Pattern Formation in Two and Three Dimensions

小行星9703684 这个项目将涉及研究分叉和模式的形成， 对称的系统这种系统自然出现在各种 应用.该提案侧重于研究 二维斑图形成的弱系统对称性破缺扰动 和三维空间。在这样做时，它力求查明 等变动力学，是鲁棒的下破缺 “不完美”。该方法有望扩展 等变分歧理论更现实的情况，是一个 迈向二、三阶段模式形成理论的重要一步 尺寸.设想了对一些连续系统的应用， 包括由法拉第和图灵不稳定性形成的图案，以及 声学驱动的液滴和气泡的形状振荡。 这个项目将涉及结构的自发形成的研究 在对称的系统中。这样的系统自然发生在各种各样的 应用，并且可以反映系统的几何形状或方便的配置。 理想化众所周知，对称性对典型的 系统的行为。结构或模式的过程 地层可以用称为振幅的简化方程来描述 方程这些方程的形式是由对称性决定的。 系统通常对称性会将这些方程限制到这样的程度 只有一种可能然后对所得方程进行分析， 提供对结构形成过程的完整理解， 可以用来预测将形成的结构类型作为参数 系统的不同。因为在自然界中假设的对称性是 通常是一种理想化的建议，旨在确定这些方面的 结构形成工艺对小的裂纹不敏感 缺陷，即，它力求确定哪些方面是健全的。A IM 因此，该建议的目的是将这种理论的适用性扩展到 在两个和三个维度的现实问题。作为例子， 方法，我们将研究图案形成的振动诱导表面 波，结构形成的化学和生物系统，通过 图灵不稳定性，以及声激发表面上的图案 滴或泡。了解这些相对简单的系统是一个 这是将技术扩展到更复杂系统的先决条件。

9703684 Knobloch This project will involve the study of bifurcations and pattern formation in systems with symmetry. Such systems arise naturally in a variety of applications. The proposal focuses on the study of the consequences of weak system symmetry--breaking perturbations for pattern formation in two and three dimensions. In doing so it seeks to identify those aspects of equivariant dynamics that are robust under symmetry-breaking "imperfections". The approach promises to extend the applicability of equivariant bifurcation theory to more realistic situations and is an important step towards a theory of pattern formation in two and three dimensions. Applications to a number of continuum systems are envisaged, including pattern formation by the Faraday and Turing instabilities, and acoustically driven shape oscillations of drops and bubbles. This project will involve the study of the spontaneous formation of structure in systems with symmetry. Such systems occur naturally in a variety of applications, and may reflect the geometry of the system or a convenient idealization. Symmetries are known to have a profound effect on the typical behavior of the system. Near onset the process of structure or pattern formation can be described by simplified equations called amplitude equations. The form of these equations is dictated by the symmetries of the system. Often the symmetries constrain these equations to such an extent that only one possibility exists. The analysis of the resulting equations then provides a complete understanding of the structure formation process, and can be used to predict the type of structure that will form as the parameters of the system are varied. Since in nature the assumed symmetries are usually an idealization the proposal seeks to identify those aspects of the structure formation process that are insensitive to small symmetry-breaking imperfections, i.e., it seeks to identify those aspects that are robust. The a im of the proposal is thus to extend the applicability of this type of theory to realistic problems in both two and three dimensions. As examples of this approach we shall study pattern formation in vibration-induced surface waves, structure formation in chemical and biological systems via the Turing instability, and the patterns on the surface of an acoustically excited drop or bubble. An understanding of these relatively simple systems is a prerequisite for extending the techniques to more complex systems.