DH 3rd floor SR

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Although the numerical methods for stochastic differential equations (SDEs) have been well studied, there are few results on the numerical solutions for SDEwMSs. The main aim of this talk is to investigate the invariant measure of numerical solutions of SDEwMSs and discuss their convergence.

The concept of the slow invariant manifold is the central idea underpinning a transition from micro to macro and model reduction in kinetic theories. We present the constructive methods of invariant manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in the most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space. The equation of motion for immersed manifolds is obtained.

Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as stability.

A collection of methods to derive analytically and to compute numerically the slow invariant manifolds is presented. The systematic use of thermodynamic structures and of the quasi-chemical representation allows us to construct approximations which are in concordance with physical restrictions.

The following examples of applications are presented: Nonperturbative derivation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for nudsen numbers Kn~1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of the list of variables) in order to gain more accuracy in description of highly nonequilibrium flows; model reduction in chemical kinetics.

We formulate and prove a Jakobson-Benedicks-Carleson type theorem on the occurrence of nonuniform hyperbolicity (stochastic dynamics) in families of one-dimensional maps, based on computable starting conditions and providing explicit, computable, lower bounds for the measure of the set of selected parameters. As a first application of our results we obtain a first ever explicit lower bound for the set of parameters corresponding to maps in the quadratic family f_{a}(x) = x^{2}-a which have an absolutely continuous invariant probability measure.