Seminar on nonlinear problems of PDE and mathematical physics on topic: “On a variational principle of finding of bifurcations for nonlinear equations”
8 November at 18:00 MSK
Speakers: Professor Y.Sh. Il’yasov (Institute of Mathematics with Computer Center of UFRC, RAS, Ufa, Russia)
Title of the talk: "On a variational principle of finding of bifurcations for nonlinear equations".
In this presentation we will discuss the problem of finding bifurcations of branches of solutions to systems of nonlinear equations. We will start with providing a brief overview of the existing approaches, common difficulties, and unsolved issues in this problem — under both finite and infinite dimensions.
In the main part of the talk, we will present a new method based on the use of a new type of variational functionals associated with the equations, the critical points of which correspond to bifurcation solutions.
This method allows to find sufficient conditions for existence of saddle-node bifurcations, and to subsequently derive a variational formula for their direct finding. The application of this method will also be illustrated on non-linear partial differential equations, finite-difference approximations of nonlinear differential equations, as well as systems of algebraic trigonometric equations that arise in finding the maximum acceptable supply voltage in electrical circuits (blackout problem).