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。Meissinstitution：科罗拉多大学Bouldertitle大学：保守系统的几何和动力学计算，用于研究主要研究者建议研究低维动力学系统的几何形状，尤其是使用计算机和分析技术的几何形状，尤其是符号和数量的图像。 尽管对二维案例有很多了解，但关于三维和高维系统的混乱的发作和发展仍然存在很多问题。 尽管大多数振荡器都是无声的（具有扭曲），但这些系统的单参数家族中发生了无扭曲的分叉。 在提案中，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叉分叉。 将通过数值研究共同点和异国无扭曲的托里的重新连接的几何形状。 这些应该在限制许多动态系统的稳定域中发挥作用。 从另一端，可以使用极限混乱的极限来盈利，这是抗积分（AI）的限制作为起点。 在此提案中，PI将使用AI极限研究地图和混乱边界的耦合系统。 接近此极限，应该发生诸如Smale Horseshoe的异国版本和其他杂斜纹缠结的结构。 保守系统中混乱的发作是通过托里的破坏来表示的。 这些已通过称为重新归一化转化的重新分析进行了研究。 对于四维系统的这种转换的结构才开始理解。 PI提出，这种转换的最新近似版本将提供有效的数值策略，以查找破坏和分析所得对象的拓扑结构。开发对保守系统动态的理解对于应用程序的应用很重要，包括粒子加速器的设计，获取速率，获取速率，从而获得速率。对于简单的化学反应，计算血浆融合设备中带电的颗粒的限制时间，了解高度预算的时代，了解高度激发的原子系统的光谱以及设计有效的航天器轨迹。 动力学是这种系统通常是混乱的，这意味着对特定轨迹的预测很困难。但是，通过明智地采用小型课程校正，可以利用混乱来提高效率，例如航天器轨迹。 混乱还会显着影响限制装置中颗粒的寿命和化学反应速率。 PI建议开发几何和计算技术，可用于解决这些问题。 此外，将我们对混乱的理解扩展到更高维度的情况将有助于在多维系统中填充混乱对象的动物园。