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四个和更高维符号映射的动力学对于理解保守物理系统中的稳定性和混乱至关重要。 在此提案中，将使用数值和分析技术的组合。 我们建议通过使用用于Tori的傅立叶级数的递归生成以及从反整合限制的cantor组来确定不变的托里的存在域。 目的是开发估计实际稳定边界并调查过渡到混乱行为的方法。 计算将确定各种频率向量的Tori的鲁棒性，从而导致贵族数字的概括，这些数字在二维中提供了最强大的频率。 对一维，共振的托里的研究也将进行 - 这些可能比两翼更持久，并且构成了运输障碍的重要组成部分。 运输四个维度。通过计算出口时间分解的数值计算各种同型类型的圆柱体的数值计算。 我们的目标是开发捕获区域和共振区域的几何描述，以及绕椭圆点周围实际稳定域的表征。 将为符号系统开发用于控制运输的新技术。 物理学的所有基本方程式均均为哈密顿动力系统。 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.梁撞到隧道墙），这需要多长时间。 通常，轨迹必须穿越诸如cantor套件和自相似分形等异国情调结构，其中一些表现出了非凡的“粘性”，以便在相位空间中移动。 这些结构的构建和可视化需要以数学见解为指导的仔细计算机研究。 一个主要的问题是，感兴趣的系统对应于四个和更高的空间 - 我们普通的三维直觉失败了。 在各种应用中，要么鼓励运输（加快反应速率）或灰心（限制颗粒）；我们将研究完成这两个任务的技术。 ***