Analysis of dimensional and recursive properties for almost periodic solutions of nonlinear partial differential equations

非线性偏微分方程几乎周期解的维数和递归性质分析

基本信息

批准号：
10640178
负责人：
NAITO Koichiro
金额：
$ 2.05万
依托单位：
KUMAMOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

项目摘要

In recent years great efforts have been made to analyze complexity or chaotic behaviors in the study of dynamical systems. In this research we investigate fractal dimensions and recursive properties of orbits for quasi-periodic dynamical systems and then, we apply the abstract results to almost or quasi-periodic solutions for nonlinear partial differential equations. In [1] (of 11.REF.) we estimate correlation dimensions of discrete quasi-periodic orbits by using the parameters derived from some algebraic properties of the irrational frequencies. On the other hand, in [2], we study recursive properties of the quasi-periodic orbits by defining recurrent dimensions and show inequality relations between the correlation dimensions and the recurrent dimensions. To estimate these dimensions we introduce new class of irrational numbers, quasi Roth numbers, quasi or weak Liouville numbers, which are classified according to badly approximable properties or (extremely) good properties for the rational approximations, respectively.Furthermore, in [2] and [3] we investigate quasi-periodic solutions of nonlinear partial differential equations with quasi periodic perturbations and estimate these dimensions of the attractors.Fractal dimensions are most essential in the sense that they show the level of complexity, or selfsimilarity or randomness. On the other hand, it is well known that periodic or almost periodic states occupy the important positions as main gateways in various routes to chaos. In the following papers (of 11.REF.) by the head and co-investigators we have shown various fundamental results, which will play important and essential roles for investigating chaotic behaviors of nonlinear dynamical models.

近年来，在动态系统的研究中，已经做出了巨大的努力来分析复杂性或混乱行为。在这项研究中，我们研究了准周期动力学系统的轨道的分形维度和递归特性，然后，我们将抽象结果应用于非线性部分偏微分方程的几乎或准周期溶液。在[1]中（11.ref。）中，我们通过使用从非理性频率的某些代数属性中得出的参数来估计离散准周期轨道的相关尺寸。另一方面，在[2]中，我们通过定义复发尺寸并显示相关维度与复发尺寸之间的不平等关系来研究准周期轨道的递归特性。 To estimate these dimensions we introduce new class of irrational numbers, quasi Roth numbers, quasi or weak Liouville numbers, which are classified according to badly approximable properties or (extremely) good properties for the rational approximations, respectively.Furthermore, in [2] and [3] we investigate quasi-periodic solutions of nonlinear partial differential equations with quasi periodic perturbations and estimate these dimensions of吸引子。范围的维度最重要的是，它们显示出复杂性，自我相似或随机性的水平。另一方面，众所周知，定期或几乎周期性的状态在各种混乱路线中占据了重要位置。在以下论文（11.ref。）中，我们已经显示了各种基本结果，这将在研究非线性动力学模型的混乱行为方面发挥重要而重要的作用。