Exploration for Complex Dynamics of Particle Patterns in Dissipative Systems

粒子模式是指由能量/物质的流入和流出之间的平衡所维持的任何空间局域化结构，以化学斑点、放电模式、形态斑点和二元对流池的形式出现。这些是由典型的三组分反应扩散系统或一对具有浓度场的复杂GL方程来模拟的。粒子模式之间的碰撞等强相互作用是一个很大的挑战，因为耗散系统没有很多保守量。与通过这些物体尾部的弱相互作用不同，到目前为止还没有系统的方法来处理它们，部分原因是在碰撞过程中图案会发生很大的变形。我们提出了一种新的方法来阐明复杂瞬态碰撞过程背后的主干结构，其关键成分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，相关的散射体网络通过异斜连接形成了整个碰撞动力学的主干。散射网络的观点似乎对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。另一个有趣的成果是，上述观点从一个不稳定模式的网络也可以很好地理解波在异质介质中的传播方式。由于非均质性导致缺陷的出现，粒子模式的传播等同于粒子模式与缺陷的碰撞问题。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，这使我们能够使大多数数值结果严格。

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.