Pattern Formation in Two and Three Dimensions

[9703684]诺布洛赫这个项目将涉及研究对称系统中的分岔和模式形成。这样的系统自然出现在各种应用程序中。该提案侧重于研究弱系统对称性的后果——二维和三维模式形成的破缺扰动。在这样做的过程中，它试图识别在对称性破坏“不完美”下具有鲁棒性的等变动力学的那些方面。该方法有望将等变分岔理论的适用性扩展到更实际的情况，是向二维和三维模式形成理论迈出的重要一步。设想将其应用于许多连续系统，包括法拉第和图灵不稳定性形成的模式，以及水滴和气泡的声学驱动形状振荡。这个项目将涉及研究对称系统中结构的自发形成。这样的系统自然地出现在各种应用中，并且可以反映系统的几何形状或方便的理想化。众所周知，对称性对系统的典型行为有着深远的影响。结构或图案形成的近起点过程可以用称为振幅方程的简化方程来描述。这些方程的形式是由系统的对称性决定的。通常对称性将这些方程限制到只有一种可能存在的程度。然后，对所得方程的分析提供了对结构形成过程的完整理解，并可用于预测随着系统参数的变化而形成的结构类型。由于在本质上假设的对称性通常是理想化的，因此提案试图确定结构形成过程中对小的对称性破坏缺陷不敏感的那些方面，也就是说，它试图确定那些强健的方面。因此，该建议的目的是将这类理论的适用性扩展到二维和三维的实际问题。作为这种方法的例子，我们将研究振动诱导的表面波中的图案形成，化学和生物系统中通过图灵不稳定性形成的结构，以及声激发的水滴或气泡表面上的图案。理解这些相对简单的系统是将这些技术扩展到更复杂的系统的先决条件。

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.