Local bifurcation analysis and global numerical pathfollowing for Turing patterns in 3D reaction--diffusion systems

3D 反应扩散系统中图灵模式的局部分岔分析和全局数值路径跟踪

基本信息

批准号：
264671738
负责人：
Professor Dr. Hannes Uecker
金额：
--
依托单位：
Institut für Mathematik (IFM)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2014
资助国家：
德国
起止时间：
2013-12-31 至 2018-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/264671738?language=en
关键词：
Local bifurcation analysis global numerical

项目摘要

Turing patterns are solutions of partial differential equations (PDE) that arise from an instability of a spatially homogeneous stationary solution, which is stable with respect to spatially homogeneous perturbations, but unstable with respect to spatially periodic perturbations. The original modeling was motivated by pattern formation in embryos. However, Turing patterns occur in a variety of systems in nature, and thus also in a variety of PDE models. The local theory is well developed in one or two spatial dimensions, and Turing patterns can be well predicted using amplitude equations near bifurcation from a homogeneous solution. However, many physically relevant systems are genuinely three dimensional (3D), and in 3D the theory becomes much more complicated and is much less developed. Moreover, also numerical calculations of Turing patterns in 3D are rather rare and not systematic. The goal of this project is to use a combination of analysis and numerics to develop tools which allow systematically to study the bifurcation scenario for 3D Turing patterns. Besides the local theory near primary bifurcations, we also aim at a more global picture of the solution space. For this, preparatory work extending the 2D software package pde2path to 3D shall be continued, to also study branches of 3D Turing patterns further away from their primary bifurcation, and to study their secondary and higher order bifurcations, including hetero--and homoclinic connections between different patterns.

图灵模式是偏微分方程 (PDE) 的解，源自空间齐次稳态解的不稳定性，该解对于空间齐次扰动而言是稳定的，但对于空间周期性扰动而言是不稳定的。最初的建模是受胚胎模式形成的启发。然而，图灵模式存在于自然界的各种系统中，因此也存在于各种偏微分方程模型中。局域理论在一维或二维空间维度上得到了很好的发展，并且可以使用齐次解的分叉附近的幅度方程很好地预测图灵模式。然而，许多与物理相关的系统实际上是三维 (3D) 的，并且在 3D 中，理论变得更加复杂并且发展也少得多。此外，3D 图灵模式的数值计算也相当罕见且不系统。该项目的目标是结合使用分析和数值来开发工具，以便系统地研究 3D 图灵模式的分叉场景。除了初级分岔附近的局部理论之外，我们还致力于更全面地了解解空间。为此，应继续将 2D 软件包 pde2path 扩展到 3D 的准备工作，以研究 3D 图灵模式的分支，远离其主要分岔，并研究其二级和更高阶分岔，包括之间的异质和同宿连接不同的图案。