Exploration for Complex Dynamics of Particle Patterns in Dissipative Systems

耗散系统中粒子模式复杂动力学的探索

基本信息

批准号：
16204008
负责人：
NISHIURA Yasumasa
金额：
$ 23.55万
依托单位：
Hokkaido University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16204008/
关键词：
reaction diilffusion systems pulse spot scattering global bifurcation heteroclinic connection

项目摘要

Particle patterns mean any spatially localized structures sustained by the balance between inflow and outflow of energy/material which arise in the form of chemical blob, discharge pattern, morphological spot, and binary convection cell. These are modeled by typically three-component reaction diffusion systems or a couple of complex GL equations With concentration field.Strong interaction such as collision among particle patterns is a big challenge, since dissipative systems do not have many conservative quantities. Unlike weak-interaction through tails of those objects, there are so far no Systematic methods to handle them partly because of large deformation of patterns during the collision process. We present a new approach to clarify a backbone structure behind the complicated transient collision process A key ingredient lies in a hidden network of unstable, solutions called scattors which play a crucial role to understand the input-output relation for collision process (namely the relation of two dynamics before and after collision).More precisely, the associated network of scattors via heteroclinic connections forms a backbone for the whole collisional dynamics. The viewpoint of scattor network seems quite useful for many classes of model systems including reaction-diffusion systems, CGLE, and binary fluid convection.Another interesting achievement is that the above viewpoint from a network of unstable patterns also works quite well in order to understand the propagating manner of waves in heterogeneous media. There appear defects induced by heterogeneities and propagation of particle patterns is equivalent to the collision problem between particle patterns and defects. In both cases, reduction PDE to finite dimensional system is possible, which allows us to make most of the numerical results rigorous.

粒子模式是指由能量/材料的流入和流出之间的平衡维持的任何空间局部结构，其以化学斑点、放电模式、形态点和二元对流细胞的形式出现。这些是通过典型的三组分反应扩散系统或几个具有浓度场的复杂 GL 方程来建模的。强相互作用（例如粒子模式之间的碰撞）是一个巨大的挑战，因为耗散系统没有很多保守量。与通过这些物体的尾部进行弱相互作用不同，迄今为止还没有系统的方法来处理它们，部分原因是碰撞过程中图案发生了很大的变形。我们提出了一种新方法来阐明复杂瞬态碰撞过程背后的主干结构。关键要素在于不稳定解的隐藏网络，称为散点图，它对于理解碰撞过程的输入输出关系（即碰撞前和碰撞后的两个动力学）。更准确地说，通过异宿连接的相关散射子网络形成了整个碰撞动力学的支柱。散射网络的观点似乎对于许多类型的模型系统非常有用，包括反应扩散系统、CGLE 和二元流体对流。另一个有趣的成就是，上述来自不稳定模式网络的观点也非常适合理解波在异质介质中的传播方式。由于不均匀性而引起缺陷的出现，粒子图案的传播相当于粒子图案与缺陷之间的碰撞问题。在这两种情况下，将偏微分方程简化为有限维系统是可能的，这使我们能够使大部分数值结果变得严格。