Instabilities in nearly integrable Hamiltonian systems

近可积哈密顿系统的不稳定性

• 批准号: 436169-2013
• 负责人: Zhang, Ke
• 金额: $ 1.75万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635202
• 关键词: Instabilities; nearly; integrable; Hamiltonian; systems

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.

太阳系是几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驱动磁场的一个激励问题。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。近可积系统是稳定的还是不稳定的？答案可能两者都有：在测量理论意义上，大多数轨道是稳定的（KAM定理）；在拓扑学意义上，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。由于稳定与不稳定并存，这类问题往往很难解答。我们特别感兴趣的是Arnold（1963）关于一个典型的近可积系统存在不稳定性和KAM现象共存的猜想。我和我的合著者已经开发出非常适合研究这些问题的新方法。我建议从以下几个方面进行研究：1。找到比阿诺德预测的更不稳定的轨道，这些轨道在某种意义上表现出混乱。2. 在更高的维度上研究阿诺德的问题。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。