Analysis on the fractal structure of quasi periodic orbits for nonlinear evolution equations

非线性演化方程准周期轨道的分形结构分析

• 批准号: 16540164
• 负责人: NAITO Koichiro
• 金额: $ 2.37万
• 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: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540164/
• 关键词: nonlinear PDE; quasi periodicity; attractor; fractal dimension; Diophantine approximation; KAM theorem; self-similarity; chaos; 準周期性; 再帰性; ディオファンタス条件

In our previous 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. To investigate various chaotic orbits we also proposed the gaps between the upper and the lower recurrent dimensions as the index parameters, which measure unpredictability levels of the orbits. In this research, classifying the irrational numbers according to the orders of goodness or badness levels of approximation by rational numbers and parametrizing the Diophantine conditions, we say d_0-(D) condition, we estimate the gaps of recurrent dimensions of quasi-periodic orbits by using these orders of the parametrizing Diophantine conditions.These results were announced by the head investigator in the international conference NACA2005 ([1]) and in Discr.Conti.Dyn.Systems ([4]) and in the other journals.Calculating the dimensions gaps of quasi-periodic dynamical systems is to measure their level of complexity and randomness. In [5], [6] the co-investigator Y.Oshima proved some related results for randomness, using probability theory. On the other hand, in [7]-[9] the co-investigator M.Misawa has shown various fundamental results on P.D.E., which will play important and essential roles for investigating chaotic behaviors of nonlinear dynamical models. In [10],[11] the co-investigator T.Sadahiro numerically estimated the fractal tiling structures of various quasi-periodic dynamical systems.

在以往的研究中，我们引入了离散动力系统的循环维数，并估计了离散准周期轨道的上、下循环维数，以分析不同类型偏微分方程拟周期解的复杂性。为了研究各种混沌轨道，我们还提出了上下循环维之间的间隙作为衡量轨道不可预测性水平的指标参数。本文将无理数按照有理数逼近的好坏等级的顺序进行分类，并将Diophantine条件参数化，我们称之为d_0-(D)条件，利用这些参数化Diophantine条件的阶数来估计准周期轨道的循环维数间隙。这些结果由首席研究员在国际会议NACA2005（[1]）和discr . ci . dyn . systems（[1]）以及其他期刊上宣布。计算准周期动力系统的维数间隙是为了测量其复杂性和随机性的程度。2010年，合作研究者Y.Oshima用概率论证明了随机性的一些相关结果。另一方面，在[7]-[9]中，共同研究者m.m asawa已经展示了关于P.D.E的各种基本结果，这些结果将对研究非线性动力学模型的混沌行为发挥重要和必不可少的作用。在b[10]， b[11]中，共同研究者T.Sadahiro对各种准周期动力系统的分形平铺结构进行了数值估计。

项目成果

Multiple points of tilings associated with Pisot numeration systems

与皮索计数系统相关的多点平铺

• 期刊: Theoretical Computer Science (to appear)
• 作者: S. Ito and T;Sadahiro;S. Ito;Mitsuo Kato;Yoshishige Haraoka;Shun Shimomura;Taizo Sadahiro
• 通讯作者: Taizo Sadahiro

Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations II : Gaps of dimensions and Diophantine conditions

非线性演化方程准周期解的循环维数 II：维数间隙和丢番图条件

• 发表时间: 2004
• 期刊: Discrete and Continuous Dynamical Systems 11
• 作者: K.Yoshino;M.Suwa;Koichiro Naito
• 通讯作者: Koichiro Naito

Existence for a Cauchy-Dirichlet problem for evolutional p-Laplacian system

演化p-拉普拉斯系统柯西-狄利克雷问题的存在性

• 发表时间: 2004
• 期刊: Appl.Math.(Warsaw) 31
• 作者: K.Yoshino;M.Suwa;Koichiro Naito;M.Misawa
• 通讯作者: M.Misawa

Existence for a Cauchy-Dirichlet problem for evolutionalp-Laplacian systems

演化p-拉普拉斯系统柯西-狄利克雷问题的存在性

• 发表时间: 2004
• 期刊: Applicationes Math. 31・3
• 作者: K.Ishige;Y.Yagisita;H.Nagai;M asashi. Misawa
• 通讯作者: M asashi. Misawa

The evolution of minimal surfaces with free boundaries in higher dimensions,

在更高维度中具有自由边界的最小表面的演变，

• 发表时间: 2005
• 期刊: "The 5th East Asia PDE Conference" T.Suzuki eds, GAKUTO Internatinal Ser.Math.Sci.Appl. 22
• 作者: M.Abe;M.Furushima;T.Shima;M Otani;M.Misawa;M.Misawa
• 通讯作者: M.Misawa