Ritz method for equations with small parameters for higher derivatives

The problem of convergence of the Ritz method is considered for positive definite operational equations of the form $$A_\varepsilon u \equiv (\varepsilon {\rm A}_1 + A_0 )u = f$$ containing small parameters ? for the principal part. For specific natural conditions it is proved that the Ritz method, used for an approximate solution to such equations, converges to an exact solution in a metric with quadratic form uniformly with respect to the parameter ?.     

Asymptotics for the solution of averaged equations for a system of coupled oscillators

The equations obtained by averaging a system of three weakly nonlinear oscillators are considered. Distinctive properties of asymptotics at infinity for the solutions of averaged equations for both constant and variable proper frequencies are studied. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 99–113, 2006.     

Existence Theorems and Estimates of Solutions for Equations of Principal Resonance

In this paper, we study systems of nonlinear, nonautonomous, ordinary differential equations that appear in the theory of averaging of nonlinear oscillations. We prove existence theorems for them and obtain conditions under which variables of the type of amplitude (or energy) are uniformly bounded with respect to time.     

Autoresonant asymptotics in an oscillating system with weak dissipation

We study a system of two first-order differential equations arising in averaging nonlinear systems over fast single-frequency oscillations. We consider the situation where the original system contains weak dissipative terms. We construct the asymptotic form of a two-parameter solution with an unbounded increasing amplitude. This result gives a key for understanding autoresonance in weak dissipative systems as a phenomenon of significant increase in the forced nonlinear oscillation initiated by a small external pumping. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 160, No. 1, pp. 102–111, July, 2009.     

Analysis of the synchronization model in an anisochronous system

A model system of two first-order differential equations that arises in the synchronization theory of nonlinear oscillations is investigated. Constraints on the parameters of the equations under which the synchronization is realized on every solution are found. A domain of the parameters in which the synchronization occurs only for a part of the solution set is determined.     

Asymptotic analysis of a multiferroic model

Theoretical and Mathematical Physics - We consider a system of nonlinear autonomous differential equations that describe the orientation of the antiferromagnetic vector in a multiferroic film and...     

Stability of equilibrium with respect to white noise

A system of ordinary differential equations with a local asymptotically stable equilibrium is considered. The problem of stability with respect to a persistent perturbation of the white noise type is discussed. Stability with given bounds is proved on a large time interval with length of the order of the squared inverse perturbation amplitude. The proof is based on the construction of a barrier function for the parabolic Kolmogorov equation associated with the perturbed dynamical system.