Instabilities in Hamiltonian systems

哈密​​顿系统的不稳定性

• 批准号: RGPIN-2019-07057
• 负责人: Zhang, Ke
• 金额: $ 1.53万
• 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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=716274
• 关键词: Instabilities; Hamiltonian; systems

Instabilities in Hamiltonian systems has been a central question in dynamical systems, since the founding of the field. This question has both been motivated by celestial mechanics (stability of the solar system) and by statistical physics (micro foundation of the thermodynamic laws). Among them, one important question is the Arnold diffusion, which asks whether a typical perturbation of a completely integrable system exhibit topological instability. This question is both difficult and deep, due to the remarkable stability enjoyed by nearly integrable systems, such as KAM and Nekhoroshev theory. With Vadim Kaloshin and other collaborators, we have been key contributors towards answering this question in two and a half degree of freedom, for smooth Hamiltonian systems. However, our understanding of the instability is still very limited. My proposed research deepens our knowledge in two separate directions. 1. Topological instability in higher degrees of freedom, and in the analytic category. With Vadim Kaloshin, we proposed a plan to study Arnold diffusion in higher degrees of freedom, using reduction to lower dimensional structures. We hope to fully answer Arnold's question in the smooth category. On the other hand, Arnold asked his original questions for analytic systems, where very little is known. With our knowledge in the smooth case, I propose to tack the analytic case, starting from simpler models. 2. Stochastic description of unstable orbit. The term "Arnold diffusion" was coined because, based on numerical evidence, the unstable exhibit a random-walk-like behavior, just like a diffusion orbit. I propose to prove stochastic limit theorems which justify the diffusion aspect. Philosophically, this aligns with the idea of "deterministic randomness", where random behavior can emerge from fully deterministic systems. The research will be pursued in two directions: in models of instability such as the a priori unstable model, one can embed a normally hyperbolic lamination, on which the dynamics is conjugate to a random dynamical systems; in slow fast system, where limit to a diffusion process can be proven. With the propose the research, we also advance our knowledge in the underlying theory, in particular weak KAM theory. These results will be applied to related fields, such as regular and random Hamilton-Jacobi equations.

自从该领域建立以来，汉密尔顿系统中的不稳定性一直是动态系统的核心问题。这个问题都是由天体力学（太阳系的稳定性）和统计物理学（热力学定律的微观基础）引起的。 其中，一个重要的问题是Arnold扩散，它询问完全可以整合系统的典型扰动表现出拓扑不稳定性。由于KAM和Nekhoroshev理论等几乎可以集成的系统所享有的显着稳定性，因此这个问题既困难又深刻。 在Vadim Kaloshin和其他合作者的陪同下，我们一直是在两个半程度上回答这个问题的主要贡献者，以实现平稳的哈密顿系统。但是，我们对不稳定的理解仍然非常有限。我提出的研究将我们的知识加深了两个单独的方向。 1。在较高的自由度和分析类别中的拓扑不稳定性。使用Vadim Kaloshin，我们提出了一项计划，以更高的自由度研究Arnold扩散，并使用还原至较低的尺寸结构。我们希望在“平滑”类别中充分回答阿诺德的问题。另一方面，阿诺德（Arnold）向分析系统提出了他的原始问题，那里知之甚少。凭借我们在平滑案例中的知识，我建议从更简单的模型开始解决分析案例。 2。不稳定轨道的随机描述。术语“ Arnold扩散”之所以创造，是因为基于数值证据，不稳定的行为表现出像扩散轨道一样。我建议证明随机限制定理，以证明扩散方面的合理性。从哲学上讲，这与“确定性随机性”的概念相吻合，其中随机行为可以从完全确定的系统中出现。该研究将在两个方向上进行：在诸如先验不稳定模型之类的不稳定性模型中，一个人可以嵌入正常的双曲线层压板，在该模型中，动力学与随机动力学系统相连；在缓慢的快速系统中，可以证明限制扩散过程。 通过提出的研究，我们还提高了基础理论的知识，特别是弱KAM理论。这些结果将应用于相关领域，例如常规和随机的汉密尔顿 - 雅各布方程。