Mathmatical study of complex systems

复杂系统的数学研究

• 批准号: 11837002
• 负责人: TSUJII Masato
• 金额: $ 2.3万
• 依托单位: Hokkaido 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 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11837002/
• 关键词: Complex system; Globally coupled maps; Piecewise expanding maps; カオス; カオス遍歴; 力学系; エルゴード理論; 間欠性; アトラクター

We study the dynamics of piecewise expanding maps in higher dimension. On the contrary to the one dimensional case, we show that a C^γ piecewise expanding map does not necessarily admit an absolutely continuous invariant measure. Also we show that if we assume that the map is piecewise analytic in 2dim or piecewise affine in higher dimension, it always admits an absolutely continuous invariant measure.We also studied the so-called global lycoupled maps from a mathematical viewpoint. We made an extensive numerical experiment for 10 globally coupled logisitic map and came to the idea that the so-called "Chaotic inenerancy" in this case is caused by a hierarchy of on-off intermittencies on the invariant subspaces.

我们研究高维分段扩展映射的动力学。与一维情况相反，我们证明 C^γ 分段扩展映射不一定接受绝对连续的不变测度。我们还表明，如果我们假设映射是二维的分段解析或更高维的分段仿射，那么它总是承认绝对连续的不变测度。我们还从数学角度研究了所谓的全局耦合映射。我们对 10 个全局耦合逻辑图进行了广泛的数值实验，得出这样的想法：这种情况下所谓的“混沌惰性”是由不变子空间上的开关间歇性层次结构引起的。

项目成果

Tsujii, Masato: "Absolutely continuous invariant measures for piecewise exponding maps"Inventiones Mathematicae. 143-2. 349-373 (2001)

Tsujii，Masato：“分段解释映射的绝对连续不变测度”数学发明。

森真: "Higher order mixing property or piecewise linear transformations"Discrete and Continuous Dynamical systems. 6. 915-934 (2000)

Makoto Mori：“高阶混合特性或分段线性变换”离散和连续动力系统。6. 915-934 (2000)。

小室元政: "Hierarchical Structure among Invariant Subspaces of a Chaotic Neural Network"Japan Journal of Industrial and Applied Mathematics. (印刷中). (2000)

Motomasa Komuro：“混沌神经网络的不变子空间的层次结构”日本工业与应用数学杂志（2000 年出版）。

辻井 正人: "A simple proof of monotonicity of entropy in quadratic family"Ergodic Theory and Dynamical Systems. (発売予定). (2000)

Masato Tsujii：“二次族中熵单调性的简单证明”遍历理论和动力系统（即将发布）。

M. TSUJII: "Absolutely continuous invariant measures for expanding piecewise linear maps"Invent. Math. 143, no. 2.. 349-373 (2001)

M. TSUJII：“用于扩展分段线性映射的绝对连续不变测度”发明。