On Reduction Principle in Stability Theory for Systems with Random Perturbations

For stochastic systems, we obtain an analog of the reduction principle that enables one to reduce the analysis of the stability of a system with random perturbations to the analysis of the stability of a deterministic system.     

Investigation of Stochastic Oscillation Systems with Toroidal Manifold

For a nonlinear stochastic system of the Itô type with invariant toroidal manifold, we establish conditions for its stability and the ergodic property of its solutions in the neighborhood of the manifold.     

Investigation of Exponential Dichotomy of Itô Stochastic Systems by Using Quadratic Forms

For linear stochastic systems, we obtain sufficient conditions for mean-square exponential dichotomy in terms of Lyapunov functions that are quadratic forms.     

On Averaging of Difference Equations on an Infinite Interval

The averaging method on a semiaxis is justified for systems of difference equations. A theorem on the closeness of the corresponding solutions of the exact and averaged systems on a semiaxis is proved. This theorem is analogous to the second Bogolyubov theorem on the averaging method for systems of difference equations.     

On the Relationship between Properties of Solutions of Difference Equations and the Corresponding Differential Equations

We establish conditions under which the existence of a periodic solution of a differential equation is preserved if a solution of the corresponding difference equation possesses the same property. We prove the convergence of periodic solutions of a system of difference equations to a periodic solution of a system of differential equations. Analogous problems are considered for bounded solutions. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 989–996, July, 2005.