Seminar on nonlinear problems of PDE and mathematical physics: “On a variational principle of finding of bifurcations for nonlinear equations”
20 December 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
Annotation: 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).