 via κ(x,y) = (x,-y). The universal unfolding of the umbilic singularities have an interesting "Russian doll" type of structure of miniversal unfoldings in all those categories. With the path formulation approach we can handle one, or more, parameter situations using the same framework. We can even consider some special parameter structure (for instance, some internal hierarchy of parameters). We classify the generic bifurcations with 1, 2 or 3 parameters that occur in those cases. Some results are known with one bifurcation parameter, but the others are new. We discuss some applications to the bifurcation of a loaded cylindrical panel. This problem has many natural parameters that provide concrete examples of our generic diagrams around the first interaction of the buckling modes.

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