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.

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。 一个突出的例子发生在哈密顿或辛的情况下，这些“不变环面”持续存在-甚至对于接近可积的运动，正如“KAM理论”所解释的那样。“二维情况下环面的破坏可以用Aubry-Mather理论和重整化结果来解释。 然而，在更高的维度上，对扰动后环面的破坏的理解已经被证明是难以捉摸的。在这个建议中，可积性的影响，由于对称性和不变量，体积保持动力学将被调查。 扰动下的可积性损失将研究分析（奥布里的反可积极限，傅立叶级数）和数值（不变流形和延续）技术的组合。 Tori是由分歧产生和破坏的，研究余维1和余维2不动点分歧的规范形将导致可能的分类现象。 运输将进行数值研究，目的是制定通量和运输分布的分析措施。 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，通过中心流形简化这些系统的系统简化，以及研究由混沌运动与规则运动的弱耦合引起的输运。保守动力学模型用于设计粒子加速器，获得简单化学反应的速率，计算等离子体聚变装置中的约束时间，理解高度激发的原子系统的光谱，以及设计有效的航天器轨道。 在这样的系统中的动力学通常是混乱的，预测单个轨迹是困难的;然而，混沌可以被有益地利用，例如，通过明智地应用小的航向修正来提高航天器轨迹的效率，或者提高约束装置中粒子的寿命和化学反应的速率。 体积保持动力学模型的流动不可压缩的流体和磁场和混沌在这些系统中的定量理解是至关重要的发展有效的混合在微型生物反应器以及预测行星尺度的天气模型。 我们目前的理论认识大多局限于二维情况下，是适合于快速旋转或薄层流体流动。 虽然这对理解诸如墨西哥湾环流圈中营养物质的捕获、臭氧洞的形成和弯曲管道中涡旋诱导混合的产生等现象很有用，但即使在这些系统中，也需要理解三维混沌诱导的运输。 PI旨在开发分析和计算方法，用于研究规则和混沌保体积运动，以广泛地促进我们对低维确定性演化行为的丰富性的基本理解，并将其与混合和运输联系起来。