Fine Structure in Hamiltonian Systems

哈密顿系统在数学物理中扮演着至关重要的角色，它们被用来对从半导体中的电子到星系中恒星和行星运动的各种物理系统的行为进行建模。这些系统的一个主要困难是，由于没有摩擦，模型中没有任何东西来平滑和隔离精细结构。相反，哈密顿系统的演化是由几何对象组织的，如平衡解、周期轨道和不变环面。这项研究项目结合了计算和分析工具，为哈密顿系统的精细结构提供了新的见解。事实上，该项目以新颖的方式结合了计算和分析技术，既提供了证明关于模型性质的数学定理的新技术，也提供了理解模型实际行为的新计算技术。这些想法被用来回答天体力学中的悬而未决的问题，并以比以往任何时候都更精细的分辨率探索粒子引力系统的行为。这项研究的具体应用包括设计/发现用于太阳系空间飞行的新的低能量转移轨道。本项目介绍了研究非线性系统相空间结构的新的分析和计算方法。强调了对哈密顿系统的应用，因为在这种情况下没有吸引子来组织动力学。该项目包括两个主要部分。第一部分是局部的，扩展了现有的计算双曲不变对象的高阶技术，以涵盖抛物线情况。第二部分更具全局性，发展了由稳定/不稳定流形的高阶图映射组成的增长不变流形地图集的计算方法。一个重要的应用是计算这类对象之间的交集，并讨论了有效的搜索策略。这些交点(或网络)在组织哈密顿系统方面起着关键作用，因为它们决定了相空间不同区域之间的传输，并产生了复杂的循环动力学。该项目涉及研究生研究活动，并促进研究人员在复兴理论和计算动力学领域的合作。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Hamiltonian systems play an essential role in mathematical physics, where they are used to model the behavior of a wide variety of physical systems ranging from electrons in semiconductors to the motions of stars and planets in galaxies. One major difficulty with these systems is that, since there is no friction, there is nothing in the model to smooth out and isolate fine structure. Instead, the evolution of a Hamiltonian system is organized by geometric objects like equilibrium solutions, periodic orbits, and invariant tori. This research project provides new insights into the fine structure of Hamiltonian systems using a blend of computational and analytical tools. Indeed, the project combines computational and analytical techniques in novel ways, providing both new techniques for proving mathematical theorems about the properties of the models, and new computational techniques for understanding their practical behavior. These ideas are used to answer outstanding questions in Celestial Mechanics, and to explore the behavior of gravitating systems of particles at a finer resolution than ever before. Explicit applications of this research include the design/discovery of new low energy transfer orbits for space flights in the solar system.This project introduces new analytical and computational methods for studying phase space structure in nonlinear systems. Applications to Hamiltonian systems are stressed, as in this case there are no attractors to organize the dynamics. The project has two main parts. The first part is local and expands existing high order techniques for computing hyperbolic invariant objects to encompass the parabolic case. The second part is more global and develops computational methods for growing invariant manifold atlases comprised of high order chart maps for stable/unstable manifolds. An important application is the computation of intersections between such objects, and efficient search strategies are discussed. These intersections (or webs) play a critical role in organizing Hamiltonian systems as they determine transport between different regions of phase space and generate complicated recurrent dynamics. The project involves graduate research activities and facilitates collaborations between researchers in the areas of resurgence theory and computational dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.