Dynamics From Forced Symmetry-Breaking

具有对称性的系统在应用中经常出现。这样的系统表现出非对称系统的典型行为。尽管这种行为可能比预期的更复杂，但来自等变分岔理论的技术允许人们分析它的起源和性质。然而，在许多应用中，对称的存在是一种理想化。因此，了解对称系统动力学的哪些方面相对于小的对称破缺缺陷是健壮的是很重要的。在许多情况下，离散和连续对称性的丧失会在动力学中引入全局分岔，从而可能出现混沌时间依赖性。本项目提出了在连续介质力学（流体流动，化学反应系统等）中产生的几种重要系统中对这种对称性破坏扰动的研究，重点是识别接近初级不稳定阈值和远离初级不稳定阈值的鲁棒行为。前者通过中心流形约简得到具有破对称的低维系统，而后者则需要研究偏微分方程。特别强调的是理解同时打破反射对称和平移不变性的影响（分别和同时）。还将研究由于参数强迫而导致时平移不变性的破坏和伽利略不变性的丧失所引起的现象。该项目的目标是深入了解理想化模型适用性的局限性，并提供典型的“近对称”系统行为类型的描述。该项目旨在了解稳健行为背后的基本原理。例如，在研究仪器对水平的小倾斜的影响时，理解反射对称性破坏的影响是必不可少的，以及从一种状态到另一种状态的转换的通流或旋转的影响。这种类型的问题出现在地球物理学（例如海洋学）的微观情况下（水平或倾斜基底上的薄膜）以及两者之间的任何地方。同样，理解空间不均匀性对波状扰动的捕获不仅在催化中具有重要意义，其中不均匀性可能在原子尺度上，而且在地形对天气模式的捕获中也具有重要意义。

Systems with symmetries arise frequently in applications. Such systems exhibit behaviorthat is not typical of nonsymmetric systems. Even though this behavior may be more complexthan expected techniques from equivariant bifurcation theory permit one to analyze itsorigin and properties. However, in many applications the presence of symmetry is anidealization. It is important therefore to understand which aspects of the dynamics of thesymmetric system are robust with respect to small symmetry-breaking imperfections. In manycases the loss of both discrete and continuous symmetries introduces global bifurcationsinto the dynamics and hence the possibility of chaotic time-dependence. This projectproposes a study of such symmetry-breaking perturbations in several important classes ofsystems arising incontinuum mechanics (fluid flow, chemically reacting systems etc) withemphasis on identifying robust behavior both near threshold of a primary instability andfurther away from it. The former gives rise to low-dimensional systems with brokensymmetry via center manifold reduction, while the latter require a study of partialdifferential equations. Special emphasis is placed on understanding the effects ofbreaking both reflection symmetry and translation invariance (both separately andsimultaneously). Phenomena arising from the breaking of time-translation invariance due toparametric forcing and the loss of Galilean invariance will also be studied. The goal ofthe project is to gain a deep understanding of the limitations on the applicability ofidealized models and to provide a description of the type of behavior that is typical of''nearly-symmetric'' systems.The project aims at understanding the basic principles behind robust behavior. Forexample, understanding the effects of broken reflection symmetry is essential whenstudying the effects of a small inclination of an apparatus to the horizontal, and theeffects of through-flow or rotation on transitions from one state to another. Problems ofthis type arise in microscopic situations (thin films on horizontal or inclinedsubstrates) in geophysics (for example in oceanography) and everywhere in between.Similarly, understanding the trapping of wavelike disturbances by spatial inhomogemeitiesis of importance not only in catalysis where the inhomogeneities may be on atomic scale,and in the trapping of weather patterns by topography.