Bifurcations of vector fields possessing the shadowing property

本文研究了一类由可微向量场组成的含有具有跟踪性质的向量场（简称SP）的单参数向量族的分支现象，SP表示具有跟踪性质的向量场的集合，并赋予其C^1-拓扑，intSP表示C^1-内部。在研究的第一阶段，利用动力系统的稳定性理论来刻画intSP中的向量场是必不可少的。2002年，我们继续对上述向量场进行了刻画，并分析了这类单参数向量族的分歧现象。刻画的主要障碍是X ∈ intSP的奇异性的存在。我们仔细地分析了积分流X_t在奇点附近的轨道的行为，发现了流在奇点附近的一种特殊的（但很自然的）性质。为了简单起见，我们称该属性为（*）。最近，已经证明了X ∈ intSP满足（*）当且仅当X同时满足公理A和强横截条件。这一结果部分回答了我们在2001年第一阶段所要解决的问题。目前，我们正在利用上述结果分析一个包含一个具有SP的向量场的单参数族的分叉现象。我们认为，这个研究项目的成功率可能达到70%。

The purpose of this research project is to analyze the bifurcation phenomena of a 1-parameter family, which contains a vector field possessing the shadowing property (abbr.SP), composed of differentiate vector fields.Denote by SP the set of vector fields possessing SP endowed with the C^1-topology, and let intSP be the C^1-interior. At the first stage of this research, it was indispensable to characterize the vector fields in intSP by making use of the stability theory of dynamical systems. In the end of 2001, however, we could not get a complete characterization for the vector fields.In 2002, we continuously proceeded to characterize the above vector fields, and to analyze the bifurcation phenomena of such 1-parameter family. The main obstacle to the characterization is the existence of singularities of X ∈ intSP. We intimately analyzed the behavior of the orbits of the integrated flow X_t in the neighborhood of singularities, and we have found a special (but very natural) property of the flow being displayed in the neighborhood. Let us call the property (*) for simplicity. Very recently, it has been proved that X ∈ intSP satisfies (*) if and only if X satisfies both Axiom A and the strong transversality condition. This result is a partial answer to the problem we aimed to solve in 2001 at the first stage. Nowadays, using the above result we are analyzing the bifurcation phenomena of a 1-parameter family which contains a vector field possessing SP. In our opinion, the achieve percentage of this research project might be evaluated 70 %.