RUI: Hamiltonian Instability

本研究致力于动力系统，重点研究阿诺德不稳定性和哈密顿系统中的混沌现象。阿诺德在1964年描述了哈密顿系统不稳定性的一个典型例子：它代表了一个简单的机械系统，由一个钟摆和两个弱耦合的旋转器组成，其轨迹从相空间的出发点“疯狂地”和“任意地”徘徊。这个例子引起了一个猜想，称为阿诺德不稳定性，指出这种行为发生在相当一般的系统中。本项目的第一个目标是在一些一般模型上研究这一猜想，并确定各种不稳定的几何和拓扑机制。将考虑可积哈密顿系统的摄动。在一定的非退化假设下，存在具有不变流形的金环族；不变流形产生了附近环面之间的连接轨道，但也出现了分离的间隙。主要目标是找到扩散轨迹，它沿着连接这些环面的连接处移动，也穿过间隙，当扰动的大小趋于零时，传播一个均匀的距离。其中一些轨迹应该能够产生混沌的漂移。本研究的方法学将基于正确排列窗口的拓扑技术，并辅以几何近似理论和变分方法。本研究的主要目标是克服大间隙问题，获得扩散时间的最优估计，放宽系统的横截性和非退化假设，研究大扰动下扩散的存在性。该项目的第二个目标是将从阿诺德不稳定性研究中获得的知识应用于三体问题。目标是了解产生混沌运动的几何机制，并在规定的行程中找到最佳轨迹。这个项目的目标将在理论上和数值上得到解决。本研究的一般背景是研究周期性地施加于稳定系统的小扰动的累积效应。这项研究源于一个可以追溯到牛顿、拉格朗日和拉普拉斯时代的问题，即在遥远的未来，太阳系是会保持现在的形态，还是会发生灾难性的变化。如果在一些受扰动的系统中，比如太阳系，稳定性占主导地位，那么在其他一些系统中，不稳定性则是典型的。阿诺德不稳定性可以看作是如何用小周期力增加物理系统能量的一个配方。应用于三体问题的阿诺德不稳定性思想在航天器动力学中有潜在的应用，在设计探索太阳系各个区域的节能太空任务中。在这个项目中概述的技术也可以用于研究双星系统的行星动力学，以及紧密双星系统的质量传递（例如英仙座斗牛座的Algol）。其他可能的应用包括心脏起搏器、等离子体约束和加速器物理。此外，该项目将提高学生和K-12教育工作者的知识和专业发展。它将直接吸引学生，包括代表性不足群体的成员，参与研究活动和数学教育。

This research is devoted to dynamical systems, with emphasis on Arnold instability and chaotic phenomena in Hamiltonian systems. A quintessential example of instability in Hamiltonian systems was described by Arnold in 1964: it represents a simple mechanical system, consisting in a pendulum and two rotators with a weak coupling, whose trajectories wander `wildly' and `arbitrarily far' from their points of departure in the phase space. This example gave rise to a conjecture, referred as Arnold instability, stating that such a behavior occurs in rather general systems. The first objective of this project is to investigate this conjecture on some general models and identify various geometric and topological mechanisms of instability. Perturbations of integrable Hamiltonian systems will be considered. Under certain non-degeneracy assumptions, there exist families of KAM tori that posses invariant manifolds; the invariant manifolds give rise to connecting orbits between nearby tori, but separating gaps also appear. The main goal is to find diffusing trajectories that move along the connections that link these tori and also move across the gaps, traveling a uniform distance as the size of the perturbation tends to zero. Some of these trajectories should be able to make chaotic excursions. The methodology of this research will be based on a topological technique of correctly aligned windows, reinforced with geometric approximation theory and variational methods. The main goals of this research are to overcome the large gap problem, obtain optimal estimates on the diffusion time, relax the transversality and non-degeneracy assumptions on the system, and study the existence of diffusion for large perturbations. The second objective of this project is to apply the knowledge gained from the study of Arnold instability to the three-body problem. The goal is to understand the geometric mechanisms that produce chaotic motions and find optimal trajectories with prescribed itineraries. The objectives of this project will be addressed both theoretically and numerically.The general context of this research is the study of the cumulative effect of small perturbations applied periodically to a stable system. This study originated from a question that goes back to Netwon, Lagrange and Laplace, whether the Solar System in the distant future will keep the same form as it is now, or will undergo a catastrophic change. If in some perturbed systems, like the Solar System, stability is predominant, in some other systems instability is typical. The Arnold instability can be viewed as a recipe on how to increase the energy of physical systems with small periodic forcing. Arnold instability ideas applied to the three-body problem have potential applications to spacecraft dynamics, in designing fuel efficient space missions exploring various regions of the Solar System. The techniques outlined in this project can also be used in studying the dynamics of planets orbiting systems of binary stars, and of mass transfers in tight binary star systems (e.g. Algol in Betta Persei). Other possible applications include heart pacemaking, plasma confinement, and accelerator physics. Additionally, this project will enhance the knowledge and professional development of students and K-12 educators. It will directly engage students, including members of underrepresented groups, in research activities and mathematics education.