Theoretical Study of the Functional Differential Equations using Computer Simulation

泛函微分方程的计算机模拟理论研究

• 批准号: 11640210
• 负责人: HARA Tadayuki
• 金额: $ 2.3万
• 依托单位: Osaka Prefecture University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640210/
• 关键词: Functional Differential Equation; Delay Differential Equation; Lotka-Volterra Equation; Prey-Predator Model; Global Asymptotic Stability; Permanence; Holling Type; Difference Equation; 時間遅れを含む微分方程式; 大域的漸近安定

Main results of our research are as follows :(I) We have improved computer software FDE2RK which was developed in our group two years ago for computer simulation of two dimensional differential equations with integral terms. In case the kernel of the integral does not depend on time variable, we improved our software and we can compute our differential(II) Using the software FDE2RK, we studied the behavior of solutions of delay differential equations in mathematical ecology and found some interesting properties of solutions. We succeeded to give mathematical proofs to these properties. The typical results are as follows :(1) Neccesary and sufficient conditions for the global asymptotic stability and the permanence of a Lotka-Volterra type differential equations with two delays(2) Siability analysis of SIR epidemic models with distributed delays(3) Stability analysis of two-dimensional neural net work with delays(III)We found neccesary and sufficient conditions for the existence of a limit cycle and the global asymptotic stability of prey-predator differential equations without delay of Holling type in mathematical ecology and succeeded to give mathematical proofs to these theorems.(VI) Oscillation theorems of the Riemann-Weber version of Euler differential equations with delay(V) Using the software DDE2E, we studied the behavior of solutions of delay differential equations with piecewise constant arguments and difference equations with delays and found some interesting properties of solutions. We succeeded to give mathematical proofs to these properties. The typical results are as follows :(1) Neccesary and sufficient conditions for the global asymptotic stability of a linear defference equation of higher order(2) Neccesary and sufficient condition for the permanence of difference equations of Lotka-Volterra type(3) Classification of the dynamics of 2-dimensional linear differential Equations with piecewise constant arguments

我们的研究的主要结果如下：（i）两年前，我们已经改进了计算机软件FDE2RK，该计算机软件是在我们组中开发的，用于对具有整体术语的二维微分方程进行计算机模拟。如果积分的内核不依赖于时间变量，我们可以改进软件，并且可以使用软件FDE2RK计算差异（ii），我们研究了数学生态学中延迟微分方程的解决方案的行为，并找到了一些有趣的解决方案。我们成功地为这些属性提供了数学证明。典型的结果如下：（1）全球渐近稳定性的必要条件和足够条件以及具有两个延迟的Lotka-volterra类型微分方程的永久性（2）具有分布式延迟的siR流行模型的可感知分析（3）二维神经网络的稳定性分析（3）与延迟的二维神经网络的稳定性分析（iiii），我们发现了（iii III）的状态（iiii）。 stability of prey-predator differential equations without delay of Holling type in mathematical ecology and succeeded to give mathematical proofs to these theorems.(VI) Oscillation theorems of the Riemann-Weber version of Euler differential equations with delay(V) Using the software DDE2E, we studied the behavior of solutions of delay differential equations with piecewise constant arguments and difference equations with delays and found some interesting解决方案的性质。我们成功地为这些属性提供了数学证明。典型的结果如下：（1）高阶的线性缩写方程的全球渐近稳定性（2）lotka-volterra类型差异方程的永久性（3）与2二维线性差分方程的差异差异（3）差异方程式的差异和足够条件（2）二维线性差分等方程的差异方程式（2），应有的和足够的条件。

项目成果

R.Ogita,H.Matsunaga and T.Hara: "Asymptotic stability condition for a class of linear delay difference equations of higher order"Journal of Mathematical Analysis and Applications. 248. 83-96 (2000)

R.Ogita、H.Matsunaga 和 T.Hara：“一类高阶线性时滞差分方程的渐近稳定性条件”数学分析与应用杂志。

Y. Takeuchi and W. Ma: "Stability analysis on a delayed SIR epidemic model with density dependent birth process"Dynamics of Continuous, 7 Discrete and Impulsive Systems. 5. 171-184 (1999)

Y. Takeuchi 和 W. Ma：“具有密度依赖性出生过程的延迟 SIR 流行病模型的稳定性分析”连续、7 个离散和脉冲系统的动力学。

J.Sugie,N.Yamaoka and Y.Obata: "Nonoscillation theorems for a nonlinear self-adjoint differential equation"Nonlinear Analysis. (accepted).

J.Sugie、N.Yamaoka 和 Y.Obata：“非线性自伴随微分方程的非振荡定理”非线性分析。

W. Ma, T. Hara and Y. Takeuchi: "Stability of a 2-dimensional neural network with time delays"Journal of Biological Systems. (in press).

W. Ma、T. Hara 和 Y. Takeuchi：“具有时间延迟的二维神经网络的稳定性”生物系统杂志。

J.Sugie: "A nonoscillation theorem for second-order nonlinear differential equations with decaying coefficients"The Bulletin of the London Mathematicat Society. (accepted).

J.Sugie：“具有衰减系数的二阶非线性微分方程的非振荡定理”伦敦数学学会公报。