Krylov-Bogolyubov Method Research Articles

Studies on invariant measures of fractional stochastic delay Ginzburg-Landau equations on Rn.

This paper is concerned with invariant measures of fractional stochastic delay Ginzburg-Landau equations on the entire space $ \mathbb{R}^n $. We first derive the uniform estimates and the mean-square uniform smallness of the tails of solutions in corresponding space. Then we deduce the weak compactness of a set of probability distributions of the solutions applying the Ascoli-Arzel$ \grave{a} $. We finally prove the existence of invariant measures by applying Krylov-Bogolyubov's method.

Calculation of technological modes of operationof vertical vibrating mill

Purpose. To build the calculation scheme of interaction between grinding chamber and process load of vertical vibrating mill. To obtain energy dependences of the grinding process as a function of the mass ratio of the working body - load, considering the used type of drive. To analyze the obtained dependences in order to recommend rational technological modes of the grinding process. The methods. Scientific research in this direction is analyzed. The methodology of solving the set problem on the basis of recording the differential equations of motion of the camera - technological load system is proposed. The Krylov-Bogolyubov method is used for their solution. Two types of drives are considered - inertial directional and rigid eccentric. The comparison is made by the interaction energy of the process load and grinding chamber at the moments of their impact interaction. Findings. To intensify the process of material grinding in vertical vibrating mill with inertial vibrating exciter it is necessary to increase the mass of grinding chamber. However, the material intensity of the system increases. In this case mills with rigid eccentric vibrating exciter have significant advantages. It is shown that the motion parameters of the process load and the energy of its interaction with the chamber do not depend on the ratio of their masses. The originality. The refined dynamic model is developed and differential equations of motion of the system grinding chamber - technological load of vertical vibrating mill are written down. New energy dependences have been obtained, which allowed to calculate the parameters of realization of vibration-impact interaction of elements of this system. Practical implementation. Application of the research results will allow to create rational design schemes of vertical vibrating mills, as well as to specify technological modes of grinding in the currently existing processes with the use of mills of this type.

Evolution systems of probability measures for nonautonomous Klein-Gordon Itô equations on [formula omitted

Invariant probability measures of time-homogeneous transition semigroups for autonomous stochastic ODEs/PDEs/lattice systems have been widely discussed in the literature. In this work we investigate evolution systems of probability measures of time inhomogeneous transition operators for a class of nonautonomous Klein-Gordon Itô equation defined on an integer set ZN. The existence, periodicity, pointwise ergodicity and tightness of evolution systems of probability measures are established in an infinite-dimensional phase space by using a generalized Krylov-Bogolyubov method developed by Da Prato and Röckner [21]. The asymptotic stability of every periodic evolution system of probability measures with respect to the noise intensity is discussed by using a recent result on this topic established by Wang, Caraballo and Tuan [49]. The uniqueness, global exponential mixing, forward strong mixing and backward strong mixing of every evolution system of probability measures are also discussed under some globally monotone conditions. The idea of uniform tail-estimate due to Wang [41] is used to overcome several difficulties caused by the lack of compactness in infinite lattices. It seems that this is the first time to study evolution systems of probability measures for nonautonomous hyperbolic stochastic equations.

Ergodicity for three-dimensional stochastic Navier–Stokes equations with Markovian switching

Asymptotic behavior of the three-dimensional stochastic Navier-Stokes equations with Markovian switching for additive noises is studied for an incompressible fluid flow in a bounded domain. The objective of this article is to prove the existence and identification of a stationary measure for such stochastic systems. To achieve this goal, a family of regularized equations is introduced in order to investigate the asymptotic behavior of their solutions. The existence of an ergodic measure for the regularized system is established by the Krylov-Bogolyubov method. The existence and identification of a stationary measure corresponding to the original system are obtained as a sequential limit point of the family of the ergodic measures that correspond to the regularized systems. The main result of this work is that under mild integrability conditions on the initial data and external force, there exists a stationary measure for the solution of the three-dimensional stochastic Navier-Stokes equations with Markovian switching.

Asymptotic stability of evolution systems of probability measures for nonautonomous stochastic systems: Theoretical results and applications

The limiting stability of invariant probability measures of time homogeneous transition semigroups for autonomous stochastic systems has been extensively discussed in the literature. In this paper we initially initiate a program to study the asymptotic stability of evolution systems of probability measures of time inhomogeneous transition operators for nonautonomous stochastic systems. Two general theoretical results on this topic are established in a Polish space by establishing some sufficient conditions which can be verified in applications. Our abstract results are applied to a stochastic lattice reaction-diffusion equation driven by a time-dependent nonlinear noise. A time-average argument and an extended Krylov-Bogolyubov method due to Da Prato and Röckner [Seminar on stochastic analysis, random fields and applications V, Birkhäuser, Basel, 2008] are employed to prove the existence of evolution systems of probability measures. A mild condition on the time-dependent diffusion function is used to prove that the limit of every evolution system of probability measures must be an evolution system of probability measures of the limiting equation. The theoretical results are expected to be applied to various stochastic lattice systems/ODEs/PDEs in the future.

Periodic measures of impulsive stochastic neural networks lattice systems with delays

This paper is concerned with the periodic measures of a class of periodic stochastic neural networks lattice models with delays and nonlinear impulses. First, by employing the idea of uniform estimates on the tails of the solutions, the technique of diadic division, and generalized Ascoli–Arzela theorem, we prove the tightness of a family of distributions of the segment solutions of the lattice systems. Then, the existence of periodic measures is established by using the Krylov–Bogolyubov method.

Non-Autonomous Hybrid Stochastic Systems with Delays

The aim of this paper is to study the dynamical behavior of non-autonomous stochastic hybrid systems with delays. By general Krylov-Bogolyubov's method, we first obtain the sufficient conditions for the existence of an evolution system of measures of the non-autonomous stochastic system and also give some easily verifiable conditions. We then prove a sufficient condition for convergence of evolution systems of measures as the delay approaches zero. As an application of the abstract theory, we first prove the existence of evolution systems of measures for stochastic system with time-vary delays, which comes from feedback control problem based on discrete-time state observations. Furthermore, when observation interval goes to zero, we show every limit point of a sequence of evolution system of measures of the non-autonomous stochastic system must be a evolution system of measures of the limiting system.

Simplification of weakly nonlinear systems and analysis of cardiac activity using them

<p style='text-indent:20px;'>The paper deals with the transformation of a weakly nonlinear system of differential equations in a special form into a simplified form and its relation to the normal form and averaging. An original method of simplification is proposed, that is, a way to determine the coefficients of a given nonlinear system in order to simplify it. We call this established method the degree equalization method, it does not require integration and is simpler and more efficient than the classical Krylov-Bogolyubov method of normalization. The method is illustrated with several examples and provides an application to the analysis of cardiac activity modelled using van der Pol equation.</p>

Invariant measures of stochastic Schrödinger delay lattice systems

In this paper, we investigate stochastic Schrödinger lattice systems with time delay, whose drift and diffusion coefficientsare locally Lipschitz continuous. Firstly, some uniform estimates of solutions areestablished which include higher-order moment estimates and uniform tail-ends estimates. Then the tightness of a family ofprobability distributions of solutions in $C([-\\rho,0];l^2)$ is proved by the Arzel${\\rm~\\grave{a}}$-Ascoli theorem and the technique of diadicdivision. Finally, the existence of invariant measures for the Markov semigroup of the system is provedby the Krylov-Bogolyubov method.

Three-dimensional integro-differential model of unstationary hydrodynamic process in a liquid-metal pool of underwater arc welding

Wet underwater arc welding is now widely used. At the same time, obtaining high-quality welds with this welding method is an urgent scientific and technical problem due to their saturation with hydrogen and oxygen and the formation of pores. One of the promising directions for solving this issue is the use of an external electromagnetic effect on the liquid metal in the weld pool in order to control the movements of the molten metal flows to improve the degassing processes of welded joints. It is possible to estimate the parameters and efficiency of external electromagnetic influence by means of mathematical modeling of related electromagnetic, hydrodynamic and thermal processes occurring in the welding installation. The article proposes a three-dimensional integro-differential model of a non-stationary hydrodynamic process occurring in the liquid metal of a weld pool in an underwater arc welding system with an external electromagnetic effect. For the equations of hydrodynamics boundary value problems are formed, which, using potential theory, are reduced to a system of integro-differential equations for the vorticity function in the volume of a liquid conductor and a simple vector layer on its surface. For a numerical solution, the resulting system of integro-differential equations is approximated by an algebraic system according to the Krylov-Bogolyubov method. This system of equations makes it possible to determine the velocity field in the liquid metal of the weld pool for any welding modes.

Physical and mathematical simulation of bridge segment oscillations

On the basis of known mathematical models describing vibrations in the gas flow of a bluff body with one degree of freedom, a model of vibrations of a body with two degrees of freedom is obtained. The Krylov–Bogolyubov method is applied. Equations for slowly varying amplitudes and phase shift of vibrations are obtained. It turned out that the differential equations written for the squares of dimensionless amplitudes of translational and rotational vibrations coincide with the well-known Lotka-Volterra equations describing competition between two species of animals that eat the same food. In a wind tunnel, the change of modes of vibrations of the bridge segment predicted by a mathematical model is studied.

Конкуренция режимов колебаний плохо обтекаемого тела в воздушном потоке

Based on well-known mathematical models describing vibrations in the gas flow of a bluff body with one degree of freedom, a model of vibrations of a body with two degrees of freedom is proposed. The equations of transverse translational vibrations and rotational vibrations of an elastically fixed body around an axis perpendicular to the velocity vector of the incoming flow are obtained. Using the Krylov-Bogolyubov method in the first approximation, the equations are reduced to equations for slowly varying amplitudes and frequencies of vibrations. It turned out that the differential equations written for the squares of dimensionless amplitudes of translational and rotational vibrations coincide with the well-known Lotka-Volterra equations describing competition between two species of animals that eat the same food. The coefficients of the equations depend on the velocity of the incoming flow. The model is verified in the experiments in the wind tunnel.

Periodic measures of reaction-diffusion lattice systems driven by superlinear noise

<abstract><p>The periodic measures are investigated for a class of reaction-diffusion lattice systems driven by superlinear noise defined on $ \mathbb Z^k $. The existence of periodic measures for the lattice systems is established in $ l^2 $ by Krylov-Bogolyubov's method. The idea of uniform estimates on the tails of solutions is employed to establish the tightness of a family of distribution laws of the solutions.</p></abstract>

Equivalent Surface Impedance of an Infinite Periodic Array of Slot Impedance Loads Based on an Equilateral Triangle Cross-Section Cavity with a Dielectric Coating

In this paper, we consider a two-dimensional problem of determining the equivalent surface impedance of an infinite array of slot impedance loads with a dielectric layer. Each element of the array contains three regions: the first region occupies the entire half-space above the interface and the primary field is excited in this region by a plane wave; the second region does not contain excitation sources and is a dielectric layer; and the third region also does not contain exciting sources and is bounded by the walls of the cavity with a equilateral triangle cross section. The second and third regions are connected through one or several slots in a perfectly conducting screen located at their border. A strip conductor is located in the opening of each slot. The problem was solved by the method of integral equations, for the numerical implementation of which the Krylov–Bogolyubov method was used. Numerical results are presented as the dependences of the equivalent surface impedance on the geometric dimensions of the structure at fixed values ​​of the array period, the incidence angle of the electromagnetic wave, and the thickness of the dielectric layer. A comparative analysis of the obtained dependences with the characteristics of a similar array without a dielectric layer is performed.

Asymptotic behavior of stochastic discrete wave equations with nonlinear noise and damping

In this article, we study the asymptotic behavior for a class of discrete wave equations with nonlinear noise and damping defined on a k-dimensional integer set. The well-posedness of the system is established when the nonlinear drift function and the nonlinear diffusion term are only locally Lipschitz continuous. The mean random dynamical system associated with the non-autonomous system is shown to possess a unique tempered weak pullback mean random attractor in L2(Ω,F,ℓ2×ℓ2). The existence of invariant measures for the autonomous system is also derived by using the Krylov–Bogolyubov method. The difficulty in proving the tightness of a family of distribution laws of the solutions is overcome by using the idea of uniform estimates on the tails of solutions.

Electrodynamic analysis of the director antenna by the method of integral equations with a reduction in the number of partitioning segments of passive conductors

It has been solved the problem task for the numerical electrodynamic analysis of a Yagi-antenna using a mathematical model developed with the Poklington integral equation and the Krylov-Bogolyubov method with a reduction in the number of partitioning segments of passive conductors. The obtained results demonstrate the high efficiency of the proposed model, due to the combination of high accuracy solutions and saving computational resources.

Equivalent Surface Impedance of an Infinite Array of Slot Impedance Loads Based on a Cavity Having the Cross-Section in the Form of an Equilateral Triangle

The two-dimensional problem about the excitation of an infinite array of slot impedance loads by a plane wave is considered. The aim is the determination of the equivalent surface impedance. Every array element is a slot in a perfectly conducting screen. The slot is loaded by a cavity having the cross-section in the form of an equilateral triangle. The problem is solved using the integral equation method, which is numerically realized with the use of the Krylov—Bogolyubov method. The dependences of the impedance on the slot dimensions, the cavity dimensions, and the plane wave incidence angle are calculated for the case of the H polarization of the incident wave. The dependences obtained for the considered construction of the impedance load contained in an infinite array are analyzed and simultaneously compared with the characteristics of a single impedance load.

Equivalent Surface Impedance of an Infinite Periodic Array of Slot Impedance Loads with Dielectric Coating

The problem of plane-wave excitation of an infinite array of slot impedance loads with a dielectric layer is considered in order to determine the equivalent surface impedance. The solution of the problem is found by the method of integral equations, solved numerically by the Krylov–Bogolyubov method, which reduces the integral equation to a system of linear algebraic equations. The numerical results are presented in the form of dependences of the equivalent surface impedance on the geometric dimensions of the structure at fixed values of the period of the array, the angle of incidence, and the thickness of the dielectric layer. It is shown that the equivalent surface impedance can be controlled by varying the width of the slot and the strip conductor. The dependences obtained are compared with the characteristics of an array of slot impedance loads without a dielectric layer.

Equivalent surface impedance of a slot impedance load on the basis of a cavity with an equilateral triangular cross section

The problem of plane-wave excitation of an impedance load on the basis of a slot in an infinite perfectly conducting screen closed from one side by a cavity with an equilateral triangular cross section is considered. The problem is solved by the integral equation method. The numerical solution of the integral equation is performed by the Krylov–Bogolyubov method. For an H-polarized incident wave, the impedance as a function of the slot and cavity sizes and the angle of incidence of a plane wave is calculated.

To the choice of collocation points for solving integro-differential equations of electromagnetics

The choice of collocation points for solving integro-differential equations of electromagnetics by the Krylov-Bogolyubov method is considered. Using the example of a simple diffraction problem, variants of the solution for two different systems of collocation points are presented. The first (traditional) system of collocation points is chosen so that the end points do not fall onto the boundaries of the interval. In the second system, the end points fall onto the boundaries. It is shown that both of the systems can be used for solving the problem but the use of the second system provides for a higher accuracy with fewer collocation points.