Geometry and Computation of Dynamics for Conservative Systems

提案#0202032PI：J.D.迈斯研究所：科罗拉多大学巨石学院题目：保守系统的几何和动力学计算摘要主要研究人员建议使用计算和分析技术来研究低维动力系统的几何，特别是辛和保体映射。虽然人们对二维情况已有了很多了解，但关于三维和更高维系统的混沌的发生和发展仍有许多问题。虽然大多数振子是非谐的(有扭转)，但在这些系统的单参数族中发生了无扭转分叉。在该提案中，将研究无扭分叉的几何形状，从而了解扭结中的折叠和尖端分叉。我们将用数值方法研究共振重联和奇异的无扭圆环的几何结构。这些都应该在限制许多动力系统的稳定域方面发挥作用。另一方面，使用极端混沌的极限，即反可积(AI)极限作为起点，可以有利可图地研究混沌的破坏。在这项提议中，PI将使用AI极限来研究地图和混沌边界的耦合系统。在这一极限附近，应该会出现结构，如奇异版本的斯梅尔马蹄，以及其他异位纠缠。保守系统中混沌的开始是由环面的毁灭发出的信号。这些已经被称为重整化变换的重标度分析研究过了。对于四维和更高维系统的这种转换的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给出有效的数值策略来寻找破坏和分析产生的物体的拓扑。发展对保守系统动力学的理解对于包括粒子加速器的设计、获得简单化学反应的速率、计算等离子体聚变设备中带电粒子的限制时间、理解高激发原子系统的光谱以及在预算较低的时代设计高效的航天器轨迹等应用非常重要。动力学是这样的系统往往是混沌的，这意味着预测具体的轨迹是困难的；然而，通过明智地应用小的航向校正，可以有益地利用混沌来提高效率，例如航天器轨迹的效率。混乱还会极大地影响限制装置中粒子的寿命和化学反应的速度。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的情况，将有助于在多维系统中填充混沌对象的动物园。

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.