Bifurcations of Dynamical Systems Satisfying the Pseudo-orbit Tracing Property

本课题的研究目的是分析含有一个满足伪轨道跟踪性质的微分同构的1参数族的分岔现象。注意，POTP也被称为阴影属性。在2000年以前，我们已经在满足POTP的微分同态的C^1内部刻画了微分同态，并在2000年利用这些结果，更准确地说，刻画了满足（伪轨道）Lipschitz阴影性质（lsp）的微分同态的稳定流形与不稳定流形的交的动力学性质。然后在2001年，我们尝试分析了在满足LSP的微分同态集合的边界上的微分同态的分岔现象。遗憾的是，我们在调查中没有取得如此辉煌的成果，但我坚信，LSP的研究结果将对解决问题起到重要作用。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，我们刻画了具有拓扑稳定性的向量场集合的C^1内部（这也是本课题的一个显著成果）。一般来说，由于拓扑稳定性比POTP强，我们不能立即表征满足POTP的向量场的动力学。但是，通过修改证明中使用的技术，在不久的将来有可能表征动力学。

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.