Instabilities in nearly integrable Hamiltonian systems

近似可积系统的一个很好的例子是太阳系。事实上，太阳系的稳定性一直是驱动该领域的一个激励性问题。自从庞加莱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逆转。 近似可积系统是稳定的还是不稳定的？答案可能是两者都有：在测度论意义上，大多数轨道是稳定的（KAM定理）; 2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。由于稳定性与不稳定性的并存，这类问题往往是非常棘手的。我们特别感兴趣的是Arnold在1963年提出的一个猜想，即对于一个典型的近可积系统，不稳定性和KAM现象是共存的。 我和我的合著者已经开发出了非常适合研究这些问题的新方法。我建议在以下主题进行研究： 1.发现比Arnold预测的更不稳定的轨道，这些轨道在一定意义上表现出混沌。2.研究阿诺德的高维问题。3.继续发展相关理论，帮助澄清稳定与不稳定共存背后的奥秘。

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.