Geometry and Computation of Dynamics for Conservative Systems

保守系统的几何和动力学计算

• 批准号: 0202032
• 负责人: James Meiss
• 金额: $ 24.5万
• 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: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2008-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202032&HistoricalAwards=false
• 关键词: Geometry; Computation; Dynamics; Conservative; Systems

Proposal #0202032PI: J.D. MeissInstitution: University of Colorado BoulderTitle: Geometry and Computation of Dynamics for Conservative SystemsABSTRACTThe principal investigator proposes to study the geometry of low-dimensional dynamical systems, especially symplectic and volume-preserving maps, using both computational and analytical techniques. While much is known about the two-dimensional case, there are still many questions about the onset and development of chaos for three- and higher-dimensional systems. While most oscillators are anharmonic (have twist), twistless bifurcations occur in one-parameter families of these systems. In the proposal, the geometry of twistless bifurcations will be studied leading to an understanding of fold and cusp bifurcations in the twist. The resulting geometry of the reconnection of resonances and exotic twistless tori will be studied numerically. These should play a role in limiting the stability domains for many dynamical systems. From the other side, the destruction of chaos can be profitably studied using a limit of extreme chaos, the anti-integrable (AI) limit as a starting point. In this proposal, the PI will use the AI limit to study coupled systems of maps and chaotic boundaries. Near this limit, structures such as exotic versions of the Smale horseshoe, and other heteroclinic tangles should occur. The onset of chaos in conservative systems is signaled by the destruction of tori. These have been studied by a rescaling analysis called the renormalization transformation. The structure of this transformation for four and higher dimensional systems is only beginning to be understood. The PI proposes that recent approximate versions of this transformation will give effective numerical strategies for finding the destruction and analyzing the topology of the resulting objects.Developing an understanding of the dynamics of conservative systems is important to applications including the design of particle accelerators, obtaining rates for simple chemical reactions, calculating confinement times for charged particles in plasma fusion devices, understanding the spectra of highly excited atomic systems, and designing efficient spacecraft trajectories in an era of lower budgets. Dynamics is such systems is often chaotic, which implies that prediction of specific trajectories is difficult; however, chaos can be profitably utilized to improve efficiency, for example of spacecraft trajectories, by judiciously applying small course corrections. Chaos can also dramatically affect the lifetimes of particles in confinement devices and the rates of chemical reactions. The PI proposes to develop geometrical and computational techniques that can be used to address these questions. In addition extending our understanding of chaos to higher dimensional cases will help populate the zoo of chaotic objects in multidimensional systems.

提案#0202032PI：J.D. Meiss机构：科罗拉多大学博尔德分校标题：保守系统动力学的几何和计算摘要主要研究者建议使用计算和分析技术来研究低维动力系统的几何形状，特别是辛和体积保持映射。 尽管人们对二维情况了解很多，但关于三维及更高维系统混沌的发生和发展仍然存在许多问题。 虽然大多数振荡器是非谐波的（有扭曲），但在这些系统的单参数族中会出现无扭曲分岔。 在该提案中，将研究无扭曲分叉的几何形状，从而了解扭曲中的折叠和尖点分叉。 将对共振重新连接和奇异的无扭转环面所产生的几何形状进行数值研究。 这些应该在限制许多动力系统的稳定性域方面发挥作用。 从另一方面来说，使用极端混沌的极限，即反可积（AI）极限作为起点，可以对混沌的破坏进行有益的研究。 在该提案中，PI将使用AI极限来研究地图和混沌边界的耦合系统。 接近这个极限时，应该会出现诸如奇异版本的 Smale 马蹄铁和其他异宿缠结之类的结构。 保守系统中混乱的开始是以托里的毁灭为标志的。 这些已经通过称为重正化变换的重新缩放分析进行了研究。 四维及更高维系统的这种变换结构才刚刚开始被理解。 PI 提出，这种变换的最新近似版本将为寻找破坏和分析所得物体的拓扑结构提供有效的数值策略。了解保守系统的动力学对于粒子加速器的设计、获得简单化学反应的速率、计算等离子体聚变装置中带电粒子的限制时间、了解高度激发原子系统的光谱以及设计高效的应用非常重要。 预算较低时代的航天器轨迹。 此类系统的动力学通常是混沌的，这意味着很难预测特定的轨迹；然而，通过明智地应用小航向修正，可以有效地利用混沌来提高效率，例如航天器轨迹。 混沌还会极大地影响限制装置中粒子的寿命和化学反应的速率。 PI 建议开发可用于解决这些问题的几何和计算技术。 此外，将我们对混沌的理解扩展到更高维度的情况将有助于填充多维系统中的混沌对象动物园。