Dynamics near Turing patterns: modulations, bifurcations, and defects

图灵模式附近的动力学：调制、分叉和缺陷

• 批准号: 0806614
• 负责人: Arnd Scheel
• 金额: $ 33.81万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0806614&HistoricalAwards=false
• 关键词: Dynamics; near; Turing; patterns; modulations

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.

该项目旨在描述不一定具有自由能的空间扩展系统中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。典型的环境是反应扩散设置，还有各种对流实验或非线性光学反馈系统。我们提出对图灵模式的扰动、图灵模式的分歧以及图灵模式中嵌入的缺陷进行系统研究。我们将尽可能以独立于模型的描述为目标，强调拓扑和几何相交理论而不是显式解决方案。一些具体问题是图灵模式中存在位错、图灵模式中对称破缺的分岔、不均匀性和参数斜坡以及与 Liesegang 结构的连接。自组织的规则模式在自然界、沙丘和自然环境中出现时非常引人注目。动物皮毛上。当它们在纳米尺度的制造过程中被利用时，它们还具有巨大的技术潜力。在这两种情况下，人们都想了解规则结构何时、如何形成，以及特征尺度如何。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精确模型和参数值不可用的情况下，模式选择中的不均匀性和缺陷的作用。最终，我们希望从宏观、普遍的原理来理解那些自组织的模式形成过程，这些原理在很大程度上独立于潜在的详细微观过程。