On the solution of the Chazy system of equations

We obtain a solution of the Chazy system that consists of nine nonlinear algebraic equations. This system gives a necessary condition for the class of nonlinear third-order differential equations with six singularities to be of P-type. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 92–98, January–March, 2009.     

On the existence of solutions and averaging of multipoint boundary-value problems for multifrequency systems with linearly transformed argument

For an m-frequency system of differential equations with linearly transformed argument, we prove the existence of a solution that satisfies initial or multipoint boundary conditions. We obtain an estimate for the error of the method of averaging over fast variables that explicitly depends on a small parameter. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 462–471, October–December, 2008.     

Invariant manifolds of one class of systems of impulsive differential equations

We consider general problems related to the existence of invariant toroidal sets for linear and weakly nonlinear systems of impulsive differential equations defined in the direct product of an m-dimensional torus and an n-dimensional Euclidean space. We investigate classes of problems for which the conditions for the existence of invariant toroidal manifolds are satisfied.     

Fokker–Planck Equations for a Free Energy Functional or Markov Process on a Graph

The classical Fokker–Planck equation is a linear parabolic equation which describes the time evolution of the probability distribution of a stochastic process defined on a Euclidean space. Corresponding to a stochastic process, there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and an entropy. In recent years, it has been shown that the Fokker–Planck equation is the gradient flow of the free energy functional defined on the Riemannian manifold of probability distributions whose inner product is generated by a 2-Wasserstein distance. In this paper, we consider analogous matters for a free energy functional or Markov process defined on a graph with a finite number of vertices and edges. If N ≧ 2 is the number of vertices of the graph, we show that the corresponding Fokker–Planck equation is a system of N nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. However, in contrast to stochastic processes defined on Euclidean spaces, the situation is more subtle for discrete spaces. We have different choices for inner products on the space of probability distributions resulting in different Fokker–Planck equations for the same process. It is shown that there is a strong connection but there are also substantial discrepancies between the systems of ordinary differential equations and the classical Fokker–Planck equation on Euclidean spaces. Furthermore, both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the Riemannian manifolds of probability distributions with different metrics. Some examples are also discussed.     

Approximation of a bundle of solutions of linear differential inclusions

We construct an approximation of a bundle of solutions of a linear impulsive differential inclusion using a system of linear impulsive differential equations with Hukuhara derivative. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 3, pp. 386–400, July–September, 2006.     

Nonlinear Differential Equations in {\mathcal{B}(\mathcal{H})} Motivated by Learning Models

We study a system of ordinary differential equations in B(H){\mathcal{B}(\mathcal{H})} , the space of all bounded linear operators on a separable Hilbert space H{\mathcal{H}} . The system considered is a natural generalization of the Oja–Cox–Adams learning models. We establish the local existence of solutions and solve explicitly the system for a class of initial conditions. For such cases, we also characterize the asymptotic behavior of solutions.     

Dynamics of a chain system of rigid bodies with gravity-friction seismic dampers: fixed supports

A design model for a chain system of N elastically linked rigid bodies with a spheroidal gravity-friction damper is proposed. The Lagrange–Painlevé equations of the first kind are used to construct nonlinear dynamical models of a mechanical system undergoing translational vibrations about the equilibrium position. The conditions under which the system moves in one plane are established. The double nonstationary phase–frequency resonance of a system with N = 2 is analyze. After the numerical integration of the systems of differential equations, the phase–frequency surfaces are plotted and examined for several combinations of system parameters under two-frequency loading     

Construction of asymptotic formulas for solutions of singularly perturbed degenerate systems of differential equations

We obtain an asymptotic solution of the Cauchy problem for a singularly perturbed degenerate system of differential equations in the case of a singular limit pencil of matrices. Translated from Neliniini Kolyvannya, Vol. 11, No. 3, pp. 408–420, July–September, 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.     

Degenerate Fredholm boundary-value problems

We obtain a criterion for the existence of solutions of degenerate inhomogeneous Fredholm boundary-value problems for a system of ordinary differential equations under the assumption that the degenerate system of differential equations can be reduced to the central canonical form. The results are illustrated by examples. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 3, pp. 303–312, July–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.     

On the structure of the set of continuously differentiable solutions of one boundary-value problem for a system of functional differential equations of neutral type

We study the structure of the set of solutions, continuously differentiable for t ∈ R ⁺ = [0; + ∞), of one limit problem for systems of nonlinear functional differential equations of neutral type with nonlinear deviations of argument that depend on an unknown function. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 2, pp. 277–289, April–June, 2007.     

On a Fluid-Structure Model in Which the Dynamics of the Structure Involves the Shear Stress Due to the Fluid

In this paper we consider a model for fluid-structure interaction. The hybrid system describes the interaction between an incompressible fluid in a three-dimensional container with interior a fixed domain and a thin elastic plate, the interface, which coincides with a flexible flat part of the surface of the vessel containing the fluid. The motion of the fluid is described by the linearized Navier–Stokes equations and the deformation of the plate by the classical plate equations for in-plane motions, modified to include the viscous shear stress which the fluid exerts on the plate as well as damping of Kelvin–Voigt type. We establish the existence of a unique weak solution of the interactive system of partial differential equations by considering an appropriate variational formulation. Uniform stability of the energy associated with the model is shown under the assumption that the potential plate energy is dominated by the dissipation induced by the viscosity of the fluid. The retention of the physical parameters in the problem is an a priori requirement in this physical condition.      

Bifurcation of solutions of an impulsive boundary-value problem

We establish constructive conditions for the bifurcation of solutions and construct an iteration procedure for finding solutions of a linear Noether boundary-value problem for a system of ordinary differential equations with pulse action in the critical case. We obtain an estimate for the range of values of a small parameter for which the iteration procedure converges. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 1, pp. 21–31, January–March, 2007.     

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.     

Domain of convergence of an iterative procedure for an autonomous boundary-value problem

We find an estimate for the range of values of a small parameter for which the convergence of an iterative procedure for the construction of solutions of an autonomous weakly nonlinear Noether boundary-value problem for a system of ordinary differential equations in the critical case is preserved. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 3, pp. 416–432, July–September, 2006.     

On Periodic Solutions of a System of Nonlinear Functional Partial Differential Equations

We obtain sufficient conditions for the existence of periodic solutions of a system of nonlinear functional partial differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 154–158, April–June, 2005.     

Construction of an asymptotic solution of the Cauchy problem for a degenerate linear system in the singular case

We propose an algorithm for the construction of an asymptotic solution of the Cauchy problem for a singularly perturbed linear system of differential equations with degenerate matrix of coefficients of derivatives in the case where the limit pencil of matrices is singular. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 271–288, April–June, 2008.     

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.     

The anisotropic and angularly inhomogeneous elastic wedge under a monomial load distribution

In this study, the generally anisotropic and angularly inhomogeneous wedge under a monomial type of distributed loading of order n of, the radial coordinate r at its external faces is considered. At first, using variable separable relations in the equilibrium equations, the strain–stress relations and the strain compatibility equation, a differential system of equations, is constructed. Decoupling this system, an ordinary differential equation is derived and the stress and displacement fields may be determined. The proposed procedure is also applied to the elastostatic problem of an isotropic and angularly inhomogeneous wedge. The special cases of loading of order n=−1 and n=−2, where the self-similarity approach is not valid, are examined and the stress and displacements fields are derived. Applications are presented for the cases of an angularly inhomogeneous wedge and in the case of a bi-material isotropic wedge.