Chaos and Bifurcations in Volume-Preserving Dynamics

体积保持动力学中的混沌和分岔

• 批准号: 0707659
• 负责人: James Meiss
• 金额: $ 51.15万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707659&HistoricalAwards=false
• 关键词: Chaos; Bifurcations; Volume; Preserving; Dynamics

AbstractRegular, quasiperiodic motion is ubiquitous in dynamical systems with sufficient symmetry. A prominent example occurs in the Hamiltonian or symplectic case, where these "invariant tori" persist---even for nearly-integrable motion, as is explained by "KAM theory." The destruction of tori in the two-dimensional case is explained by Aubry-Mather theory and renormalization results. However, a concomitant understanding of the destruction of tori upon perturbation in higher dimensions has proved elusive. In this proposal, the implications of integrability, due to symmetries and invariants, of volume-preserving dynamics will be investigated. The loss of integrability under perturbation will be studied by a combination of analytical (Aubry's anti-integrable limit, Fourier series) and numerical (invariant manifold and continuation) techniques. Tori are both created and destroyed by bifurcations, and a study of the normal forms for codimension-one and two bifurcations of fixed points will lead to classification possible phenomena. Transport will be investigated numerically with the goal of developing analytical measures of flux and transport distributions. In a second project, the PI will investigate bifurcations in nonsmooth systems appropriate to the modeling of chemical reactions, the systematic simplification of these systems by center manifold reduction, as well as the study of transport caused by weak coupling of chaotic motion to regular motion.Conservative dynamical models are used in designing particle accelerators, obtaining rates for simple chemical reactions, calculating confinement times in plasma fusion devices, understanding the spectra of highly excited atomic systems, and designing efficient spacecraft trajectories. Dynamics in such systems is often chaotic and prediction of individual trajectories is difficult; nevertheless, chaos can be profitably utilized, for example, to improve efficiency of spacecraft trajectories, by judiciously applying small course corrections, or to enhance the lifetimes of particles in confinement devices and the rates of chemical reactions. Volume-preserving dynamics models the flow of incompressible fluids and magnetic fields and a quantitative understanding of chaos in these systems is crucial for the development of efficient mixing in microscale bioreactors as well as of predictive planetary scale weather models. Most of our current theoretical understanding is limited to the two-dimensional case that is appropriate for flows in rapidly rotating or thin layers of fluid. While this has been useful in the understanding of such phenomena as the trapping of nutrients in gulf stream rings, the formation of the ozone hole and the creation of vortex-induced mixing in sinuous tubes, even in these systems, three-dimensional, chaos-induced transport needs to be understood. The PI seeks to develop analytical and computational methods for the study of regular and chaotic volume-preserving motion both to contribute broadly to our fundamental understanding of the richness of the behavior of low-dimensional deterministic evolution, and, to relate it to mixing and transport.

摘要在具有充分对称性的动力系统中，规则的准周期运动是普遍存在的。一个突出的例子出现在哈密顿或辛的情况下，其中这些“不变环面”仍然存在-甚至对于近乎可积的运动，正如“KAM理论”所解释的那样。用Aubry-Mather理论和重整化结果解释了二维环的破坏。然而，对环面在高维微扰下的破坏的理解已被证明是难以捉摸的。在这个方案中，将研究由于对称性和不变量而导致的体积保持动力学的可积性的含义。微扰下的可积性损失将通过分析(Aubry反可积极限，傅立叶级数)和数值(不变流形和连续)技术的组合来研究。环面既有由分叉产生的，也有由分叉破坏的，对不动点的余维一分叉和二分叉的范式的研究将导致分类可能的现象。将对输送进行数值研究，目的是发展通量和输送分布的分析措施。在第二个项目中，PI将研究适合于化学反应建模的非光滑系统中的分叉，通过中心流形缩减对这些系统的系统简化，以及研究混沌运动与规则运动的弱耦合引起的输运。保守的动力学模型被用于设计粒子加速器，获得简单化学反应的速率，计算等离子体聚变设备中的限制时间，了解高激发原子系统的光谱，以及设计有效的航天器轨迹。这类系统中的动力学往往是混乱的，很难预测个别轨迹；然而，混沌可以被有益地利用，例如，通过明智地应用小航向校正来提高航天器轨迹的效率，或延长限制装置中粒子的寿命和化学反应的速率。体积守恒动力学模拟了不可压缩流体和磁场的流动，对这些系统中混沌的定量理解对于发展微尺度生物反应器中的有效混合以及预测行星尺度的天气模式至关重要。我们目前的大多数理论理解仅限于二维情况，适用于快速旋转或薄层流体中的流动。虽然这有助于理解湾流环中营养物质的捕获、臭氧空洞的形成和弯曲管中涡旋诱导的混合等现象，但即使在这些系统中，也需要理解三维的、由混沌引起的传输。PI试图发展分析和计算方法来研究规则和混沌的体积保持运动，这既有助于我们对低维确定性演化行为的丰富性的基本理解，也有助于将其与混合和运输联系起来。