积分微分方程线性多步方法的散逸性

研究一类积分微分方程线性多步方法（p,σ）的散逸性．当积分项用复合求积公式逼近时,证明了线性多步方法是有限维散逸的．这说明该方法很好地继承了系统本身所具有的重要性质．这一结论为数值求解这一类微分方程提供了更多的选择．     

延迟动力系统线性θ-方法的散逸性

１．引言 科学与工程中的许多问题具有散逸性,即系统具有一有界吸引集,从任意初始条件出发的解经过有限时间后进入该吸引集并随后保持在里面．如 ２维的 Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程、Ｌｏｒｅｎｚ方程等许多重要系统都是散逸的．散逸性研究一直是动力系统研究中的重要课题（参见Ｔｅｍａｍ［７］）．当数值求解这些系统时,自然希望数值方法能保持系统的该重要特性．１９９４年, Ｈｕｍｐｈｒｉｅｓ和 Ｓｔｕａｒｔ［６］首次研究了 Ｒｕｎｇｅ－Ｋｕｔｔａ方法对有限维系统的散逸性．１９９７年Ｈｉｌｌ［２］研究了其无穷维散逸性…     

随机种群模型数值解的均方散逸性

讨论了随机种群模型数值解的均方散逸性,基于步长受限制和无限制的两种条件,利用补偿的和无补偿的数值方法研究了随机种群模型数值解的均方散逸性.从而得出补偿的数值算法更适合解决随机种群模型数值解的均方散逸性问题.     

非线性中立型延迟积分微分方程线性多步法的散逸性

考虑非线性中立型延迟积分微分方程数值方法的散逸性,把一类线性多步法应用到以上问题中,当积分项用复合求积公式逼近时,证明该数值方法在满足一定条件下具有散逸性.     

Cahn-Hilliard方程的高阶保能量散逸性方法

能量散逸性是物理和力学中某些微分方程一项重要的物理特性.构造精确地保持微分方程能量散逸性的数值格式对模拟具有能量散逸性的微分方程具有重要的意义.本文利用四阶平均向量场方法和傅里叶谱方法构造了Cahn-Hilliard方程高阶保能量散逸性格式.数值结果表明高阶保能量散逸性格式能很好地模拟Cahn-Hilliard方程在不同初始条件下解的行为,并且很好地保持了Cahn-Hilliard方程的能量散逸特性.     

非线性多比例延迟微分方程向后Euler方法的散逸性

本文讨论了多比例延迟微分方程的散逸性，证明了应用向后Euler方法求解多比例延迟微分方程数值解仍保持散逸性，它可视为文献[9]中相应结果的推广。     

分数Brown运动时变随机种群收获系统数值解的均方散逸性

讨论了一类带分数Brown运动时变随机种群收获系统数值解的均方散逸性.在一定条件下,利用It公式和Bellman-Gronwall-Type引理,研究了方程(1)具有均方散逸性.分别利用带补偿的倒向Euler方法和分步倒向Euler方法讨论数值解的均方散逸性存在的充分条件,并通过数值算例对所给出的结论进行了验证.     

基于补偿倒向Euler方法带分数Brown运动的随机固定资产模型数值解的均方散逸性

讨论了一类带分数Brown运动随机固定资产模型数值解的均方散逸性.在漂移系数和扩散系数满足单边Lipschitz条件和有界条件下,建立了随机固定资产模型补偿倒向Euler法数值解均方散逸性的判定准则.最后通过数值算例对结论进行了验证.     

HILBERT空间中散逸动力系统一般线性方法的散逸稳定性

１．引言 １９９４年,Ｓｔｕａｒｔ与 Ｈｕｍｐｈｒｉｅｓ［４,５］首先考察了用 Ｒｕｎｇｅ－Ｋｕｔｔａ方法求解 Ｒｍ中的散逸动力系统（２．１）－（２．２）时数值解是否继承真解具有的散逸稳定性,并表明代数稳定且不可约的 Ｒｕｎｇｅ－Ｋｕｔｔａ方法是散逸稳定的且有一有界吸引集．１９９６年,本文作者［１］把这一工作推广到了两类特殊的一般线性方法．１９９７年,Ｈｉｌｌ在［３］中证明了Ａ－稳定是单支方法散逸稳定的充要条件,在［２］中又把文［４,５］的工作推广到了 Ｈｉｌｂｅｒｔ空间中的散逸动力系统（２．１）－（…     

多比例延迟微分方程的散逸性

本文讨论多比例延迟微分方程的散逸性，给出了多比例延迟微分方程是散逸的充分条件，它可视为文献[8]中相应结果的推广。     

Dissipativity and asymptotic stability of nonlinear neutral delay integro-differential equations

This paper is concerned with the dissipativity and asymptotic stability of the theoretical solutions of a class of nonlinear neutral delay integro-differential equations (NDIDEs). We first give a generalization of the Halanay inequality which plays an important role in the study of dissipativity and stability of differential equations. Then, we apply the generalization of the Halanay inequality to NDIDEs and the dissipativity and the asymptotic stability results of the theoretical solution of NDIDEs are obtained. From a numerical point of view, it is important to study the potential of numerical methods in preserving the qualitative behavior of the analytical solutions. Therefore, the results, presented in this paper, provide the theoretical foundation for analyzing the dissipativity and the asymptotic stability of the numerical methods when they are applied to these systems.     

Dissipativity of Runge-Kutta methods for Volterra functional differential equations

This paper is concerned with the numerical dissipativity of nonlinear Volterra functional differential equations (VFDEs). We give some dissipativity results of Runge-Kutta methods when they are applied to VFDEs. These results provide unified theoretical foundation for the numerical dissipativity analysis of systems in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs), Volterra delay integro-differential equations (VDIDEs) and VFDEs of other type which appear in practice. Numerical examples are given to confirm our theoretical results.     

Dissipativity of Runge–Kutta methods for neutral delay integro-differential equations

This paper is concerned with the numerical dissipativity of a class of nonlinear neutral delay integro-differential equations. The dissipativity results are obtained for algebraically stable Runge–Kutta methods when they are applied to above problems.     

Functional differential inclusions and dynamic behaviors for memristor-based BAM neural networks with time-varying delays

In this paper, we formulate and investigate a class of memristor-based BAM neural networks with time-varying delays. Under the framework of Filippov solutions, the viability and dissipativity of solutions for functional differential inclusions and memristive BAM neural networks can be guaranteed by the matrix measure approach and generalized Halanay inequalities. Then, a new method involving the application of set-valued version of Krasnoselskii’ fixed point theorem in a cone is successfully employed to derive the existence of the positive periodic solution. The dynamic analysis in this paper utilizes the theory of set-valued maps and functional differential equations with discontinuous right-hand sides of Filippov type. The obtained results extend and improve some previous works on conventional BAM neural networks. Finally, numerical examples are given to demonstrate the theoretical results via computer simulations.     

On dissipativity and stabilization of time-delay stochastic systems with switching control

This paper investigates dissipativity and stabilization problems for a class of stochastic systems with time delays. Several sufficient conditions on exponential dissipativity with respect to quadratic supply rates are derived via a Lyapunov functional approach. Passive and non-expansive property of time delay stochastic systems is also characterized. In addition, a switching controller has been developed for the time delay stochastic systems so that exponential stabilization can be achieved. A simulation example is finally given to illustrate the theoretical results.     

STABILITY OF THEORETICAL SOLUTION AND NUMERICAL SOLUTION OF NONLINEAR DIFFERENTIAL EQUATIONS WITH PIECEWISE DELAYS

This paper is concerned with the stability of theoretical solution and numerical solutionof a class of nonlinear differential equations with piecewise delays.At first,a sufficientcondition for the stability of theoretical solution of these problems is given,then numericalstability and asymptotical stability are discussed for a class of multistep methods whenapplied to these problems.     

Asymptotic stability of exact and discrete solutions for neutral multidelay-integro-differential equations

This paper concerns the long-time behavior of the exact and discrete solutions to a class of nonlinear neutral integro-differential equations with multiple delays. Using a generalized Halanay inequality, we give two sufficient conditions for the asymptotic stability of the exact solution to this class of equations. Runge–Kutta methods with compound quadrature rule are considered to discretize this class of equations with commensurate delays. Nonlinear stability conditions for the presented methods are derived. It is found that, under suitable conditions, this class of numerical methods retain the asymptotic stability of the underlying system. Some numerical examples that illustrate the theoretical results are given.     

Exact and numerical stability analysis of reaction-diffusion equations with distributed delays

This paper is concerned with the stability analysis of the exact and numerical solutions of the reaction-diffusion equations with distributed delays. This kind of partial integro-differential equations contains time memory term and delay parameter in the reaction term. Asymptotic stability and dissipativity of the equations with respect to perturbations of the initial condition are obtained. Moreover, the fully discrete approximation of the equations is given. We prove that the one-leg θ-method preserves stability and dissipativity of the underlying equations. Numerical example verifies the efficiency of the obtained method and the validity of the theoretical results.     

Dissipativity of multistep runge-kutta methods for nonlinear volterra delay-integro-differential equations

This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations.We investigate the dissipativity properties of (k,l)algebraically stable multistep Runge-Kutta methods with constrained grid and an uniform grid.The finitedimensional and infinite-dimensional dissipativity results of (k,l)-algebraically stable Runge-Kutta methods are obtained.