Analysis on the structure of quasi periodic attractors for nonlinear partial differential equations

非线性偏微分方程拟周期吸引子结构分析

• 批准号: 14540182
• 负责人: NAITO Koichiro
• 金额: $ 2.62万
• 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: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540182/
• 关键词: nonlinear PDE; quasi periodicity; attractor; fractal dimension; Diophantine approximation; KAM theorem; self-similarity; chaos; 偏微分方程式; 準周期性; 再帰性; ディオファンタス条件

In recent years great efforts have been made to analyze complexity or chaotic behaviors in various fields. In this research we introduced recurrent dimensions of discrete dynamical systems and we have estimated the upper and lower recurrent dimensions of discrete quasi-periodic orbits to analyze complexity of quasi-periodic solutions given by various types of partial differential equaitions. We also proposed the gaps between the upper and the lower recurrent dimensions as the index parameters, which measure unpredictability levels of the orbits. We show that the gaps of recurrent dimensions of quasi-periodic orbits take positive values when the irrational frequencies are weak Liouville numbers with sufficiently large orders of goodness levels of approximation by rational numbers. These results were announced by the head investigator in the international conference NACA2003 ([1], [2]) and will appear in Discr. Conti. Dyn. Systems ([3]).Calculating the dimensions of the attractors is to measure their level of complexity and randomness. In [5], [6], [7] the co-investigator Y. Oshima proved some related results for randomness, using probability theory. On the other hand, in [8] -[11] the co-investigator M. Misawa have shown various fundamental results on P.D.E., which will play important and essential roles for investigating chaotic behaviors of nonlinear dynamical models.

近年来，人们对复杂性或混沌行为的分析在各个领域都取得了很大的进展。在本研究中，我们引入了离散动力系统的回归维数，我们估计了离散拟周期轨道的上，下回归维数，以分析各种类型的偏微分方程的拟周期解的复杂性。我们还提出了上、下回归维之间的间隙作为衡量轨道不可预测性水平的指标参数。我们证明，当无理频率是弱Liouville数且有理数逼近的优度水平足够大时，准周期轨道的回归维度间隙取正值。这些结果由首席研究员在NACA 2003国际会议上宣布（[1]，[2]），并将出现在Discr上。康帝Dyn.系统（[3]）。计算吸引子的维数是为了衡量它们的复杂性和随机性。在[5]、[6]、[7]中，共同研究者Y. Oshima利用概率论证明了随机性的一些相关结果。另一方面，在[8] -[11]中，共同研究者M.三泽已经展示了关于P.D.E.的各种基本结果，这对于研究非线性动力学模型的混沌行为具有重要的意义。

项目成果

Y.Oshima: "On a construction of diffusion processes on moving domains"Potential Anal.. 20. 1-31 (2004)

Y.Oshima：“关于移动域上扩散过程的构造”Potential Anal.. 20. 1-31 (2004)

M.Misawa: "L^q estimates of gradients for evolutional p-Laplacian systems"Ark.Mat.. (掲載予定). (2004)

M.Misawa：“进化 p-拉普拉斯系统的梯度的 L^q 估计”Ark.Mat..（即将出版）。

M.Misawa: "Partial regularity results for evolutional p-laplacian systems with natural growth"Manuscripta Mathematica. 109(4). 419-455 (2002)

M.Misawa：“具有自然增长的进化 p-拉普拉斯系统的部分规律性结果”Manuscripta Mathematica。

Koichiro Naito: "Recurrent dimensions of quasi-periodic orbits with multiple frequencies : Extended common multiples and Diophantine conditions"Proc.NACA03. (to appear).

Koichiro Naito：“具有多个频率的准周期轨道的循环维度：扩展公倍数和丢番图条件”Proc.NACA03。

Koichiro Naito: "Recurrent Dimensions of Quasi-Periodic Solutions for Nonlinear Evolution Equations II : Gaps of Dimensions and Diophantine Conditions"Disc.Conti.Dyn.Sys.. (to appear).

Koichiro Naito：“非线性演化方程的准周期解的循环维数 II：维数间隙和丢番图条件”Disc.Conti.Dyn.Sys..（待出版）。