Logarithmic structure of the generalized bifurcation set

S. Janeczko

Displaying similar documents to “Logarithmic structure of the generalized bifurcation set”

Path formulation for multiparameter $𝔻_{3}$ -equivariant bifurcation problems

Jacques-Élie Furter, Angela Maria Sitta (2010)

Annales de l’institut Fourier

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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...

On the topological triviality along moduli of deformations of $J_{k, 0}$ singularities

Piotr Jaworski (2000)

Annales Polonici Mathematici

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It is well known that versal deformations of nonsimple singularities depend on moduli. However they can be topologically trivial along some or all of them. The first step in the investigation of this phenomenon is to determine the versal discriminant (unstable locus), which roughly speaking is the obstacle to analytic triviality. The next one is to construct continuous liftable vector fields smooth far from the versal discriminant and to integrate them. In this paper we extend the results...

Bifurcation theorems of Rabinowitz type for certain differential operators of the fourth order

Jolanta Przybycin (1992)

Annales Polonici Mathematici

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This paper was inspired by the works of P. H. Rabinowitz. We study nonlinear eigenvalue problems for some fourth order elliptic partial differential equations with nonlinear perturbation of Rabinowitz type. We show the existence of an unbounded continuum of nontrivial positive solutions bifurcating from (μ₁,0), where μ₁ is the first eigenvalue of the linearization about 0 of the considered problem. We also prove the related theorem for bifurcation from infinity. The results obtained...

On Halphen’s Theorem and some generalizations

Alcides Lins Neto (2006)

Annales de l’institut Fourier

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Let $M^{n}$ be a germ at $0 \in ℂ^{m}$ of an irreducible analytic set of dimension $n$ , where $n \geq 2$ and $0$ is a singular point of $M$ . We study the question: when does there exist a germ of holomorphic map $φ : (ℂ^{n}, 0) \to (M, 0)$ such that $φ^{- 1} (0) = {0}$ ? We prove essentialy three results. In Theorem 1 we consider the case where $M$ is a quasi-homogeneous complete intersection of $k$ polynomials $F = (F_{1}, ..., F_{k})$ , that is there exists a linear holomorphic vector field $X$ on $ℂ^{m}$ , with eigenvalues $λ_{1}, ..., λ_{m} \in ℚ_{+}$ such that $X (F^{T}) = U \cdot F^{T}$ , where $U$ is a $k \times k$ matrix with entries in $𝒪_{m}$ . We prove that if...

On the versal discriminant of $J_{k, 0}$ singularities

Piotr Jaworski (1996)

Annales Polonici Mathematici

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It is well known that the versal deformations of nonsimple singularities depend on moduli. The first step in deeper understanding of this phenomenon is to determine the versal discriminant, which roughly speaking is an obstacle to analytic triviality of an unfolding or deformation along the moduli. The versal discriminant of the Pham singularity ( $J_{3, 0}$ in Arnold’s classification) was thoroughly investigated by J. Damon and A. Galligo [2], [3], [4]. The goal of this paper is to continue their...

Nonlinear eigenvalue problems for fourth order ordinary differential equations

Jolanta Przybycin (1995)

Annales Polonici Mathematici

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This paper was inspired by the works of Chiappinelli ([3]) and Schmitt and Smith ([7]). We study the problem ℒu = λau + f(·,u,u',u'',u''') with separated boundary conditions on [0,π], where ℒ is a composition of two operators of Sturm-Liouville type. We assume that the nonlinear perturbation f satisfies the inequality |f(x,u,u',u'',u''')| ≤ M|u|. Because of the presence of f the considered equation does not in general have a linearization about 0. For this reason the global bifurcation...

On some formula in connected cocommutative Hopf algebras over a field of characteristic 0

Piotr Wiśniewski (2000)

Colloquium Mathematicae

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Let H be a cocommutative connected Hopf algebra, where K is a field of characteristic zero. Let $H^{+} = K e r$ and $h^{+} = h - (h)$ for $h \in H$ . We prove that $d_{h} = \sum_{r = 1}^{\infty} ({(- 1)}^{r + 1} / r) \sum h_{1}^{+} . . . h_{r}^{+}$ is primitive, where $\sum h_{1} \otimes . . . \otimes h_{r} = Δ_{r - 1} (h)$ .

A study of the Rimmer bifurcation of symmetric fixed points of reversible diffeomorphisms.

Petrişor, Emilia (1996)

Balkan Journal of Geometry and its Applications (BJGA)

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Existence, multiplicity, and bifurcation in systems of ordinary differential equations.

Ward, James R. jun. (2007)

Electronic Journal of Differential Equations (EJDE) [electronic only]

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A characterization of linear automorphisms of the Euclidean ball

Hidetaka Hamada, Tatsuhiro Honda (1999)

Annales Polonici Mathematici

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Let B be the open unit ball for a norm on $ℂ^{n}$ . Let f:B → B be a holomorphic map with f(0) = 0. We consider a condition implying that f is linear on $ℂ^{n}$ . Moreover, in the case of the Euclidean ball , we show that f is a linear automorphism of under this condition.

The complementing condition and its role in a bifurcation theory applicable to nonlinear elasticity.

Negrón-Marrero, Pablo V., Montes-Pizarro, Errol (2011)

The New York Journal of Mathematics [electronic only]

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