Numerical Dissipativity of Runge-Kutta Methods for Delay Differential Equations

【作者】 沈爱莉；

【导师】 项家祥；

【作者基本信息】 上海师范大学 ， 计算数学， 2006， 硕士

【摘要】 滞时微分动力系统在神经网络、光学、生态学、自动控制等许多领域具有广泛的应用。目前在解的基本理论、稳定性理论、周期解理论、分歧理论等方面都取得了许多重要的成果。 一般来说，滞时微分动力系统的精确解及其动力学性质是很难精确获得的。数值模拟已经成为了解滞时微分动力系统动力学性质的主要手段之一。数值方法的古典收敛性只能保证在有限求积区间上数值解序列是收敛的，而无法保证数值解序列具有原始微分系统的动力学性质。因此，研究滞时微分动力系统数值方法的动力学性质具有非常重要的理论和实际意义。 本学位论文旨在研究Runge-Kutta方法求解滞时微分动力系统的数值耗散性。主要内容包括： ①Runge-Kutta方法求解变系数线性滞时微分动力系统的数值耗散性。首先给出线性变系数滞时微分动力系统耗散的一个充分条件，并讨论了将Runge-Kutta方法结合Lagrange插值求解耗散动力系统的数值耗散性。最后给出几个数值例子验证我们的理论结果。 ②Runge-Kutta方法求解非线性非自治滞时微分动力系统的数值耗散性。在已有的非线性滞时微分动力系统耗散的条件下，讨论了(k，l)-代数稳定的Runge-Kutta方法求解此耗散系统的数值耗散性。最后同样给出数值例子验证我们的结论。更多还原

【Abstract】 Delay differential dynamical systems have appeared in areas of applications as diverse as neural networks, optics, biology, and automatic control theory. There are many significant achievements on basic theory of solution, stability, periodic solution and bifurcation of delay differential dynamical systems.Numerical simulation is one of the most important methods to understand dynamical properties of delay differential dynamical systems. A numerical method which is convergent in a finite interval does not yield the same asymptotic behavior as the underlying differential system. Therefore, it is very important to understand the dynamics of numerical methods for solving delay differential equations.The purpose of this thesis is to study dissipativity property of Runge-Kutta methods solving delay differential equations. It contains① Numerical dissipativity of Runge-Kutta methods for linear delay differential equations with variable coefficients. A sufficient condition is given to ensure that a class of linear delay differential equations is dissipative. We then apply Runge-Kutta methods combined with Lagrange interpolation procedure to these dissipative equations and investigate the numerical dissipativity of numerical solution. Also, we give several examples to illustrate our theoretical result.② Numerical dissipativity of Runge-Kutta methods for nonlinear nonautonomous delay differential equations with variable coefficients. We study the numerical dissipativity of (k, l)-algebraically stable Runge-Kutta methods and give some numerical examples to demonstrate our theory.更多还原