Fine Structure in Hamiltonian Systems

哈密顿系统在数学物理学中起着至关重要的作用，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几何对象组织的，如平衡解，周期轨道和不变环面。这个研究项目提供了新的见解精细结构的哈密顿系统使用的计算和分析工具的混合。 事实上，该项目以新颖的方式结合了计算和分析技术，既提供了证明有关模型属性的数学定理的新技术，又提供了理解其实际行为的新计算技术。 这些想法被用来回答天体力学中悬而未决的问题，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。 该研究的明确应用包括设计/发现太阳系空间飞行的新的低能量转移轨道。该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。应用程序的哈密顿系统强调，在这种情况下，没有吸引子组织的动态。该项目有两个主要部分。第一部分是本地和扩展现有的高阶技术计算双曲不变的对象，包括抛物线的情况下。第二部分是更全球性的，并制定计算方法的增长不变的流形图集组成的高阶图表映射稳定/不稳定的流形。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，并产生复杂的循环动力学。 该项目涉及研究生的研究活动，并促进研究人员在复苏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知识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.