Chaos and Bifurcations in Volume-Preserving Dynamics

正则、准周期运动在具有充分对称性的动力系统中普遍存在。一个突出的例子出现在哈密顿或辛的情况下，这些“不变环面”持续存在——甚至对于几乎可积的运动，正如“KAM理论”所解释的那样。用奥布里-马瑟理论和重整化结果解释了二维情况下环面的破坏。然而，对高维扰动下环面破坏的相关理解已被证明是难以捉摸的。在本建议中，由于对称性和不变量，将研究保体积动力学的可积性的含义。在摄动下的可积性损失将通过解析（奥布里的反可积极限，傅立叶级数）和数值（不变流形和延拓）技术的结合来研究。环面既有分岔的产生，也有分岔的破坏，对不动点的共维分岔和二维分岔的正规形式的研究将导致可能的分类现象。将对输运进行数值研究，目的是制定通量和输运分布的分析措施。在第二个项目中，PI将研究适合化学反应建模的非光滑系统中的分岔，通过中心流形约简对这些系统进行系统简化，以及研究混沌运动与规则运动弱耦合引起的输运。保守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，计算等离子体聚变装置的约束时间，理解高激发原子系统的光谱，以及设计有效的航天器轨迹。这类系统的动力学通常是混沌的，很难预测单个轨迹；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、不可压缩流体和磁场的流动以及对这些系统中的混沌的定量理解对于开发微尺度生物反应器中的有效混合以及预测行星尺度天气模型至关重要。我们目前的大多数理论认识都局限于二维情况，这适用于快速旋转或薄层流体的流动。虽然这在理解诸如墨西哥湾流环中营养物质的捕获、臭氧空洞的形成和弯曲管道中涡诱导混合的产生等现象方面是有用的，但即使在这些系统中，三维的、混沌诱导的运输也需要理解。PI寻求发展规则和混沌保体积运动研究的分析和计算方法，以广泛地促进我们对低维确定性演化行为丰丰性的基本理解，并将其与混合和输运联系起来。

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.