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

耗散系统中复杂时空模式的全局分叉方法

基本信息

批准号：
13440027
负责人：
NISHIURA Yasumasa
金额：
$ 9.6万
依托单位：
Hokkaido University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

项目摘要

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

在远离平衡的状态中，出现了多种复杂的模式，例如自我复制、时空模式以及类粒子模式之间的碰撞，理解这些事物的有力工具之一是动力系统理论，但它的理论如何。由于相空间的高维性和解的大变形，天真的应用通常不起作用。我们需要改变我们的思维方式，即“让我们”。思考指导的几何结构换句话说，我们应该尝试表征无限维相空间的几何结构，在该空间中，解轨道的行为变得很容易检测到。观点来看，我们主要完成了以下两件事。其他方面的成就请参阅已发表的论文。1.广义异宿循环的展开意味着时空混沌。嵌合方法，例如AUTO，给出了我们获得了有关不稳定解及其不稳定流形的行为的大量信息，其中连接多个稳态模式的异斜循环被认为是理解格雷-斯科特模型的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”在类粒子图案碰撞过程中的作用。耗散系统中类粒子图案的散射受到了各个领域的广泛关注。在行进脉冲或光斑相互作用强烈的正面碰撞中如何控制输入输出关系的问题。由于碰撞点处图案的大变形，这仍然是一个悬而未决的问题。我们发现了特殊类型的不稳定稳态。或称为散射子的时间周期解及其稳定和不稳定流形引导轨道的交通流。即使在 ID 情况下，此类散射子通常也高度不稳定，这会通过散射过程导致各种输入输出关系。通过使用复杂的 Ginzburg-Landau 方程、Gray-Scott 模型和气体放电现象中出现的三分量反应扩散模型，我们发现了散射体的普遍性。