Mathematical Sciences: Transition to Chaos in Multidimensional Hamiltonian Systems

四维及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。在这个建议中，数值和分析技术的结合将被使用。我们提出用环面傅里叶级数的递推生成和从反积极限出发的康托集的延拓来确定不变环面存在的定义域。目标是发展估计实际稳定边界和研究过渡到混沌行为的方法。计算将决定各种频率矢量的环面鲁棒性，导致在二维中提供最鲁棒频率的高尚数字的泛化。对一维共振环面也将进行研究——它们可能比二维环面更持久，并形成运输障碍的重要组成部分。四维运输。通过数值计算研究了不同同伦类型柱体的出口时间分解。我们的目标是发展捕获区和共振区的几何描述，以及椭圆点周围实际稳定域的表征。将为辛系统开发控制运输的新技术。物理学的所有基本方程都被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（例如加速器中的受限光束）进化到另一种状态（例如光束撞击隧道壁），以及这需要多长时间。为了在相空间中移动，轨迹通常必须穿过像康托集和自相似分形这样的奇异结构，其中一些结构表现出显著的“粘性”。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或更高维度的空间——我们普通的三维直觉就失效了。在各种应用中，要么鼓励输送（加快反应速率），要么阻止输送（限制颗粒）；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. ***