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态系统的全球结构和动力学系统的全球结构和动力学系统，并特别强调了诸如PDES和耦合系统的动力学和系统中的混乱复杂和不可预测的行为，并获得了许多有趣的结果。 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,获得了一种在适当的环境下显示在此类系统中周期性和混乱解决方案存在的方法，并将其应用于几个具体问题。 Shishikura研究了复杂的分析动力学系统，尤其是为抛物线固定点开发了一种重新归一化的理论，这将是研究此类系统的结构和分叉的新型且非常强大的工具。 Asaoka研究了具有一种称为Project Anosov结构的双波利度的动力系统，并在3维流量的情况下完成了分类。 Arai将严格的计算与Conley指数理论（例如Conley指数理论）相结合，在亨逊地图中获得了几个有趣的结果。 Tsujii从厄贡理论的角度研究了动力系统，并获得了二维部分双曲线系统中良好不变测量的一般结果。 Nishiura研究了在某种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，并使用动力学系统理论阐明了其机制。具有较大自由度的系统的其他结果包括Komuro对全球耦合地图中混沌效率的详细分析。