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Logarithmic structure of the generalized bifurcation set

100%
Janeczko S.
Annales Polonici Mathematici
|
1996
|
tom 63
|
nr 2
187-197
EN
Let $G: ℂ^{n} × ℂ^{r} → ℂ$ be a holomorphic family of functions. If $Λ ⊂ ℂ^{n} × ℂ^{r}$, $π_r: ℂ^{n} × ℂ^{r} → ℂ^{r}$ is an analytic variety then   $Q_{Λ}(G) = {(x,u) ∈ ℂ^{n} × ℂ^{r}: G(·,u)$ has a critical point in $Λ ∩ π_{r}^{-1}(u)} is a natural generalization of the bifurcation variety of G. We investigate the local structure of $Q_{Λ}(G)$ for locally trivial deformations of $Λ₀ = π_{r}^{-1}(0)$. In particular, we construct an algorithm for determining logarithmic stratifications provided G is versal.
2
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On Arnold’s singularities of type \(B_k, C_k, F_4\)

100%
Janeczko S.
Commentationes Mathematicae
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1986
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tom 26
|
nr 1
EN
The article contains no abstract
3
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On implicit Lagrangian differential systems

100%
Janeczko S.
Annales Polonici Mathematici
|
2000
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tom 74
|
nr 1
133-141
EN
Let (P,ω) be a symplectic manifold. We find an integrability condition for an implicit differential system D' which is formed by a Lagrangian submanifold in the canonical symplectic tangent bundle (TP,ὡ).
4
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Systems of rays in the presence of distribution of hyperplanes

100%
Janeczko S.
Banach Center Publications
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1995
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tom 32
|
nr 1
245-260
EN
Horizontal systems of rays arise in the study of integral curves of Hamiltonian systems $v_H$ on T*X, which are tangent to a given distribution V of hyperplanes on X. We investigate the local properties of systems of rays for general pairs (H,V) as well as for Hamiltonians H such that the corresponding Hamiltonian vector fields $v_H$ are horizontal with respect to V. As an example we explicitly calculate the space of horizontal geodesics and the corresponding systems of rays for the canonical distribution on the Heisenberg group. Local stability of systems of horizontal rays based on the standard singularity theory of Lagrangian submanifolds is also considered.
5
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On Martinet's singular symplectic structures

64%
Domitrz W., Janeczko S.
Banach Center Publications
|
1996
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tom 33
|
nr 1
51-59
6
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Bifurcations in symplectic space

64%
Ishikawa G., Janeczko S.
Banach Center Publications
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2008
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tom 82
|
nr 1
111-124
EN
In this paper we take new steps in the theory of symplectic and isotropic bifurcations, by solving the classification problem under a natural equivalence in several typical cases. Moreover we define the notion of coisotropic varieties and formulate also the coisotropic bifurcation problem. We consider several symplectic invariants of isotropic and coisotropic varieties, providing illustrative examples in the simplest non-trivial cases.
7
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Normal forms of symplectic structures on the stratified spaces

64%
Domitrz W., Janeczko S.
Colloquium Mathematicum
|
1995
|
tom 68
|
nr 1
101-119
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