Study of Global Structures and Bifurcations of Dynamical Systems including Systems with Large Degrees of Freedom

包括大自由度系统在内的动力系统的整体结构和分岔研究

• 批准号: 21340035
• 负责人: KOKUBU Hiroshi
• 金额: $ 10.98万
• 依托单位: Kyoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009 至 2012
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-21340035/
• 关键词: 力学系; 分岐; 大域的構造; 不変集合; 計算機援用証明; 位相計算的方法; 大自由度系; 計算機援用解析; カオス; Morse分解; グラフアルゴリズム; 計算機援用; 位相計算; 精度保証付き計; ホモロジー; グラフ・アルゴリズム

A great improvement of the Conley-Morse graph method, a computer-assisted method that analyzes global dynamics and bifurcations by combining a topological method and validated numerical computation has been done by developing a computer-algorithm for generating so-called clutching graphs which describe relation between Conley-Morse graphs on adjacent parameter domains. Another improvement is a method of non-uniform grid decomposition of the phase space in order to substantially decrease computational cost of the Conley-Morse graph method.A new approach to the bifurcation theory of dynamical systems from topological-computation theory viewpoint has been proposed which captures bifurcation of dynamics from changes of the corresponding Conley-Morse graphs. From this viewpoint, we have obtained a new mathematical framework for the crisis bifurcation.Moreover, we have studied the global dynamics of a Coupled Map Lattice system and a Coupled Oscillator system by using the Conley-Morse graph method, and have obtained the creation and bifurcations of some global dynamics such as unstable invariant tori and their connecting orbits.

Conley-Morse图法是对Conley-Morse图法的极大改进。Conley-Morse图法是一种将拓扑学方法和有效的数值计算相结合来分析全局动力学和分叉的计算机辅助方法，它开发了一种计算机算法，用于生成描述相邻参数域上Conley-Morse图之间的关系的所谓离合器图。另一种改进方法是对相空间进行非均匀网格分解，以大大降低Conley-Morse图方法的计算量。从拓扑计算理论的观点出发，提出了一种从相应Conley-Morse图的变化来捕捉动力学分叉的新方法。利用Conley-Morse图方法研究了耦合映象格子系统和耦合振子系统的全局动力学，得到了不稳定不变环面及其连通轨道等全局动力学的产生和分支。

项目成果

Graph-based topological approximation of saddle-node bifurcation in maps

地图中鞍节点分叉的基于图的拓扑近似

• 发表时间: 2012
• 期刊: RIMS Kokyuroku Bessatsu B31
• 作者: Zin Arai;Marcio Gameiro;Tomas Gedeon;Hiroshi Kokubu;Konstantin Mischaikow;Hiroe Oka
• 通讯作者: Hiroe Oka

Combinatorial-topological framework for the analysis of global dynamics

用于分析全局动力学的组合拓扑框架

• DOI: 10.1063/1.4767672
• 发表时间: 2012
• 期刊: Chaos
• 影响因子: 2.9
• 作者: Justin Bush;Marcio Gameiro;Shaun Harker;Hiroshi Kokubu;Konstantin Mischaikow;Ippei Obayashi;Pawel Pilarczyk
• 通讯作者: Pawel Pilarczyk

Topological-computational methods for analyzing global dynamics and bifurcations

用于分析全局动力学和分岔的拓扑计算方法

• 发表时间: 2011
• 作者: Shuto Kakimoto;Katsuhiro Ota;Yoshiaki Oda;Kenta Ozeki;Hiroshi Kokubu
• 通讯作者: Hiroshi Kokubu

A method for constructing databases for global dynamics of multi-parameter systems

一种多参数系统全局动力学数据库构建方法

• 发表时间: 2009
• 作者: T. Yamamoto;T. Chigai;H. Kimura;K. K. Tanaka;T. Ozawa and K. Yamauchi;舟木直久;國府寛司
• 通讯作者: 國府寛司

Topological Computation Methods for Global Dynamics and Bifurcations Part 1: Theory, Part 2: Application

全局动力学和分岔的拓扑计算方法第 1 部分：理论，第 2 部分：应用

• 发表时间: 2012
• 作者: 岩根秀直;樋口博之;穴井宏和;Seiya Negami;Rei-ichiro KAWAI;J. Byeon and K. Tanaka;M.Idzumi;Hitoshi Nakada;Y. Takei;Hiroshi Kokubu
• 通讯作者: Hiroshi Kokubu