Lipschitz quasistability of impulsive differential equations

In the present paper, sufficient conditions for uniform Lipschitz stability of an arbitrary solution of an impulsive system of differential equations with unfixed moments of impulse effect are obtained. The results obtained are used for the investigation of the uniform Lipschitz stability of a given solution of a linear impulsive system of differential equations     

Investigation of the lipschitz stability via limiting equations

This work is dedicated to an effective method for investigation of the stability of solutions of ordinary differential equations. The notions of strong Lipschitz stability and strong uniform Lipschitz stability are introduced. Two theorems are proved. The first contains sufficient conditions under which the strong Lipschitz stability of the zero solution of each of the respective limiting equations implies stability of the zero solution of the initial equation. In the second theorem sufficient conditions are given under which from the uniform Lipschitz stability of the zero solution of the initial equation there follows uniform Lipschitz stability of the zero solution of each one of the respective limiting equations     

Stability of impulsive delay differential equations with impulses at variable times

This paper considers the stability of the zero solution of impulsive delay differential equations with impulses at variable times. By means of Lyapunov functions and the Razumikhin technique, some sufficient conditions of uniform stability and uniform asymptotic stability for the delay differential equation with impulses at variable times are obtained.     

具拟线性扩散系数的脉冲中立型抛物系统的(强)振动性

研究一类具拟线性扩散系数的脉冲中立型抛物偏微分系统解的(强)振动性, 直接利用振动的定义、Green公式和Newmann边值条件将这类脉冲中立型抛物系统的振动问题转化为脉冲中立型微分不等式不存在最终正解的问题, 并利用最终正解的定义和脉冲中立型微分不等式, 获得了该类系统(强)振动的充分判据. 所得结果充分反映了脉冲和时滞在振动中的影响作用.     

一类非线性脉冲中立型时滞抛物方程组的振动准则

本文研究一类非线性脉冲中立型时滞抛物方程组的振动性,利用一阶脉冲中立型微分不等式,获得了该类方程组在两类不同边界条件下所有解振动的若干充分条件。所得结果充分反映了脉冲和时滞在振动中的影响作用。     

B空间中无限时滞随机泛函微分方程解的存在唯一性(英文)

本文研究抽象空间B中无限时滞随机泛函微分方程解的存在唯一性,在弱化的线性增长条件和一致Lipschitz条件下,得到无限时滞随机泛函微分方程在区间[0,∞)上存在唯一解,进而,得到近似解与精确解之间的误差估计。     

一类固定时刻脉冲微分系统的Φ-变差稳定性

本文借助Φ-有界变差函数理论,以及一类不连续系统的Φ-有界变差解的稳定性理论,讨论了一类固定时刻脉冲微分系统的Φ-变差稳定性,给出了该类脉冲微分系统的Φ-变差稳定、Φ-变差吸引以及渐近变差稳定的定义,建立了该类脉冲微分系统Φ-有界变差解的Φ-变差稳定性和渐近Φ-变差稳定性的Lyapunov型定理.     

On the global stability of sets for impulsive differential systems by Lyapunov's direct method

In the present paper the question of global stability of sets of sufficiently general type with respect to systems of impulsive differential equations is considered

It is proved that the existence of piecewise continuous functions of the type of Lyapunov's functions with certain properties is a sufficient condition for various types of global asymptotic stability     

一类具有脉冲的向量型时滞微分系统的振动性

研究了一类向量型时滞脉冲微分方程系统的振动性问题,通过作变最变换,将常系数脉冲微分方程系统变为变系数非脉冲微分方程系统,得到了模型非振动解的渐近性态和方程任意解振动的充分条件,利用留数理论,得到了方程广义振动和广义非振动的充分条件。     

Convergence of schemes for stochastic differential equations

In many applications of stochastic calculus, like stochastic dynamical systems, stochastic differential equations are involved, the coefficients of which are not globally, but only locally Lipschitz. For instance, in order to study technics using one trajectory of a process defined by differential equations of oscillators associated to structures submitted to a white noise excitation, such as the random decrement, one need to simulate a trajectory for such a process. Different schemes are proposed to numerically solve such stochastic differential equations: Euler, Milshtein or Newmark schemes for example. In this paper, the almost sure convergence of some of the most important schemes is studied under locally Lipschitz assumptions and a speed of convergence is established.     

Nonautonomous attractors of switching systems

Switching systems formed by switching between a finite number of autonomous ordinary differential equations are formulated as abstract nonautonomous dynamical systems driven by the shift operator on the space of switching controls. A metric is proposed for such switching controls and the space of switching controls with a given positive dwell time is shown to be compact. This allows known results on pullback attractors for nonautonomous dynamical systems to be applied to switching systems including conditions which ensure that a pullback attractor exists and is also a forward attractor. In particular, this provides a conceptual framework for investigating the asymptotic behaviour of switching systems beyond the usual case in which the asymptotic stability of the zero solution under switching is considered.     

一类非线性多时滞脉冲抛物型方程解的振动性质

本文研究一类非线性多时滞脉冲抛物型方程在齐次Dirichlet和Neumann边界条件下解的振动性质.利用分析技巧,给出一个脉冲微分不等式无最终正解(或最终负解)的条件.然后,利用平均法,将该方程解振动性问题转化为相应脉冲时滞微分不等式有无最终正解(或最终负解)问题,进而在两类齐次边界条件下获得了判别该类方程解振动的充...     

Dynamic coupled thermoelastic problems in micropolar theory—I

General solution of the dynamic micropolar coupled thermoelastic equations has been obtained for arbitrary distribution of body forces, body couples and heat sources in an infinite body by the use of Laplace-Fourier transforms. Short time solutions have been obtained for the cases of impulsive body force, body couple and heat source acting at a point. The corresponding classical coupled thermoelastic solutions have been derived by letting the parameter α approach zero. Some numerical results have been illustrated graphically.     

Stability results for impulsive differential systems with applications to population growth models

This paper establishes some stability criteria for impulsive differential systems. It is shown that impulses do contribute to yield stability properties even when the corresponding differential system without impulses does not enjoy any stability behavior. As an application, these results are applied to some population growth models     

Linear dynamical analysis of fractionally damped beams and rods

The aim of this study is to develop a general model for beams and rods with fractional derivatives. Fractional time derivatives can represent the damping term in dynamical models of continuous systems. Linear differential operators with spatial derivatives make it possible to generalize a wide range of problems. The method of multiple scales is directly applied to equations of motion. For the approximate solution, the amplitude and phase modulation equations are obtained in terms of the operators. Stability boundaries are derived from the solvability condition. It is shown that a fractional derivative influences the stability boundaries, natural frequencies, and amplitudes of vibrations. The solution procedure may be applied to many problems with linear vibrations of continuous systems.     

The initial motions for holonomic and nonholonomic mechanical systems

Whittaker first put forward a new approach, called the initial motions, to solve the differential equations of motion aimed at holonomic systems. Since most of the differential equations of motion for mechanical systems are nonlinear ordinary ones, which are difficult to find the analytic solutions. Fortunately, the concept of initial motions can manage these situations and study its subsequent motions. This work is devoted to discuss and investigate the initial motions for mechanical systems, particularly for nonholonomic systems. First, the differential equations for holonomic systems are formulated, and the formulation and solution of initial motions of the systems are proposed. Second, the differential equations of motion for nonholonomic systems are established, based on the new method of initial motions to obtain the initial values of high-order derivatives of generalized velocities, the formulation and solution of initial motions are introduced in the general nonholonomic systems and Chaplygin systems. The methods and results obtained are illustrated by a number of classical examples, both for holonomic and nonholonomic systems.     

Normed linear spaces and optimal design of discrete systems

Summary The behaviour of discrete mechanical or electrical systems under the action of disturbances shall be weighted. In a space which contains the solutions of the corresponding differential equations appropriate norms are introduced. The optimal design problem can then be interpreted as an approximation problem for the zero solution. A method for the delivering from the initial conditions is proposed.     

Some anomalies of the numerical solutions of the Euler equations

Summary The numerical solution of the unsteady Euler equations for compressible flow over a circular cylinder is obtained using standard numerical techniques. The equations, written in cylindrical coordinates, are discretized on an orthogonal grid via central differences for spatial derivatives, using a simple second order artificial viscosity form and a special treatment of the boundary conditions. Backward differences in time are employed resulting in a large system of nonlinear difference equations at each step. A direct solver (LAPACK), based on an efficient Gaussian elimination procedure for banded matrices, is used to solve the linearized system of equations. The stability of the nonunique solutions of the steady Euler equations is investigated. It is demonstrated that the symmetric solutions, with zero circulation, are not stable. For a certain Mach number range, a periodic solution is obtained where the shock oscillation persists. If a periodic circulation (within a certain frequency range) is enforced in the far field, an irregular solution emerges with unpredictable shock motions. For such a solution, the Lyapunov exponent is shown to be greater than zero, indicating the appearance of chaos.     

Exact solutions for the in-plane vibrations of rectangular Mindlin plates using Helmholtz decomposition

Deriving frequency equations for in-plane vibration of a rectangular plate with different boundary conditions and uniform thickness in the elastic range is the goal of this research. To derive frequency equations, the kinetic and potential energy for in-plane behavior initially are obtained by using the stress–strain–displacement expressions according to the theory of Mindlin plates in Cartesian coordinates by applying the Hamilton’s principle, which leads to five sets of highly coupled differential equations for the equations of motion. Replacement of Helmholtz decomposition for the coupled differential equations creates uncoupled equations of motion. The hypothesis of a harmonic solution for the uncoupled equations lead to wave equations. The general solutions for the wave equations are obtained by using the separation of variables. Finally, the application of boundary conditions yields the frequency equations for the rectangular plate. The natural frequencies are compared and validated by finite element analysis and previously reported results.