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

在远离平衡的区域，出现了各种复杂的模式，如自我复制，时空模式，粒子间的碰撞等。理解这些事情的有力工具之一是动力系统理论，但其朴素的应用一般不起作用，部分原因是相空间的高维性和解的大变形。我们应该从何入手来理解这些行为呢？我们需要改变我们的思维方式，即“让我们思考引导解决方案轨道的几何结构，创造这样一个混乱的动态，而不是详细跟踪解决方案的变形”。换句话说，我们应该尝试描述无限维相空间的几何结构，在这种结构中，解轨道的行为变得容易检测。从这个观点出发，我们主要做了以下两件事。其他方面请参考已发表的论文。 ...更多信息 1.广义异宿环的开折蕴含着时空混沌，奇异方法（如奇异值分解）可以提供大量关于不稳定解及其不稳定流形的信息。异宿环是理解Gray-Scott模型时空混沌等复杂行为的关键。该机制本身对其他模型系统具有更广泛的适用性。2.“散射子”在类粒子斑图碰撞过程中的作用耗散系统中类粒子斑图的散射受到了各个领域的关注。本文主要研究了在两个运动脉冲或运动光斑发生强烈相互作用的正面碰撞中，由于碰撞点处的斑图会发生较大的变形，如何控制输入输出关系一直是一个未解决的问题。我们发现，一类特殊的不稳定的定常解或时间周期解（称为scattors）及其稳定流形和不稳定流形控制着轨道的交通流，即使在ID情形下，这类scattors也是高度不稳定的，通过散射过程产生了各种各样的输入输出关系。我们通过使用复杂的Ginzburg-Landau方程，Gray-Scott模型和气体放电现象中产生的三组分反应扩散模型来说明scattors的普遍性。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