Coherent Structures: Interaction and Propagation of Defects

我们将使用独立于模型的动态系统方法来研究复杂系统中的相干结构。我们将研究的现象包括不稳定性的扩散、可激发和振荡介质中波列的动力学、二维介质中的线和角缺陷以及相干结构的相互作用。我们利用光谱的详细描述来从背景场中分离缺陷的特性。我们孤立组织中心的性质的介质变化和分叉发生。重要的应用范围从生物学到材料科学到流体动力学和湍流建模。突出的例子是心脏组织中的螺旋波，神经轴突中的尖峰和爆裂，超导体中的巨大漩涡，结晶过程中的界面传播，或裂纹尖端运动。在许多这样的例子中，人们对确切的微观机制知之甚少，或者仅仅是系统的大小无法从微观规律中准确预测宏观行为。相干结构是复杂系统中粗糙统计方法的主要障碍。我们打算用解析方法探索相干结构的性质，并计划开发可靠的数值工具。这些结果将进一步提高我们预测复杂系统在远离平衡的参数范围内的行为的能力，通过将基于平均场理论的大规模计算与基于相干结构的结合分析和数值描述的小规模信息相结合。

We will investigate coherent structures in complex systems using a model-independent, dynamical systems approach. Among the phenomena that we will study are the spread of instabilities, the dynamics of wave trains in excitable and oscillatory media, line and corner defects in two-dimensional media, and the interaction of coherent structures. We exploit a detailed description of spectra to separate properties of defects from the background field. We isolate organizing centers where properties of the medium change and bifurcations occur.Important applications range from biology through material science to fluid dynamics and turbulence modeling. Prominent examples are spiral waves in cardiac tissue, spiking and bursting in nerve axons, giant vortices in superconductors, interface propagation in crystallization processes, or crack tip motion. In many of these examples, little is known about the exact microscopic mechanisms, or the mere size of the system does not allow for an accurate prediction of macroscopic behavior from microscopic laws. Coherent structures are the major obstruction to crude statistical approaches in complex systems. We intend to explore the properties of coherent structures using analytical methods and we project to develop reliable numerical tools. The results will further our ability to predict the behavior of complex systems in parameter ranges far from equilibrium by joining large-scale computations, based on a mean field theory, with small-scale information, based on a combined analytical and numerical description of coherent structures.