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The Generalized Saddle-Node Bifurcation of Degenerate Solution

100%
Liu P., Wang Y.-W.
Commentationes Mathematicae
|
2005
|
tom 45
|
nr 2
EN
In this paper we discuss the bifurcation problem for the abstract operator equation of the form \(F (u, \lambda) = \theta\) with a parameter \(\lambda\), where \(F\colon X \times R \to Y\) is a \(C^1\) mapping, \(X, Y\) are Banach spaces. By the bounded linear generalized inverse \(A^+\) of \(A = F_u (u_0 , \lambda_0 )\), an abstract bifurcation theorem for the case \(\operatorname{dim}N (F_u (u_0 , \lambda_0 )) \geq \operatorname{codim} R(F_u (u_0 , \lambda_0 )) = 1\) has been obtained.
2
Content available remote

Versal deformations of $D_q$-invariant 2-parameter families of planar vector fields

100%
Świrszcz G.
Annales Polonici Mathematici
|
1995
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tom 62
|
nr 3
265-281
EN
The paper deals with 2-parameter families of planar vector fields which are invariant under the group $D_q$ for q ≥ 3. The germs at z = 0 of such families are studied and versal families are found. We also give the phase portraits of the versal families.
3
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Generic saddle-node bifurcation for cascade second order ODEs on manifolds

88%
Medveď M.
Annales Polonici Mathematici
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1998
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tom 68
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nr 3
211-225
EN
Cascade second order ODEs on manifolds are defined. These objects are locally represented by coupled second order ODEs such that any solution of one of them can represent an external force for the other one. A generic saddle-node bifurcation theorem for 1-parameter families of cascade second order ODEs is proved.
4
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Local properties of the solution set of the operator equation in Banach spaces in a neighbourhood of a bifurcation point

75%
Janczewska J.
Open Mathematics
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2004
|
tom 2
|
nr 4
561-572
EN
In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×R k →Y is a C 2-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R 1 that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ0)∈X×R k and we describe the solution set of the studied equation in a small neighbourhood of this point.
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