Chaos control associated to topoiogical dynamics

混沌现象，粗略地说，是不可预测的。这些问题通过各种模拟进行了检验，但其理论并不容易在实际中应用。最近，Ott， Grebogi和Yorke提出了混沌控制，具有实际应用的可能性。在本研究中，我们尝试用数学的方法来表述混沌控制，并从数学和实际的角度对其进行分析。在数学研究方面，我们从与Lyapunov指数相关的无限小行为的角度考虑混沌现象，研究了纤维不变测度、Ruelle不变量、保护流的提升、投影Anosov流的分支点、保护流的链循环集、余维2的例外极小集。通过对流导数的正交投影进行投影，得到了纤维束的流，其纤维与投影空间同胚。这种流称为投影流。李亚普诺夫指数表示无穷小膨胀。射影流意味着沿轨道的无限小的扭曲。通过这些结果，表示无穷小扭转的Ruelle不变量和Lyapunov不变量对于混沌行为的确定是有效的。鲁埃尔不变量在数学意义上易于处理。因此，它们似乎是有价值的。对于数学理论在实际现象中的应用，我们从数学上阐述了多智能体系统、切换到达系统、混沌磨坊和大气现象。例如，对于混沌磨坊，我们从它们衍生出Markus磨坊，并发现它们的混沌行为来自于Parry地图。招架地图是数学定义的，它们的混沌控制可以自然地形成。从这个角度出发，对混沌控制进行了理论分析。

Chaotic phenomena are, roughly speaking, unpredictable ones. These subjects are examined by various simulations, but their theory is not easy to be used practically. Recently Ott, Grebogi and Yorke proposed the chaos control, which has a possibility of practical use. In this study, we tried to formulate the chaos control mathematically and to analyze it from both mathematical and practical points of view.As for mathematical study, we considered chaos phenomena in term of infinitesimal behavior related to Lyapunov exponent, and studied fiberwise invariant measures, Ruelle invariants, lifts of protective flows, branch points of projectively Anosov flows, chain recurrent sets of protective flows, exceptional minimal sets of codimension two. By projectivizing the othogonal projection for the derivative of a flow, we obtain a flow of a fiber bundle whose fiber is homeomorphic to the projective space. This flow is called a projective flow. Lyapunov exponent represents the infinitesimal dilatation. The projective flow means the infinitesimal twist along the orbits. By these results, Ruelle invariant, the characteristic representing the infinitesimal twist, is valid for the determination of chaotic behavior as well as Lyapunov invariant. Ruelle invariant is easy to be treated in mathematical sense. Thus they seem to be valuable.As for applications of mathematical theory to real phenomena, we formulated mathematically multi-agents systems, switching arrival systems, chaotic mills and atmospheric phenomena. For example, for chaotic mills, we derived Markus mills from them, and found that their chaotic behaviors come from Parry maps. Parry maps are mathematically defined and their chaos control can be naturally formulated. In this point of view, we stepped out for the theoretical analysis of the chaotic control.