Global Bifurcational Approach to Complex Spatio^temporal Patterns in Dissipative Systems

在远离平衡的状态下，出现了各种各样的复杂图案，如自我复制、时空图案，以及粒子样图案之间的碰撞。动力系统理论是理解这些问题的有力工具之一，然而，由于相空间的高维性和解的大变形，其幼稚的应用一般不起作用。我们理解这种行为的线索应该是什么？我们需要改变我们的思维方式，即“让我们思考引导解的轨道的几何结构，从而产生如此混乱的动力，而不是详细跟踪解的变形”。换句话说，我们应该尝试刻画无限维相空间的几何结构，在这种结构中，解的轨道行为变得容易检测到。本着这一观点，我们主要做了以下两件事。有关…的其他方面，请参阅已发表的论文更多的成果：1.广义异宿循环的展开蕴含着时空的混沌，化学方法，如AUTO，给我们提供了大量关于不稳定解及其不稳定流形行为的信息。将多个平稳模式连接起来的异宿环被认为是理解Gray-Scott模型中时空混沌等复杂行为的关键。这种机制本身对其他模型系统具有更广泛的适用性。2.散射体在类粒子图案间碰撞过程中的作用。耗散系统中类粒子图案的散布引起了各个领域的关注。我们重点研究了在行进脉冲或光斑强烈相互作用的迎头碰撞中如何控制输入输出关系的问题，但由于碰撞点处图案的大变形，这一直是一个悬而未决的问题。我们发现，特殊类型的不稳定稳定或时间周期解称为散射子，它们的稳定和不稳定流形指导着轨道的交通流，即使在ID情况下，这种散射子通常也是高度不稳定的，这通过散射过程导致了不同的输入输出关系。我们用复数Ginzburg-Landau方程、Gray-Scott模型和气体放电现象中的三组分反应扩散模型说明了散射体的普遍存在。较少

In a regime of far-from equilibrium there appears a diversity of complex patterns such as self-replication, spatio-temporal patterns, and collisions among particle-like patterns. One of the powerful tools to understand these things is dynamical system theory, however its naive application in general does not work partly due to the high dimensionality of phase space and large deformation of solutions. What should be the clue for us to start with in understanding such behaviors? We need to alter our way of thinking, namely "Let us think about the geometric structures that guide solution orbits creating such a chaotic dynamism, rather than keeping track of the deformations of solutions in detail". In other words, we should try to characterize geometric structures of the infinite dimensional phase space in which behaviors of solution orbits become easily detectable. Taking this viewpoint, we accomplished the following two main things. Please refer to the published papers for other aspect o … More f achievements.1.Unfolding of generalized heteroclinic cycle implies spatio-temporal chaos.Chimerical methods, such as AUTO, give us a great amount of information on an unstable solution, as well as on the behavior of its unstable manifold. Heteroclinic cycle connecting several stationary patterns was identified as a key to understand the complex behaviors like spatio-temporal chaos for the Gray-Scott model. The mechanism itself has much wider applicability to other model systems.2.Role of "Scattors" for collision process among particle-like patterns.Scattering of particle-like patterns in dissipative systems has much attention from various fields. We focused on the issue how the input-output relation is controlled at a head-on collision where traveling pulses or spots interact strongly.It had remained an open problem due to the large deformation of patterns at a colliding point. We found that special type of unstable steady or time-periodic solutions called scattors and their stable and unstable manifolds direct the traffic flow of orbits.Such scattors are in general highly unstable even in ID case which causes a variety of input-output relations through the scattering process. We illustrate the ubiquity of scattors by using the complex Ginzburg-Landau equation, the Gray-Scott model and a three-component reaction diffusion model arising in gas-discharge phenomena. Less