Pattern Formation and Spatiotemporal Complex Dynamics in Extended Anisotropic Systems

该项目研究了在向列型液晶、通过离子束侵蚀压印在表面上的纳米结构、催化表面反应中化学波的传播、外延生长以及熔体凝固中的有序图案等例子中观察到的图案有趣特征背后的数学原理。这些模式是由一种现象形成的，这种现象被描述为各向异性介质在外力的驱使下失去平衡，并促使了广泛的实验研究，以寻找产生这些模式的物理机制。来自向列相液晶电对流的研究结果的反馈将指导肯特州立大学一个实验液晶组的研究人员进行实验，并将促进对复杂流体中图案形成及其技术应用的理解。了解雪面--大气和地球之间的界面--的时空变化可以增进对积雪内部的水文过程及其对融化动力学以及对环境气候学的影响的知识。该项目开发的粗糙度方法将对地球科学家，特别是那些在气候研究领域工作的科学家，在可视化空间和时间变量方面非常有用。绘制和解释动态物理表面，如雪面，将有助于制定采样和监测战略，包括对遥感信息进行充分的地面真实处理。该项目的目标是通过研究幅度和相位方程，开发一种全面和系统的理论方法，用于分析各向异性系统中复杂时空图案的形成和动力学的具体机制和特征。需要解决的关键问题是混沌吸引子的对称性破缺在时空混沌的产生中的作用和通向时空混沌的途径，波的非线性相互作用在时空复杂性的产生中的作用以及它们的发生涉及哪些各向异性。这项研究的新奇之处在于，通过对振幅和相位方程的分叉分析，将定性的理论预测与用于研究复杂动力学的传统定量诊断方法相结合。此外，还提出了拓扑数据分析和粗糙度度量等新的创新工具来定量表征各向异性图案。这些结果将为理解物理实验所需的直觉提供坚实的基础，并可能提出有趣的进一步实验问题。

This project studies the mathematics behind the intriguing features of patterns observed in examples such as nematic liquid crystals, nanostructures imprinted on surfaces through ion-beam erosion, the propagation of chemical waves in catalytic surface reactions, epitaxial growth, and ordered patterns in the solidification from a melt. These patterns are formed through a phenomenon described as anisotropic media being driven out of equilibrium through external forces and have motivated a wide range of experimental studies to find the physical mechanisms generating them. The feedback from the research findings on electroconvection in nematic liquid crystals will guide researchers of an experimental liquid crystal group at Kent State University in their experiments and will advance the understanding of pattern formation in complex fluids and their technological applications. Understanding the spatiotemporal changes in the snow surface - the interface between the atmosphere and the Earth - can improve the knowledge of hydrologic processes within the snowpack and their impact on melt dynamics as well as on the climatology of the environment. Roughness methods developed in the project will be useful for earth scientists, especially those working in the realm of climate studies, in visualizing variables over space and time. Mapping and interpretation of dynamic physical surfaces, such as the snow surface, will help inform sampling and monitoring strategies, including adequate ground-truthing of remotely sensed information.The goal of the project is to develop a comprehensive and systematic theoretical approach, through the study of amplitude and phase equations, for the analysis of specific mechanisms and features of the formation and dynamics of complex spatiotemporal patterns in anisotropic systems. Key questions to be addressed are the role of symmetry breaking of a chaotic attractor in the creation of spatiotemporal chaos and the routes to it and the role of nonlinear interactions of waves in the creation of spatiotemporal complexity and which anisotropies are involved in their occurrence. The novelty of this research lies in a combination of qualitative theoretical predictions resulting from a bifurcation analysis of amplitude and phase equations with traditional quantitative diagnostic methods used to study complex dynamics. Moreover, new and innovative tools such as topological data analysis and roughness measures are proposed for the quantitative characterization of anisotropic patterns. The results will provide a firm foundation for the intuition required to make sense of physical experiments and may suggest interesting further questions to test experimentally.