Chaos control associated to topoiogical dynamics

与拓扑动力学相关的混沌控制

• 批准号: 11554001
• 负责人: YOSHIDA Toshio
• 金额: $ 2.3万
• 依托单位: HIROSHIMA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11554001/
• 关键词: topological dynamics; chaos control; ローレンツ水車; ルエル不変量; 回転数

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.

粗略地说，混乱现象是不可预测的。这些课题通过各种模拟进行了检验，但他们的理论并不容易应用于实际。最近，Ott，Grebogi和York等人提出了混沌控制，具有实际应用的可能性。在数学研究方面，我们用与Lyapunov指数相关的无穷小行为来研究混沌现象，并研究了纤维不变量、Ruelle不变量、保护流的升力、射影Anosov流的分支点、保护流的链递归集、余维2的例外极小集。通过射影一个流的导数的三角投影，我们得到了一个纤维丛的流，它的纤维与射影空间同胚。这种流被称为投影流。李雅普诺夫指数表示无限小的膨胀。投影流意味着沿轨道的无限微小的扭曲。由这些结果可知，Ruelle不变量和Lyapunov不变量同样适用于混沌行为的判定。Ruelle不变量很容易在数学意义上处理。对于数学理论在实际现象中的应用，我们建立了多智能体系统、切换到达系统、混沌磨坊和大气现象的数学模型。例如，对于混沌磨坊，我们从它们导出了Markus磨坊，并发现它们的混沌行为来自于Parry映射。帕里地图是数学定义的，它们的混沌控制可以自然地表达出来。从这个角度出发，我们对混沌控制进行了理论分析。

项目成果

T.Kobayashi: "Extendibility, stable extendibility and span of some vector bundles over real projective spaces"Mem. Fac. Sci. Kochi Univ. (Math.). 23. 45-56 (2002)

T.Kobayashi：“真实射影空间上某些向量束的可扩展性、稳定可扩展性和跨度”Mem。

T.Kobayashi: "Stable extendibility of vector bundles over real projective spaces and determination of the Schwarzenberger number β(k)"Mem. Fac. Sci. Kochi Univ. (Math.). 24. 19-35 (2003)

T.Kobayashi：“实投影空间上向量束的稳定可扩展性和施瓦岑贝格数 β(k)”Mem。高知大学 (Math.)。

T. Kobayaishi: "Extendibiiity and stable extendibiiity of the power of the normal bundle associated to an immersion of the lens space mod 4"Mem. Fac. Sci. Kochi Univ. (Math.). 22. 45-57 (2001)

T. Kobayaishi：“与透镜空间 mod 4 的浸没相关的法向束光焦度的可扩展性和稳定可扩展性”Mem。

T. Kobayashi: "Extendibiiity, stable extendibiiity and span of some vector bundles over real projective spaces"Mem. Fac. Sci; Kochi Univ. (Math.). 23. 45-56 (2002)

T. Kobayashi：“真实射影空间上某些向量束的可扩展性、稳定可扩展性和跨度”Mem。

T. Kobayashi: "Stable extendibiiity of vector bundles over real projective spaces and determination of the Schwarzenberger number β (k)"Mem. Fac. Sci. Kochi Univ. (Math.). 24. 19-35 (2003)

T. Kobayashi：“实射影空间上向量束的稳定延展性和 Schwarzenberger 数 β (k)”Mem。高知大学 (Math.)。