Instabilities in nearly integrable Hamiltonian systems

太阳系是近乎可积系统的一个很好的例子。事实上，太阳系的稳定性一直是推动这一领域的一个鼓舞人心的问题。自从彭加莱的S证明平面三体问题不可积以来，稳定性问题的答案经历了几次颠倒。 近可积系统是稳定的还是不稳定的？答案可能是两者兼而有之：在测度论意义上，大多数轨道是稳定的(Kam定理)；在拓扑意义上，大多数系统可能是不稳定的。我们研究的主要目的是了解不稳定性。由于稳定和不稳定并存，这样的问题往往很难解决。我们特别感兴趣的是Arnold在1963年提出的猜想，即对于一个典型的近可积系统，不稳定性和KAM现象共存。 我和我的合著者开发了非常适合研究这些问题的新方法。我建议在以下主题中进行研究： 1.发现比Arnold预测更多的不稳定轨道，这些轨道在某种意义上表现出混沌。2.从更高的维度研究阿诺德的问题。3.继续发展有助于解开稳定与不稳定共存背后奥秘的相关理论。

A good example of nearly integrable systems is the solar system. Indeed, the stability of the solar system has been a motivating question that drives the field. Since Poincaré's proof that the planar three body question is not integrable, the answer to the stability question has gone through several reversals. Are nearly integrable systems stable or unstable? The answer is probably both: in the measure theoretical sense, most orbits are stable (KAM Theorem); in the topological sense, most systems are probably unstable. The main goal of our research is to understand the instabilities. Due to the coexistence of stability and instability, such question are often very difficult. We are, in particular, interested in the conjecture of Arnold in 1963, that for a typical nearly integrable system, there is coexistence of instability and KAM phenomenon. My coauthors and I have developed new approaches well suited to study these problems. I propose research in the following topics: 1. Find even more unstable orbits than Arnold predicted, these orbits exhibits chaos in a certain sense. 2. Investigate Arnold's question for higher dimensions. 3. Continue to develop the related theories that help clarify the mysteries behind the coexistence of stability and instability.