Fine Structure in Hamiltonian Systems

哈密​​顿系统在数学物理学中发挥着重要作用，它们被用来模拟各种物理系统的行为，从半导体中的电子到星系中恒星和行星的运动。 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可以平滑和隔离精细结构。 相反，哈密顿系统的演化是由平衡解、周期轨道和不变环面等几何对象组织的。该研究项目结合使用计算和分析工具，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，提供了证明有关模型属性的数学定理的新技术，以及理解其实际行为的新计算技术。 这些想法用于回答天体力学中的突出问题，并以比以往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/发现太阳系空间飞行的新低能转移轨道。该项目引入了研究非线性系统中相空间结构的新分析和计算方法。强调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。该项目有两个主要部分。第一部分是局部的，扩展了用于计算双曲不变对象的现有高阶技术以涵盖抛物线情况。第二部分更加全局化，开发了用于增长不变流形图集的计算方法，该图集由稳定/不稳定流形的高阶图表映射组成。一个重要的应用是计算这些对象之间的交集，并讨论了有效的搜索策略。这些交叉点（或网络）在组织哈密顿系统中发挥着关键作用，因为它们决定相空间不同区域之间的传输并产生复杂的循环动力学。 该项目涉及研究生研究活动，并促进复兴理论和计算动力学领域的研究人员之间的合作。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值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.