Pattern Formation and Spatiotemporal Complex Dynamics in Extended Anisotropic Systems

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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.