On invariant tori of weakly connected systems of differential equations

We consider a family of systems of differential equations depending on a sufficiently small parameter, whose zero value corresponds to a couple of independent systems. We use the method of Green-Samoilenko function for the construction of an invariant manifold of the perturbed system and present some examples of application. Published in Neliniini Kolyvannya, Vol. 8, No. 4, pp. 468–489, October–December, 2005.     

Existence of an invariant torus for one class of systems of differential equations

We consider the problem of the existence of an asymptotically stable toroidal set for a system of linear differential equations defined on an m-dimensional torus. We establish conditions under which a nonlinear system of differential equations has an invariant toroidal manifold. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 520–529, October–December, 2008.     

On approximation of systems of differential-difference equations of neutral type by systems of ordinary differential equations

We construct a scheme of approximation of a system of differential-difference equations of neutral type by systems of ordinary differential equations and investigate the convergence conditions of this scheme. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 3, pp. 328–335, July–September, 2007.     

Invariant recording of elasticity theory equations

An invariant (with respect to rotations) formalization of equations of linear and nonlinear elasticity theory is proposed. An equation of state (in the form of a convex generating potential) for various crystallographic systems is written. An algebraic approach is used, which does not require any geometric constructions related to the analysis of symmetry in crystals. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 127–142, September–October, 2008.     

Invariant Sets of Systems of Stochastic Differential Equations with Jumps

We introduce the notion of invariant surfaces for inhomogeneous stochastic differential equations with jumps. The results obtained enable one to determine invariant surfaces for stochastic differential equations of the type indicated. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 234–240, April–June, 2005.     

Regular submodels of types (1,2) and (1,1) of the equations of gas dynamics

Partially invariant solutions of types (1, 2) and (1, 1) for gas-dynamic equations are regularly divided into two classes: for the first class, the invariant independent variable is the time, i.e., this class contains barochronic solutions, and for the second class, the invariant variable necessarily depends on spatial coordinates. The barochronic submodel of gas-dynamic equations, as well as a passive subsystem for solutions of the second class, is integrated in finite form. In the latter case, the invariant subsystem is reduced to an ordinary differential equation and quadratures. Integration of the submodels is illustrated by a number of examples. The following common properties of barochronic gas flows are described: rectilinear trajectories of gas particles, the possibility of collapse of density on a manifold, and stratification of the space of events. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 40–49, March–April, 1999.     

Exact solutions of the axisymmetric equations of motion of a viscous heat-conducting perfect gas described by systems of ordinary differential equations

All invariant solutions of rank 1 of the two-dimensional equations of motion of a heat-conducting perfect gas with a polytropic equation of state are described. A sufficient condition for reducibility of regular, partially invariant solutions of rank 1 and defect 1 to invariant solutions is given. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 51–54, September–October, 1999.     

On the decomposition of a degenerate singularly perturbed linear system of differential equations

We study the problem of decomposition of degenerate singularly perturbed systems of differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 3, pp. 401–415, July–September, 2006.     

Investigation of even systems of nonlinear functional differential equations with restrictions

We establish consistency conditions for even systems of nonlinear functional differential equations with restrictions and substantiate the applicability of an iterative method to these problems. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 252–260, April–June, 2008.     

Setting up Lyapunov functions for the class of systems with quasiperiodic coefficients

The paper proposes a method to set up a matrix-valued Lyapunov function for a system of differential equations with quasiperiodic coefficients. This function is used to establish asymptotic stability conditions for a class of linear systems Translated from Prikladnaya Mekhanika, Vol. 44, No. 12, pp. 121–130, December 2008.     

New approach to the investigation of the existence of periodic solutions of systems of differential equations and their construction

Methods developed for the solution of general equations with restrictions are applied to the construction of periodic solutions of systems of differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 1, pp. 55–70, January–March, 2007.     

Elements of Lyapunov stability theory for dynamic equations on time scale

Stability of dynamic equations on time scale is analyzed. The main results are new conditions of stability, uniform stability, and uniform asymptotic stability for quasilinear and nonlinear systems On the occasion of the 150th birthday of A. M. Lyapunov __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 3–27, September 2007.     

Even systems of functional differential equations with restrictions and methods for their solution

We propose an approach to the investigation of even systems of functional differential equations with restrictions and control. According to this approach, the investigation of the consistency of the considered problem is reduced to the investigation of the solvability of a system of integral equations. We substantiate the application of the iteration and projection-iterative methods to this problem. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 1, pp. 113–125, January–March, 2007.     

Asymptotics of solutions of nonautonomous second-order ordinary differential equations close to linear equations

We find asymptotic representations for certain classes of solutions of nonautonomous second-order differential equations that are close, in a certain sense, to linear equations. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 230–241, April–June, 2008.     

Bounded solutions of linear differential equations in a banach space

For a linear inhomogeneous differential equation in a Banach space, we find a criterion for the existence of solutions that are bounded on the entire real axis under the assumption that the homogeneous equation admits an exponential dichotomy on the semiaxes. This result is a generalization of the Palmer lemma to the case of infinite-dimensional spaces. We consider examples of countable systems of ordinary differential equations that have bounded solutions. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 3–14, January–March, 2006.     

On the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operators

We establish new efficient conditions sufficient for the unique solvability of the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operators. Published in Neliniini Kolyvannya, Vol. 10, No. 4, pp. 560–573, October–December, 2007.     

Asymptotic properties of solutions of systems of nonlinear functional differential equations of neutral type

We establish sufficient conditions for systems of nonlinear functional differential equations of neutral type to have solutions that are continuously differentiable and bounded for t ∈ ℝ (together with their first derivatives) and investigate the asymptotic properties of these solutions. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 20–26, January–March, 2009.     

Some types of kinetic equations reducible to partial differential equations

The possibility of passing from the kinetic equation to a partial differential equations is rigorously mathematically proved for the case of nearly elastic scattering processes. Some examples are considered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 12–16, September–October, 2007.     

A version of the projection-iterative method for systems of linear differential equations with delay of neutral type and restrictions

We establish consistency conditions for systems of linear differential equations with constant delay of neutral type and restrictions. The applicability of the projection-iterative method to these problems is justified. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 4, pp. 564–573, October–December, 2006.     

Invariant and Partially Invariant Solutions of the Green-Naghdi Equations

All invariant and partially invariant solutions of the Green-Naghdi equations are obtained that describe the second approximation of shallow water theory. It is proved that all nontrivial invariant solutions belong to one of the following types: Galilean-invariant, stationary, and self-similar solutions. The Galilean-invariant solutions are described by the solutions of the second Painleve equation, the stationary solutions by elliptic functions, and the self-similar solutions by the solutions of the system of ordinary differential equations of the fourth order. It is shown that all partially invariant solutions reduce to invariant solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 26–35, November–December, 2005.