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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组(PDE)的解，它源于空间齐次定常解的不稳定性，该定常解相对于空间均匀扰动是稳定的，但相对于空间周期扰动是不稳定的。最初的建模是由胚胎中的图案形成驱动的。然而，图灵模式在自然界中存在于各种系统中，因此也存在于各种PDE模型中。局域理论在一维或两维空间中得到了很好的发展，从齐次解出发，利用分叉附近的振幅方程可以很好地预测图灵图案。然而，许多与物理相关的系统是真正的三维(3D)，而在3D中，理论变得更加复杂，也远远落后于发展。此外，图灵图案在3D中的数值计算也相当罕见，而且不是系统的。这个项目的目标是使用分析和数值相结合的方法来开发工具，使之能够系统地研究3D图灵图案的分叉场景。除了初等分支附近的局部理论外，我们还着眼于解空间的更全局的图景。为此，应继续开展将2D软件包pde2Path扩展到3D的准备工作，研究远离其初级分叉的3D图灵图案的分支，并研究其二次和高阶分叉，包括不同图案之间的异位和同宿联系。