Dynamics near Turing patterns: modulations, bifurcations, and defects

该项目旨在描述空间扩展系统中不一定具有自由能的图灵模式附近的动力学。类图灵模式是以自组织方式出现的空间周期性和静止结构。典型的背景是反应扩散设置，但也有各种对流实验，或非线性光学反馈系统。我们提出了一个系统的调查图灵模式的扰动，从图灵模式的分叉，并嵌入在图灵模式的缺陷。我们的目标是尽可能地独立于模型的描述，强调拓扑和几何相交理论，而不是明确的解决方案。一些具体的问题是存在的位错在图灵模式，从图灵模式，不均匀性和参数斜坡，并与Liesegang结构的连接。自组织，规则的模式是惊人的，当它们出现在自然界中，在沙丘，动物的外套。当在纳米级的制造过程中被利用时，它们也具有巨大的技术潜力。在这两种情况下，人们都希望了解规则结构何时以及如何形成，以及在何种特征尺度上形成。我们计划在分析中使用几何和拓扑工具来研究这些问题。这些工具可以揭示在没有精确模型和参数值的情况下，不均匀性和缺陷在模式选择中的作用。最终，我们希望从宏观的、普遍的原则来理解这些自组织模式的形成过程，这些原则在很大程度上独立于底层的详细微观过程。

This project aims at a description of dynamics near Turing patterns in spatially extended systems that do not necessarily possess a free energy. Turing-like patterns are spatially periodic and stationary structures that emerge in a self-organized fashion. Typical contexts are reaction-diffusion settings, but also various convection experiments, or nonlinear optical feedback systems. We propose a systematic investigation of perturbations of Turing patterns, bifurcations from Turing patterns, and defects embedded in Turing patterns. We will aim at model-independent descriptions whenever possible, emphasizing topological and geometric intersection theory rather than explicit solutions. Some specific problems are the existence of dislocations in Turing patterns, symmetry-breaking bifurcations from Turing patterns, inhomogeneities and parameter ramps, and connections with Liesegang structures.Self-organized, regular patterns are striking when they appear in nature, in sand dunes, and on animal coats. They also carry great technological potential when exploited in manufacturing processes at nanoscales. In both cases, one would like to understand when and how regular structures form, and at which characteristic scale. We plan to study those questions using geometric and topological tools in the analysis. Those tools can shed light on the role of inhomogeneities and defects in the selection of patterns in situations where precise models and parameter values are unavailable. Ultimately, we would like to understand those self-organized pattern formation processes from macroscopic, universal principles that are largely independent of the underlying detailed microscopic processes.