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Research on comparisons of Global Properties of solutions of Non-linear Difference Equations and solutions of Nonlinear Phenomena.

非线性差分方程解与非线性现象解的全局性质比较研究。

基本信息

批准号：
15540217
负责人：
SUZUKI Mami
金额：
$ 1.98万
依托单位：
Aichi-gakusen University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

1.There is no general existence theorem for solutions for nonlinear difference equations, so we must prove the existence of solutions in accordance with models one by one. In our work, we found theorems for the existence of analytic solutions of nonlinear second order difference equations.2.One of the main work of this research is obtaining representations of analytic general solutions which converge to an equilibrium point of nonlinear second order difference equations with new methods of complex analysis.3.For analytic solutions of a nonlinear first order difference equation, T.Kimura studied the cases in which eigenvalue equal to 1,furthermore N.Yanagihara studied the cases in which the absolute value of the eigenvalue equal to 1. For nonlinear second order difference equation in which the absolute values of the all eigenvalues equal to 1, we consider analytic solutions of it in this study.4.Furthermore we apply the our theory of nonlinear higher order difference equations to Dynamics of Economic systems, for example, Nonlinear Cournot Dynamics with Heterogenous Duopolists. Generally, economist consider only equilibrium points for Mathematical Models. They call the equilibrium points as "solution of the model", and ignore orbits of the true solutions of the Models. However the equilibrium points are not "solution" in mathematical means, furthermore the orbits are important in the systems. Therefore in this study, we derive the true solutions and represent of solutions of Cournot Dynamics.5.In other studies, we treat economic Models with time delay. One of them is a dynamic economic model in isolated island. This Model is nonlinear differential equation with delayed terms, and studied stability of it. Next we treat an singular perturbation problem for economic models of R.Goodwin.

1.非线性差分方程的解没有普遍的存在性定理，因此必须根据模型一一证明解的存在性。在我们的工作中，我们发现了非线性二阶差分方程解析解存在性的定理。2.本研究的主要工作之一是用复分析的新方法获得收敛于非线性二阶差分方程平衡点的解析通解的表示。3.对于非线性一阶差分方程的解析解，T.Kimura研究了特征值等于的情况 1、N.Yanagihara还研究了特征值绝对值等于1的情况。对于所有特征值的绝对值等于1的非线性二阶差分方程，我们在本研究中考虑了它的解析解。4.此外，我们将我们的非线性高阶差分方程理论应用到经济系统动力学中，例如，非线性古诺动力学与异质双寡头主义者。一般来说，经济学家只考虑数学模型的均衡点。他们将平衡点称为“模型的解”，并忽略模型真实解的轨道。然而，平衡点并不是数学意义上的“解”，而且轨道在系统中很重要。因此，在本研究中，我们推导了古诺动力学的真解和解的表示。5.在其他研究中，我们将经济模型视为具有时滞的。其中之一是孤岛的动态经济模式。该模型是带有延迟项的非线性微分方程，并研究了其稳定性。接下来我们处理 R.Goodwin 经济模型的奇异摄动问题。