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

在远面平衡的制度中，出现多种复杂模式，例如自我复制，时空模式以及类似粒子样模式之间的碰撞。理解这些内容的强大工具之一是动态系统理论，但是通常，其天真的应用并不能部分归功于相位空间的高维度和解决方案的较大变形。我们应该从理解这种行为开始的线索是什么？我们需要改变思维方式，即“让我们思考引导解决方案轨道创造这种混乱动态的几何结构，而不是详细跟踪解决方案的变形”。换句话说，我们应该尝试表征无限尺寸相空间的几何结构，其中溶液轨道的行为很容易被检测到。从这个角度来看，我们完成了以下两件事。请参阅已发表的论文，以了解其他方面。1。杂物杂斜周期的未包装意味着时空的混乱。Chimeric方法（例如自动），为我们提供了有关不稳定解决方案的大量信息，以及其不稳定歧管的行为。连接多个固定模式的杂斜周期被确定为了解灰色 - 斯科特模型的时空混乱等复杂行为的关键。该机制本身对其他模型系统具有更大的适用性。2。“ Scattor”在类似粒子的模式之间碰撞过程。耗散系统中类似粒子样模式的散射过程引起了各个领域的关注。我们专注于在正面碰撞中如何控制输入输出关系的问题，在该碰撞中，旅行脉冲或斑点相互作用很强。由于在碰撞点上模式的较大变形，因此仍然存在一个空旷的问题。我们发现，特殊类型的不稳定的稳定或时间周期性解决方案称为Scattors及其稳定且不稳定的歧管指导轨道的交通流量。通常，即使在ID情况下，这种扫描仪也通常是高度不稳定的，即使在ID情况下也会通过散射过程引起各种输入输出关系。我们通过使用复杂的Ginzburg-landau方程，灰色 - 斯科特模型和在气体 - 分离现象中产生的三组分反应扩散模型来说明史瓦的无处不在。较少的