Exploration for Complex Dynamics of Particle Patterns in Dissipative Systems

粒子图样是指由能量/物质的流入和流出之间的平衡所维持的任何空间局部化结构，这些结构以化学斑点、放电模式、形态斑点和二元对流单元的形式出现。这些模型通常由三组分反应扩散系统或几个具有浓度场的复杂GL方程来模拟。由于耗散系统没有太多的保守量，强相互作用(如粒子图案之间的碰撞)是一个巨大的挑战。与通过这些物体的尾巴进行弱相互作用不同，到目前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的大变形。我们提出了一种新的方法来阐明复杂的瞬时碰撞过程背后的主干结构一个关键因素在于一个被称为散射子的不稳定解的隐藏网络，它对理解碰撞过程的输入-输出关系(即碰撞前后两个动力学的关系)起着至关重要的作用。更准确地说，通过异宿连接关联的散射子网络构成了整个碰撞动力学的主干。散射网络的观点对于包括反应扩散系统、CGLE和二元流体对流在内的许多类模型系统似乎非常有用。另一个有趣的成果是，上述来自不稳定模式网络的观点也适用于理解波在非均匀介质中的传播方式。存在由非均质引起的缺陷，颗粒图案的传播等价于颗粒图案与缺陷的碰撞问题。在这两种情况下，将偏微分方程组简化为有限维系统是可能的，这使得我们可以使大多数数值结果变得严格。

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.