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Bifurcations of Dynamical Systems Satisfying the Pseudo-orbit Tracing Property

满足伪轨道追迹性质的动力系统的分岔

基本信息

批准号：
11640217
负责人：
SAKAI Kazuhiro
金额：
$ 0.9万
依托单位：
Kanagawa University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640217/
关键词：
dynamical systems theory bifurcation pseudo-orbit tracing property Axiom A transversality condition

项目摘要

The purpose of this research project is to analyze the bifurcation phenomena of 1-parameter family containing a diffeomorphism which satisfies the pseudo-orbit tracing property (abbr.POTP). Remark that the POTP is also well known as the shadowing property. Before 2000, we had characterized diffeomorphisms in the C^1 interior of diffeomorphisms satisfying the POTP, and in 2000, by making use of those results the dynamics, more precisely, the property of the intersection of the stable and unstable manifolds of diffeomorphisms satisfying the (pseudo-orbit) Lipschitz shadowing property (abbr.LSP) was characterized.Then, in 2001 we tried to analyze the bifurcation phenomena of diffeomorphisms lying in the boundary of the set of diffeomorphisms satisfying the LSP.Unfortunately, we could not produce so splendid achievements in the investigation, but I am strongly convinced that the results on the LSP will play an important role to solve the problem. Furthermore, in 2001 we had noticed that our method in this project also work for C^1 vector fields. Actually, by Hayashi's connecting lemma we have characterized the C^1 interior of the set of vector fields having the topological stability (this is also a remarkable result of this project). In general, since the topological stability is stronger than the POTP, we cannot characterize the dynamics of vector fields satisfying the POTP at once. But, by modifying the techniques used in the proof it might be possible to characterize the dynamics in the near future.

本研究项目的目的是分析满足赝轨道追迹性质（简称POTP）的含有微分同胚的1参数族的分岔现象。请注意，POTP 也称为阴影属性。在 2000 年之前，我们已经描述了满足 POTP 的微分同胚的 C^1 内部的微分同胚，并且在 2000 年，通过利用这些结果，更准确地描述了满足（伪轨道）Lipschitz 阴影性质（缩写 LSP）的微分同胚的稳定流形和不稳定流形的交集的性质。 2001年我们试图分析满足LSP的微分同胚集合边界上的微分同胚的分岔现象。不幸的是，我们在研究中未能取得如此辉煌的成果，但我坚信LSP的结果将对解决这个问题发挥重要作用。此外，在 2001 年，我们注意到我们在这个项目中的方法也适用于 C^1 向量场。实际上，通过Hayashi的连接引理我们已经刻画了具有拓扑稳定性的向量场集合的C^1内部（这也是该项目的一个显着成果）。一般来说，由于拓扑稳定性比 POTP 更强，所以我们不能立即表征满足 POTP 的矢量场的动力学。但是，通过修改证明中使用的技术，也许可以在不久的将来描述动态特征。