Dynamics From Forced Symmetry-Breaking

具有对称性的系统在实际应用中经常出现。这样的系统表现出的行为是不典型的非对称系统。即使这种行为可能比预期的更复杂，等变分叉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，对称性的存在是一种理想化。因此，重要的是要了解哪些方面的动态的对称系统是强大的，相对于小的破坏性缺陷。在许多情况下，离散和连续对称性的丧失将全局分叉引入动力学中，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）中产生的几类重要系统中研究这种不稳定性破坏扰动，重点是识别在主不稳定性阈值附近和远离主不稳定性阈值的鲁棒行为，前者通过中心流形约化产生对称性破坏的低维系统，而后者需要研究偏微分方程。特别强调的是放在理解的影响打破反射对称性和平移不变性（分别和同时）。由于参数强迫和伽利略不变性的损失，时间平移不变性的破坏所产生的现象也将被研究。该项目的目标是深入了解理想化模型的适用性限制，并提供一个典型的“近似鲁棒”系统的行为类型的描述。该项目旨在了解鲁棒行为背后的基本原则。例如，理解反射对称性破缺的影响是必不可少的，当研究的影响，一个小的倾斜度的设备水平，和thefround流或旋转的影响，从一个国家过渡到另一个。这类问题出现在微观情况下（水平或倾斜衬底上的薄膜），出现在微物理学（例如海洋学）以及两者之间的任何地方。同样，理解空间不均匀性对波动扰动的捕获不仅在催化方面（不均匀性可能是原子尺度的），而且在地形对天气模式的捕获方面也很重要。

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.