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

从不同的角度研究了动力系统的全局结构和分岔，特别强调了大自由度动力学系统（如偏微分方程和耦合系统）中的混沌复杂和不可预测行为，并获得了许多有趣的结果。作为本项目的一些主要成果，Kokubu(1)证明了在Lorenz系统及其类似系统中存在一个称为“奇异简并异斜环”的奇异不变量集，并证明了几何Lorenz型混沌吸引子的分岔；(2)利用称为Conley指数的拓扑不变量发展了描述奇异摄动向量场结构的理论；在适当的条件下，给出了该系统周期解和混沌解存在性的一种证明方法，并将其应用于若干具体问题。Shishikura研究了复杂解析动力系统，特别是发展了抛物不动点的重整化理论，这将是研究这类系统结构和分岔的一个新的有力工具。Asaoka研究了一种称为投影ansov结构的双曲动力系统，并在三维流动的情况下完成了分类。将严格的计算与拓扑方法如Conley指标理论相结合，Arai在Henon映射中的双曲和全局分岔方面得到了几个有趣的结果。Tsujii从遍历理论的角度研究了动力系统，得到了二维部分双曲型系统中存在良好不变测度的一般结果。Nishiura研究了一些pde的自复制和自毁灭模式中观察到的复杂有趣的瞬态行为，并用动力系统理论阐明了其机制。其他关于大自由度系统的结果包括Komuro对全局耦合映射中混沌迭代的详细分析。

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.