Graduate Students' Seminar

Title: A Brief Tour of Bifurcation Theory.

Abstract:  We are often interested in studying nonlinear equations dependent on a parameter of the form f(x, a)=0. This talk offers a concise introduction to bifurcation theory for nonlinear problems. After motivating the study of parameter-dependent solution branches, we review the key necessary conditions for the occurrence of bifurcation. We then outline the Lyapunov–Schmidt reduction and its use in simplifying infinite-dimensional problems, together with Morse's lemma, which clarifies the local structure near critical points. The talk concludes with a brief overview of a perturbation method and a discussion of Krasnoselskii’s and Rabinowitz's theorem.