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Bifurcation theoretical approach to chaotic dynamics and to systems with large degrees of freedom

混沌动力学和大自由度系统的分岔理论方法

基本信息

批准号：
14340055
负责人：
KOKUBU Hiroshi
金额：
$ 9.02万
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：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

项目摘要

Global structures and bifurcations of dynamical systems, with special emphasis on chaos-complicated and unpredictable behavior in dynamics-and systems of large degrees of freedom such as PDEs and coupled systems, are studied from various different points of view and many interesting results are obtained. As some of main results in this project, Kokubu (1)showed the existence of a singular invariant set called "singularly degenerate heteroclinic cycle" in the Lorenz system and its alike, from which a chaotic attractor of geometric Lorenz type is proven to bifurcate, (2)developed a theory describing the structure of singularly perturbed vector fields with using a topological invariant called Conley index, obtained a method to show the existence of periodic and chaotic solutions in such systems under suitable setting, and applied it to several concrete problems. Shishikura studied complex analytic dynamical systems, and in particular developed a renormalization theory for parabolic fixed points, which will be a new and very powerful tool for studying the structure and bifurcation of such systems. Asaoka studied dynamical systems with a sort of hyperbolicity called projectively Anosov structure and completed a classification in the case of 3-dimensional flows. Combining rigorous computation with topological methods such as the Conley index theory, Arai obtained several interesting results on hyperbolicity and global bifurcations in the Henon maps. Tsujii studied dynamical systems from ergodic theory viewpoint and obtained a general result on the existence of good invariant measures in 2-dimensional partially hyperbolic systems. Nishiura studied complicated interesting transient behavior observed in some kinds of PDEs called self-replicating and self-destruction patterns and clarified its mechanism by using dynamical system theory. Other results on systems with large degrees of freedom include Komuro's detailed analysis on chaotic itenerancy in globally coupled maps.

从各种不同的角度研究了动力系统的全局结构和分叉，特别强调了动力学中的混沌-复杂和不可预测的行为-以及大自由度系统，如偏微分方程和耦合系统，并得到了许多有趣的结果。作为该项目的一些主要结果，Kokubu（1）证明了Lorenz系统及其类似系统中存在一个称为“奇异退化异宿环”的奇异不变集，并证明了几何Lorenz类型的混沌吸引子可以从中分叉，（2）发展了一个理论使用称为Conley指数的拓扑不变量描述奇扰动向量场的结构，在适当的条件下，得到了一种证明这类系统周期解和混沌解存在性的方法，并将其应用于几个具体问题。Shishikura研究了复杂的解析动力系统，特别是开发了抛物不动点的重整化理论，这将是研究此类系统的结构和分叉的新的非常强大的工具。Asaoka研究了具有一种称为投影Anosov结构的双曲性的动力系统，并在三维流动的情况下完成了分类。结合严格的计算与拓扑方法，如Conley指数理论，Arai获得了一些有趣的结果双曲性和全球分歧的Henon地图。Tsujii从遍历理论的角度研究了动力系统，得到了2维部分双曲系统良好不变测度存在性的一般结果。Nishiura研究了在一些被称为自我复制和自我毁灭模式的PDE中观察到的复杂有趣的瞬态行为，并利用动力系统理论阐明了其机制。其他结果系统的大自由度包括小室的详细分析混沌itenerancy在全球耦合地图。

项目成果

期刊论文数量（34）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Linearity of exceptional set for maps of P_k(C)

P_k(C) 映射的异常集线性

DOI：
发表时间：
2004
期刊：
Math.Annalen 330
影响因子：
0
作者：
J.-Y.Briend;S.Cantat;M.Shishikura
通讯作者：
M.Shishikura

Physical measures for partially hyperbolic surface endomorphisms

DOI：
10.1007/bf02392516
发表时间：
2003-01
期刊：
Acta Mathematica
影响因子：
3.7
作者：
M. Tsujii
通讯作者：
M. Tsujii

Pattern dynamics of self-replication and self-destruction

自我复制和自我毁灭的模式动力学

DOI：
发表时间：
2003
期刊：
影响因子：
0
作者：
L.Accardi;A.Ben Ghorbal;N.Obata;A.Nakamoto;西浦廉政;Toshiyuki Koto;T.Aoki et al.;N.Kuno;M.Ozawa;T.Iguchi;Abazajian 他;Yasumasa Nishiura
通讯作者：
Yasumasa Nishiura

M.Kuwamura, E.Yanagida: "The Eclhaus and zigzag instability criteria in gradient/skew gradient dissipative systems"Physica D. 175. 185-195 (2003)

M.Kuwamura、E.Yanagida：“梯度/斜梯度耗散系统中的 Eclhaus 和锯齿形不稳定准则”Physica D. 175. 185-195 (2003)