[ad_1]

**Brief Description : **https://ocw.mit.edu/courses/18-965-geometry-of-manifolds-fall-2004/8a7e4dd837d1bdd6988e0330babb8c5e_lecture16_17.pdf

**Bifurcation Theory for Fredholm Operators(****arXiv****)**

**Author : **Julián López-Gómez, Juan Carlos Sampedro

**Abstract : **This paper consists of four parts. It begins by using the authors’s generalized Schauder formula, cite{JJ}, and the algebraic multiplicity, χ, of Esquinas and López-Gómez cite{ELG,Es,LG01} to package and sharpening all existing results in local and global bifurcation theory for Fredholm operators through the recent author’s axiomatization of the Fitzpatrick — Pejsachowicz — Rabier degree, cite{JJ2}. This facilitates reformulating and refining all existing results in a compact and unifying way. Then, the local structure of the solution set of F(λ,u)=0 at a simple degenerate eigenvalue is ascertained by means of some concepts and devices of Algebraic Geometry and Galois Theory, which establishes a bisociation between Bifurcation Theory and Algebraic Geometry. Further, we combine the theorem of structure of analytic manifolds with a brilliant idea of Buffoni and Toland cite{BT} to show that the solution sets of the most paradigmatic one-dimensional boundary value problems with analytic nonlinearities actually consist of global analytic arcs of curve. Finally, the unilateral theorems of cite{LG01,LG02}, as well as the refinement of Xi and Wang cite{XW}, are substantially generalized. This paper also analyzes two important examples to illustrate and discuss the relevance of the abstract theory. The second one studies the regular positive solutions of a multidimensional quasilinear boundary value problem of mixed type related to the mean curvature operator

**2.The Index Bundle for Selfadjoint Fredholm Operators and Multiparameter Bifurcation for Hamiltonian Systems (****arXiv****)**

**Author : **Robert Skiba, Nils Waterstraat

**Abstract : **The index of a selfadjoint Fredholm operator is zero by the well-known fact that the kernel of a selfadjoint operator is perpendicular to its range. The Fredholm index was generalised to families by Atiyah and Jänich in the sixties, and it is readily seen that on complex Hilbert spaces this so called index bundle vanishes for families of selfadjoint Fredholm operators as in the case of a single operator. The first aim of this note is to point out that for every real Hilbert space and every compact topological space X there is a family of selfadjoint Fredholm operators parametrised by X×S1 which has a non-trivial index bundle. Further, we use this observation and a family index theorem of Pejsachowicz to study multiparameter bifurcation of homoclinic solutions of Hamiltonian systems, where we generalise a previously known class of examples.

**3. An analytic bifurcation principle for Fredholm operators(****arXiv****)**

**Author :**Matthias Stiefenhofer

**Abstract :** Smooth Equations of the form G[z]=0 are investigated in Banach spaces with the aim of continuing the basic solution G[0]=0 to a solution curve of G[z]=0 with the implicit function theorem. If the linearization is surjective, then the transversality condition of the implicit function theorem can be satisfied in a straightforward way, yielding a regular solution curve, whereas otherwise the equation G[z]=0 has to be extended appropriately for reaching a surjective linearization accessible to the implicit function theorem. This extension process, implying in the first step the standard bifurcation theorem of simple bifurcation points, is continued arbitrarily, yielding a sequence of bifurcation results presumably being applicable to bifurcation points with finite degeneracy.

[ad_2]

Source link