Mathematical Sciences: From Tori to Cantori: Symplectic Mappings

数学科学：从 Tori 到 Cantori：辛映射

• 批准号: 9305847
• 负责人: James Meiss
• 金额: $ 6.8万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-15 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9305847&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Tori; Cantori; Symplectic

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. Typically 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. ***

9305847 Meiss 四维及更高维辛映射的动力学对于理解保守物理系统中的稳定性和混沌至关重要。 在本提案中，将结合使用数值和分析技术。 我们建议通过使用环面的傅里叶级数的递归生成以及通过从反可积极限延续康托集来确定不变环面的存在域。 目标是开发估计实际稳定性边界和研究向混沌行为转变的方法。 计算将确定各种频率向量的环面的鲁棒性，从而产生在二维中提供最鲁棒频率的高贵数的概括。 还将对一维共振环面进行研究——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。 四个维度的运输。将通过各种同伦类型圆柱体的退出时间分解的数值计算来研究。 我们的目标是开发捕获区域和共振区域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传输的新技术。 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，其动机是理解“输运”问题。这在粒子加速器约束、化学反应速率、流体混合、磁聚变中的等离子体约束等领域至关重要。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）演化到另一种状态（例如光束撞击隧道壁），以及这需要多长时间。 通常，轨迹必须穿过奇异的结构，例如康托集和自相似分形，其中一些表现出显着的“粘性”，以便在相空间中移动。 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。 一个主要问题是，感兴趣的系统对应于四个或更高维度的空间——我们普通的三维直觉失败了。 在各种应用中，要么鼓励传输（加快反应速率），要么阻止传输（限制颗粒）；我们将研究完成这两项任务的技术。 ***