Slawomir Rybicki

Nicolaus Copernicus Univeristy, Torum, Polonia

Basic facts on topological bifurcation theory

Resumen: Many of the  problems of mathematics and mechanics reduces to the study of zeros of continuous families of mappings $f : \bR^n \times \bR \to \bR^n.$ Bifurcation theory is a branch of mathematics investigating a structure of the set of solutions of the equation \beq \label{eq} \tag{E} f(x,\lambda) =0 \eeq under changing  parameter $\lambda \in \bR.$

More precisely speaking, assume that $f : \bR^n \times \bR \to \bR^n$ is a continuous map such that  $f(0,\lambda)=0$ for all $\lambda \in \bR.$
The purpose of my  talk is to study solutions of \eqref{eq}  satisfying $x \neq 0.$

A point $(0,\lambda_0) \in \bR^n \times \bR$ is said to be a bifurcation point of solutions of \eqref{eq} if any sufficiently small neighborhood of $(0,\lambda_0)$ contains a solution $(x,\lambda_1)$ such that $x \neq 0.$

We are going to formulate necessary and sufficient conditions for the existence of   bifurcation points of solutions of  \eqref{eq}.

Finally we will formulate some versions of the famous Krasnosel'skii local bifurcation theorem and Rabinowitz' global bifurcation theorem