Nonlocal problem with integral conditions for a system of hyperbolic equations in characteristic rectangle

We consider a nonlocal problem with integral conditions for a system of hyperbolic equations in rectangular domain. We investigate the questions of existence of unique classical solution to the problemunder consideration and approaches of its construction. Sufficient conditions of unique solvability to the investigated problem are established in the terms of initial data. The nonlocal problem with integral conditions is reduced to an equivalent problem consisting of the Goursat problem for the system of hyperbolic equations with functional parameters and functional relations. We propose algorithms for finding a solution to the equivalent problem with functional parameters on the characteristics and prove their convergence. We also obtain the conditions of unique solvability to the auxiliary boundary-value problem with an integral condition for the system of ordinary differential equations. As an example, we consider the nonlocal boundary-value problem with integral conditions for a two-dimensional system of hyperbolic equations.     

On the uniqueness of a solution of the fourier problem for a system of sobolev-gal’pern type

We establish conditions for the uniqueness of a solution of the problem for a system of equations unresolved with respect to the time derivative without initial conditions in a noncylindrical domain. The system considered, in particular, contains pseudoparabolic equations.     

Isochronicity of centers in a switching Bautin system

In this paper isochronicity of centers is discussed for a class of discontinuous differential system, simply called switching system. We give some sufficient conditions for the system to have a regular isochronous center at the origin and, on the other hand, construct a switching system with an irregular isochronous center at the origin. We give a computation method for periods of periodic orbits near the center and use the method to discuss a switching Bautin system for center conditions and isochronous center conditions. We further find all of those systems which have an irregular isochronous center.     

Charpit System of Partial Differential Equations with Algebraic Constraints

In this paper the Charpit system of partial differential equations with algebraic constraints is considered. So, first the compatibility conditions of a system of algebraic equations and also of the Charpit system of partial differential equations are separately considered. For the combined system of equations of both types sufficient conditions for the existence of a solution are found. They lead to an algorithm for reducing the combined system to a Charpit system of partial differential equations of dimension less than the initial system and without algebraic constraints. Moreover, it is proved that this system identically satisfies the compatibility conditions if so does the initial system.     

Battle-outcome prediction for an extended system of Lanchester-type differential equations

Battle-outcome-prediction conditions are given for an extended system of Lanchester-type differential equations for two different types of battle-termination conditions: (a) fixed-force-level-breakpoint battles, and (b) fixed-force-ratio-breakpoint battles. Necessary and sufficient conditions for predicting battle outcome are given in the former case for a fight to the finish, while sufficient conditions are given in the latter case. The former results are equivalent to those for the problem of classical analysis of determining (explicitly as a function of the initial conditions) the occurrence of a zero point for the solution to this extended system, although such results as given here have not appeared previously for nonoscillatory (in the strict sense) solutions.     

具有三个成长阶段的单种群时滞模型的永久持续生存和全局稳定性

该文研究具有分布时滞和三个成长阶段的单种群模型,得到了系统永久持续生存的充分条件;同时通过构造Lyapunov 函数得到了系统全局渐近稳定的充分条件;最后建立具体模型说明所得结果的可行性.     

一类五次多项式系统无穷远点等时中心条件与极限环分支

研究一类五次系统无穷远点的中心、拟等时中心条件与极限环分支问题.首先通过同胚变换将系统无穷远点转化成原点,然后求出该原点的前8个奇点量,从而导出无穷远点成为中心和最高阶细焦点的条件,在此基础上给出了五次多项式系统在无穷远点分支出8个极限环的实例.同时通过一种最新算法求出无穷远点为中心时的周期常数,得到了拟等时中心的必要条件,并利用一些有效途径一一证明了条件的充分性.     

On global existence for the Vlasov-Poisson system in a half space

We prove global existence for the Vlasov-Poisson system in a half space in the plasma physics case assuming homogeneous Dirichlet boundary conditions for the electric potential. Our proof uses an adaptation of Pfaffelmoser's method and it provides also a different proof of the previously known global existence results for the Vlasov-Poisson system in a half space with Neumann boundary conditions for the electric potential. We also obtain a large class of one-dimensional stationary solutions for the problem with Neumann boundary conditions.     

Solving some problems for systems of linear ordinary differential equations with redundant conditions

Numerical methods are proposed for solving some problems for a system of linear ordinary differential equations in which the basic conditions (which are generally nonlocal ones specified by a Stieltjes integral) are supplemented with redundant (possibly nonlocal) conditions. The system of equations is considered on a finite or infinite interval. The problem of solving the inhomogeneous system of equations and a nonlinear eigenvalue problem are considered. Additionally, the special case of a self-adjoint eigenvalue problem for a Hamiltonian system is addressed. In the general case, these problems have no solutions. A principle for constructing an auxiliary system that replaces the original one and is normally consistent with all specified conditions is proposed. For each problem, a numerical method for solving the corresponding auxiliary problem is described. The method is numerically stable if so is the constructed auxiliary problem.     

Stabilisation frontière de problèmes de Ventcel

We study Ventcel's problems for the wave equation and the isotropic linear elastodynamic system. The boundary observability and the exact controlabillity were established in [3]. We prove the energy decay for the elastodynamic system with stationary Ventcel's conditions. We also give a boundary feedback leading to arbitrarily large energy decay rates for the elastodynamic system with evolutive Ventcel's conditions. A spectral study proves, finally, that the “natural” feedback is not sufficient for the exponential decay in the case of the wave equation with Ventcel's conditions.     

Solving a system of linear ordinary differential equations with redundant conditions

A system of linear ordinary differential equations is examined under the assumption that, in addition to the basic conditions, which in general are nonlocal and are specified by a Stieltjes integral, certain redundant (and possibly also nonlocal) conditions are imposed. Generically, such a problem has no solution. A principle for constructing an auxiliary system is proposed. This system replaces the original one and is normally consistent with all the conditions prescribed. A method for solving this auxiliary problem is analyzed. The method is numerically stable if the auxiliary problem is numerically stable.     

On the singular nonlinear self-adjoint eigenvalue problem for differential algebraic systems of equations

The general nonlinear self-adjoint eigenvalue problem for a differential algebraic system of equations on a half-line is examined. The boundary conditions are chosen so that the solution to this system is bounded at infinity. Under certain assumptions, the original problem can be reduced to a self-adjoint system of differential equations. After certain transformations, this system, combined with the boundary conditions, forms a nonlinear self-adjoint eigenvalue problem. Requirements for the appropriate boundary conditions are clarified. Under the additional assumption that the initial data are monotone functions of the spectral parameter, a method is proposed for calculating the number of eigenvalues of the original problem that lie on a prescribed interval of this parameter.     

一类Lotka-Volterra竞争生态系统的周期解

讨论一类特殊的n种群LotkaVolterra竞争生态系统的周期解,应用拓扑度理论中的延拓定理和Lyapunov泛函方法,得到了这类系统周期解的存在性和全局渐近稳定性的充分判据.     

一类具有阶段结构和时滞的捕食系统的持续生存和稳定性

我们提出和研究一类带有阶段结构和时滞的捕食模型,得到了种群持续生存的充分条件.研究了阶段结构和时滞对系统稳定性的影响,获得了系统发生Hopf分支和稳定性的条件以及轨道渐进稳定的周期解的存在性.     

Bifurcation of limit cycles at the equator

This paper studies center conditions and bifurcation of limit cycles from the equator for a class of polynomial differential system of order seven. By converting real planar system into complex system, we established the relation of focal values of a real system with singular point quantities of its concomitant system, and the recursion formula for the computation of singular point quantities of a complex system at the infinity. Therefore, the first 14 singular point quantities of a complex system at the infinity are deduced by using computer algebra system Mathematica. What’s more, the conditions for the infinity of the real system to be a center or 14 degree fine focus are derived, respectively. A system of order seven that bifurcates 12 limit cycles from the infinity is constructed for the first time.     

Application of the duality theory in modeling hydraulic systems with flow regulators

We consider mathematical models of the operation of pipelines with automatic flow regulators installed on several sections. We present a system of equations and inequalities that describes the flow distribution in such pipelines. We prove that the stated system gives optimality conditions for two mutually dual convex programming problems. This enables us to obtain conditions for the unique existence of a solution to this system.     

Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems

Several results regarding the stability and the stabilization of linear impulsive positive systems under arbitrary, constant, minimum, maximum and range dwell-time are obtained. The proposed stability conditions characterize the pointwise decrease of a linear copositive Lyapunov function and are formulated in terms of finite-dimensional or semi-infinite linear programs. To be applicable to uncertain systems and to control design, a lifting approach introducing a clock-variable is then considered in order to make the conditions affine in the matrices of the system. The resulting stability and stabilization conditions are stated as infinite-dimensional linear programs for which three asymptotically exact computational methods are proposed and compared with each other on numerical examples. Similar results are then obtained for linear positive switched systems by exploiting the possibility of reformulating a switched system as an impulsive system. Some existing stability conditions are retrieved and extended to stabilization using the proposed lifting approach. Several examples are finally given for illustration.     

On Impulsive Lotka–Volterra Systems with Diffusion

We study a two-dimensional Lotka–Volterra system with diffusion and impulse action at fixed moments of time. We establish conditions for the permanence of the system. In the case where the coefficients of the system are periodic in t and independent of the space variable x, we obtain conditions for the existence and uniqueness of periodic solutions of the system.     

An inclusion principle for hereditary systems

Given two hereditary dynamic systems having different dimensions, the conditions are provided under which a part of the motion of the larger system is reproduced by the smaller system, that is, the larger system “includes” the smaller one. The conditions for inclusion are useful in applying the concept of vector Liapunov functions to stability analysis of systems composed of overlapping subsystems. By expanding the systems into a larger space the overlapping subsystems appear as disjoint and standard methods can be used to conclude stability of the expanded system. Under the inclusion conditions, stability of the expansion implies stability of the original system. An example is provided to show stability where the standard disjoint decompositions fail.