Mathematical Sciences: Transition to Chaos in Multidimensional Hamiltonian Systems

小行星9305847 四维和高维辛映射的动力学对于理解保守物理系统的稳定性和混沌具有重要意义。在本提案中，将使用数值和分析技术相结合的方法。我们建议使用递归生成的傅立叶级数的环面，并继续康托集的反可积极限，以确定域的存在不变环面。其目标是开发用于估计实际稳定性边界和用于调查向混沌行为的过渡的方法。 计算将确定各种频率向量的环面的鲁棒性，从而导致在二维中提供最鲁棒频率的高贵数的泛化。 一个一维的研究，共振环面也将被证明，这些可能是更持久的比两个环面，并形成一个重要的组成部分的障碍，运输。四维交通。将研究的出口时间分解的各种同伦类型的圆柱的数值计算。我们的目标是发展一个几何描述的捕获区域和共振区和一个椭圆点周围的实际稳定域的表征。新的技术控制的运输将开发辛系统。 所有的物理学基本方程都可以表述为哈密顿动力系统。我们建议研究这些系统的轨道结构，其动机是为了理解“运输”问题。“这在粒子加速器约束、化学反应速率、流体混合、磁聚变装置中的等离子体约束、小行星和行星环稳定性等领域至关重要，基本问题是：系统如何从一种状态（例如加速器中的约束束）演变到另一种状态（例如束撞击隧道壁），以及这需要多长时间。典型的轨迹必须通过奇特的结构，如康托集和自相似分形，其中一些表现出显着的“粘性”，以通过相空间移动。这些结构的构建和可视化需要在数学洞察力的指导下进行仔细的计算机研究。一个主要的问题是，感兴趣的系统对应于四维和更高维的空间-我们普通的三维直觉失败。在各种应用中，运输要么被鼓励（加速反应速率），要么被劝阻（限制粒子）;我们将研究完成这两项任务的技术。 ***

9305847 Meiss The dynamics of four and higher dimensional symplectic mappings is of fundamental importance to understanding stability and chaos in conservative physical systems. In this proposal a combination of numerical and analytical techniques will be used. We propose to determine the domain of existence of invariant tori both by using recursive generation of the Fourier series for the tori, and by continuation of the Cantor sets from the anti-integrable limit. The goal is to develop methods for estimating practical stability boundaries and for investigating the transition to chaotic behavior. Computations will determine the robustness of the tori of various frequency vectors, leading to a generalization of the noble numbers that provide the most robust frequencies in two dimensions. A study of one dimensional, resonant tori will also be undertaken-these may be more persistent than two-tori, and form an important component of the barriers to transport. Transport in four dimensions. will be studied by numerical computation of exit time decompositions for cylinders of various homotopy types. Our goal is the development of a geometrical description of trapping regions and resonance zones and a characterization of the practical stability domain around an elliptic point. New techniques for control of transport will be developed for symplectic systems. All of the fundamental equations of physics are formulated as Hamiltonian dynamical systems. We propose to study the structure of the orbits of these systems with the motivation being to understand the problem of "transport." This is of primary importance in such areas as particle accelerator confinement, chemical reaction rates, fluid mixing, plasma confinement in magnetic fusion devices, asteroid and planetary ring stability, etc. The basic question is: how does a system evolve from one state (e.g. a confined beam in an accelerator), to another (e.g. beam hits the tunnel wall), and how long does this take. Typi cally trajectories must wend their way through exotic structures such as Cantor sets and self-similar fractals, some of which exhibit a remarkable "stickiness", in order to move through the phase space. The construction and visualization of these structures requires careful computer study guided by mathematical insight. A major problem is that the systems of interest correspond to four and higher dimensional spaces--our ordinary three-dimensional intuition fails. In various applications transport is either to be encouraged (speeding up reaction rates) or discouraged (confining particles); we will investigate techniques for accomplishing both tasks. ***