Il'yasov Yavdat Shavkatovich

• Position:  Principal Research Scientist, Professor,
• Degree: Doctor Phys.-Math. Sci.
• E-mail: ilyasov02@gmail.com
• Research interests: 
• My research interests are broad and vary with time but I am particularly fascinated by the study of bifurcations in infinite-dimensional equations defined by the evolution operator of non-linear PDEs.

  In science and technology, bifurcation corresponds to the limiting behavior of systems when the properties undergo sudden change under a small variation of the parameters. The bifurcations often become a trigger to systems’ chaotic and critical behavior. Шmportant examples with bifurcations include models describing neural and biological networks, static voltage collapse (blackout) in power systems, chemical reactions, plasma physics, quantum physics, etc. The associated dynamic equations with bifurcations could have rarefaction and shock waves and exhibit catastrophes and hysteresis behavior. This makes the study of bifurcations of paramount importance in understanding and predicting the behavior of systems in technology and science.

My research in the field is aimed at the problem of the finding singularities to parametrized families of non-linear equations and it focuses on the following issues:

• Finding explicit variational formulations for determining singularity points of a given type to parametrized families of nonlinear equations.
• Finding general necessary and sufficient conditions under which parametrized families of non-linear equations has a singularity, including the saddle-node bifurcations, Hopf bifurcations, state, and switching limits values, and extremal values of applicability of methods.
• Development of innovative ideas for new high-performance algorithms to numerically find singular values.
• Development of new methods in the solvability theory for nonlinear PDEs based on the theory of singularities of non-linear equations.

To solve these issues, I introduced a concept that is based on the idea that the solutions can be obtained due to appropriate non-linear generalizations of the well-known linear and finite-dimensional theory:

• Poincaré and Courant-Weyl principles.
• minimax Collatz–Wielandt formula.

In this way, I have succeeded to introduce two general approaches:

• A generalization of the Poincaré and Courant-Weyl principles to the nonlinear problems which I called a nonlinear generalized Rayleigh quotient method. It provides a new tool for studying the geometry of nonlinear equations and thus allows to solve geometrically more complex and open problems
• The so-called extended functional method that makes it possible to find bifurcations of non-linear equations using a new type of variational principle, which is essentially a generalization of the Collatz-Wielandt variational principle to the nonlinear problems. In general, the method is a new paradigm in the theory of bifurcations and in the theory of differential equations.
• Furthermore, it turns out that the extended functional method makes it possible to construct a novel method for numerically finding bifurcations of nonlinear problems, which is more efficient and less time-consuming in comparison with known methods.

My research in the field covers a wide range of topics including 1) applied problems such as voltage stability of the power systems, design of chemical reactions, orbital stability of models in the plasma and quantum physics; 2) development of new numerical approaches such as iterative and gradient methods for studying nonsmooth functionals, new algorithms for the inverse spectral problem; 3) purely theoretical research on the nonlinear analysis, variational theory, the theory of solvability and stability of solutions to PDEs, spectral theory, inverse problems, the theory of infinite-dimensional stochastic equations.