The stability of a specific class of mechanical systems

A special class of mechanical systems is considered, the linearized equations of which either belong to the class of time-varying systems, reducible to stationary systems using constructive Lyapunov transformations or to systems close to these. A method of decomposing of the matrices of a system, which differs from the traditional method, is proposed for investigating of the stability of motion. It is shown that the conclusions concerning the stability are more complete in the case of this decomposition of the system matrix. A number of problems on the stability of motion of various mechanical systems is considered as examples.     

Stability of equilibrium points of demand-inventory model with stock-dependent demand

A model of demand and inventory of a product in one echelon of supply chain is considered. The model is formulated as a system of difference equations, in which every equilibrium point is nonhyperbolic. A positive invariant set of the system is constructed. An analysis of properties of equilibrium points of the system is based on the Lyapunov method or reducing it to the family of systems of difference equations with hyperbolic equilibrium points.     

Default Times in a Continuous Time Markov Chain Economy

Abstract

A continuous time financial market is considered where randomness is modelled by a finite state Markov chain. Using the chain, a stochastic discount factor is defined. The probability distributions of default times are shown to be given by solutions of a system of coupled partial differential equations.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

Synthesis of a control in a non-linear dynamical system based on decomposition

A non-linear controllable dynamical system with many degrees of freedom, described by Lagrange equations of the second kind, is considered. Geometric constraints are imposed on the magnitudes of the controls. It is assumed that, in the equations of motion, the kinetic energy matrix is close to a certain constant diagonal matrix. It is possible, for example, to reduce the equations of motion of robots, the drives of which have large gear ratios, to a system of this kind. A problem is formulated on the transfer of a system in a finite time from a specified initial state to a final state with zero velocities. The method of decomposition [1] is used to construct the equations. Sufficient conditions are found subject to which the maximum values of the non-linear terms in the equations of motion do not exceed the permissible magnitudes of the controls. In this case, non-linearities are treated as limited perturbations and the system is decomposed into independent, linear, second-order subsystems. A feedback control is specified for these subsystems which guarantees that each of them is brought into the final state for any permissible perturbations. The control has a simple structure. Applications of the proposed approach to problems in the control of manipulating robots are considered.     

Analysis of non-linear dynamic systems excited by intense non-white noise

A method of finding the stationary moments of the solution of non-linear stochastic equations with additive Gaussian random action is proposed, based on the use of matrix continued fractions. The method imposes no a priori limitations on the intensity and correlation time of the noise. Two methods of constructing such fractions are considered, namely, based on a chain of equations for the combined moments or a chain of equations for the combined cumulants of the solution and a random force.     

Analysis of a Prey-predator Model with Disease in Prey

In this paper, a system of reaction-diffusion equations arising in ecoepidemiological systems is investigated. The equations model a situation in which a predator species and a prey species inhabit the same bounded region and the predator only eats the prey with transmissible diseases. Local stability of the constant positive solution is considered. A number of existence and non-existence results about the nonconstant steady states of a reaction diffusion system are given. It is proved that if the diffusion coefficient of the prey with disease is treated as a bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steadystate solution under some conditions.     

The use of splitting of the equations of motion in the numerical solution of problems of the dynamics of elastic systems

Linear elastic systems with a finite number of degrees of freedom, the initial equations of motion of which are constructed using the finite element method or other discretization methods, are considered. Since, in applied dynamics problems, the motions are usually investigated in a frequency range with an upper bound, the degrees of freedom of the initial system of equations are split into dynamic and quasi-dynamic degrees. Finally, the initial system of equations is split into a small number of differential equations for the dynamic degrees of freedom and into a system of algebraic equations for determining the quasi-static displacements, represented in the form of a matrix series. The number of terms of the series taken into account depends on the accuracy required.     

A numerical method for a system of equations with a small parameter and a point source on an infinite interval

A method for solving a boundary-value problem on an infinite interval is considered for a linear system of second-order ordinary differential equations with a small parameter at the highest derivatives and a point source. The question is addressed of reduction of this problem to a finite interval. A mesh, condensing in the boundary layer, is used for numerical solution of a system of singularly perturbed equations on a finite interval.     

Method of boundary integral equations as applied to the numerical solution of the three-dimensional Dirichlet problem for the laplace equation in a piecewise homogeneous medium

A Dirichlet problem is considered in a three-dimensional domain filled with a piecewise homogeneous medium. The uniqueness of its solution is proved. A system of Fredholm boundary integral equations of the second kind is constructed using the method of surface potentials, and a system of boundary integral equations of the first kind is derived directly from Green’s identity. A technique for the numerical solution of integral equations is proposed, and results of numerical experiments are presented.     

Group Classification and Generalized Conditional Symmetry Reduction of the Nonlinear Diffusion–Convection Equation with a Nonlinear Source

The generalized conditional symmetry method, which can be considered a generalization of the conditional symmetry method, is used to study the nonlinear diffusion–convection equations with a nonlinear source. In particular, exponential and power law diffusivities are examined and we obtain mathematical forms of the convective term and the source term, which permit the generalized conditional symmetry reductions. A number of examples are considered and some exact solutions are constructed via the compatibility of the generalized conditional symmetry and the considered equation.     

Global existence in a food chain model consisting of two competitive preys,one predator and chemotaxis

A model for the spatio-temporal evolution of three biological species in a food chain model consisting of two competitive preys and one predator with intra-specific competition is considered. Besides diffusing, the predator species moves toward higher concentrations of a chemical substance produced by the prey. The prey, in turn, moves away from high concentrations of a substance secreted by the predators. The resulting reaction–diffusion system consists of three parabolic equations along with three elliptic equations describing the diffusion of the chemical substances. The local existence of nonnegative solutions is proved. Then uniform estimates in Lebesgue spaces are provided. These estimates lead to boundedness and global well-posedness for the system. Numerical simulations are presented and discussed.     

Solving a parabolic PDE with nonlocal boundary conditions using the Sinc method

In this paper, the problem of solving the parabolic partial differential equations subject to given initial and nonlocal boundary conditions is considered. We change the problem to a system of Volterra integral equations of convolution type. By using Sinc-collocation method, the resulting integral equations are replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the condition number of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. Some examples are considered to illustrate the ability of this method.     

Investigation of a powerful analytical method into natural convection boundary layer flow

The natural convection boundary layer flow modeled by a system of nonlinear differential equations is considered. By means of similarity transformation, the non-linear partial differential equations are reduced to a system of two coupled ordinary differential equations. The series solutions of coupled system of equations are constructed for velocity and temperature using homotopy analysis method (HAM). Convergence of the obtained series solution is discussed. Finally some figures are illustrated to show the accuracy of the applied method and assessment of various prandtl numbers on the temperature and the velocity is undertaken.     

The analytical solution of two interesting hyperbolic problems as a test case for a finite volume method with a new grid refinement technique

A finite volume method with grid adaption is applied to two hyperbolic problems: the ultra-relativistic Euler equations, and a scalar conservation law. Both problems are considered in two space dimensions and share the common feature of moving shock waves. In contrast to the classical Euler equations, the derivation of appropriate initial conditions for the ultra-relativistic Euler equations is a non-trivial problem that is solved using one-dimensional shock conditions and the Lorentz invariance of the system. The discretization of both problems is based on a finite volume method of second order in both space and time on a triangular grid. We introduce a variant of the min-mod limiter that avoids unphysical states for the Euler system. The grid is adapted during the integration process. The frequency of grid adaption is controlled automatically in order to guarantee a fine resolution of the moving shock fronts. We introduce the concept of “width refinement” which enlarges the width of strongly refined regions around the shock fronts; the optimal width is found by a numerical study. As a result we are able to improve efficiency by decreasing the number of adaption steps. The performance of the finite volume scheme is compared with several lower order methods.     

Method of perturbations in vector electromagnetic problems of diffraction by a wedge

A vector electromagnetic problem of diffraction by a wedge-shaped region is reduced to a system of coupled functional equations by using Sommerfeld integrals. This system of functional equations is solved by the perturbation method, and the convergence of the related series is analyzed. The system of functional equations is further reduced to linear equations with contracting operators, and the solution is represented in the form of a Neumann series. Reduction to a system with compact operators is also considered. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 138–156. Translated by M. A. Lyalinov.     

A control problem under incomplete information for a linear stochastic differential equation

A problem of guaranteed closed-loop control under incomplete information is considered for a linear stochastic differential equation (SDE) from the viewpoint of the method of open-loop control packages worked out earlier for the guidance of a linear control system of ordinary differential equations (ODEs) to a convex target set. The problem consists in designing a deterministic open-loop control providing (irrespective of a realized initial state from a given finite set) prescribed properties of the solution (being a random process) at a terminal point in time. It is assumed that a linear signal on some number of realizations is observed. By the equations of the method of moments, the problem for the SDE is reduced to an equivalent problem for systems of ODEs describing the mathematical expectation and covariance matrix of the original process. Solvability conditions for the problems in question are written.     

A System of Poisson Equations for a Nonconstant Varadhan Functional on a Finite State Space

Given a discrete-time Markov chain with finite state space and a stationary transition matrix, a system of "local" Poisson equations characterizing the (exponential) Varadhan's functional J(·) is given. The main results, which are derived for an arbitrary transition structure so that J(·) may be nonconstant, are as follows: (i) Any solution to the local Poisson equations immediately renders Varadhan's functional, and (ii) a solution of the system always exist. The proof of this latter result is constructive and suggests a method to solve the local Poisson equations.     

A steplike contrast structure in a singularly perturbed system of elliptic equations

A singularly perturbed boundary value problem for a system of elliptic equations in a two-dimensional region is considered. The asymptotics and existence of a solution with an internal transition layer are studied. The asymptotics is justified by the method of differential inequalities.     

Three-Dimensional Maneuvering and Control of Flexible Robots

This paper develops a general approach to the three-dimensional maneuver and vibration control of a robot in the form of a chain of flexible links. The equations for the rigid-body maneuvering motions are derived by means of Lagrange equations in terms of quasi-coordinates and the equations for the elastic deformations by means of ordinary Lagrange equations. The equations of motion are derived for the full system simultaneously, using recursive equations to relate the motions of a given link to the motions of the preceding links in the chain. The maneuver is carried out by means of joint torques and the vibration is suppressed by means of point actuators dispersed throughout the links. The controls are designed by the Liapunov direct method. A numerical example demonstrates the theoretical developments.