Dynamic bifurcation of patterns through spatio-temporal heterogeneity

这个项目的重点是如何自然发生的，人为的空间模式与时空异质性相互作用。在这种情况下，模式是指经常性的几何空间结构，并出现在各种物理域。 在自然和实验环境中，时空异质性，如杂质，外部强迫，动态淬火和系统参数的缓慢演变已被证明可以选择系统中形成的结构类型并介导缺陷的形成。这个项目的动机和组织由三个例子：在光敏化学反应系统中的定向淬火，在流体对流辊中的缓慢空间斜坡，以及在缓慢淬火系统中的缺陷形成。它旨在严格理解动态异质性与重要和相关数学模型中模式的相互作用，并将重点关注异质性如何诱导在空间均匀设置中未观察到的新行为。 该项目的结果将有助于理解许多其他科学领域的图案形成，包括动物手指形成，皮肤图案，组织形成，半干旱气候中的植被图案，早期宇宙中的结构形成以及功能材料中结晶相的缓慢冷却。 该项目的结果还可以帮助设计和组装各种长度尺度的功能材料。 该项目将通过本科生研究经验和研究生研究项目促进早期职业研究人员的发展。该项目旨在开发新的数学工具，以严格研究相干结构及其与原型偏微分方程模型中动态异质性的相互作用。它的重点是如何在空间扩展系统的异质性可以诱导动态分岔。它将开发和应用新技术，从有限和无限维动力系统理论，功能分析和数值计算在三个项目领域，研究模式和准模式在淬火系统，前线和模式存在缓慢变化的空间斜坡，并通过缓慢的时间淬火模式的动态分叉。第一个领域将证明淬火模式的模空间在实验相关的反应扩散系统中的使用。它还试图将空间动力学技术扩展到2维和3维领域，这些领域没有一个独特的无界方向。除了深入了解淬火和其他不均匀性对这些领域中图案的影响外，这些技术还将在非线性波和相干结构领域的许多不同环境中广泛使用。它还将研究异质性如何影响很少研究的超晶格和准模式的形成。第二和第三个项目领域也将以几种方式促进最近蓬勃发展的偏微分方程动态分叉领域。第二个领域将研究如何缓慢的空间斜坡选择和控制模式，而第三个将揭示新的前线，模式，并通过这种缓慢变化的时间淬火缺陷形成现象。这两个项目都将研究偏微分方程中的新型动态分叉，并开发无限维几何奇异摄动理论的新工具，如慢不变流形和中性连续谱存在下的系统几何爆破。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is focused on how naturally occurring, and man-made spatial patterns interact with spatio-temporal heterogeneities. In this context, patterns refer to recurrent geometric spatial structures, and occur in a variety of physical domains. In both natural and experimental settings, spatio-temporal heterogeneities, such as impurities, external forcing, dynamic quenching, and slow evolution of system parameters have been shown to select the type of structure formed in a system and mediate the formation of defects. This project is motivated and organized by three examples: directional quenching in light-sensitive chemical reaction systems, slow spatial ramping in fluid convection rolls, and defect formation in slowly quenched systems. It seeks to rigorously understand the interaction of dynamic heterogeneities with patterns in important and relevant mathematical models, and will focus on how heterogeneities can induce novel behavior not observed in spatially homogeneous settings. The results of this project will aid in the understanding of pattern formation in many other scientific domains, including animal digit formation, skin patterning, tissue formation, vegetation patterning in semi-arid climates, structure formation in the early universe, and slow cooling of crystalline phases in functional materials. The results of this project could also aid in the design and assembly of functional materials at various length scales. The project will foster the development of early career researchers through undergraduate research experiences and graduate research projects. The project seeks to develop new mathematical tools to rigorously study coherent structures and their interactions with dynamic heterogeneities in prototypical partial differential equation models. It focuses on how such heterogeneities can induce dynamic bifurcations in spatially-extended systems. It will develop and apply novel techniques from finite and infinite dimensional dynamical systems theory, functional analysis, and numerical computation in three project areas, studying patterns and quasi-patterns in quenched systems, fronts and patterns in the presence of slowly-varying spatial ramps, and dynamic bifurcation of patterns via slow temporal quenching. The first area will evidence the use of the moduli space of quenched patterns in an experimentally relevant reaction-diffusion system. It also seeks to extend spatial dynamics techniques to 2- and 3-dimensional domains which do not have a single distinguished unbounded direction. In addition to gaining insight into the effect of quenching and other heterogeneities on patterns in such domains, these techniques will be widely useful in many different settings across the field of nonlinear waves and coherent structures. It will also study how heterogeneities affect the formation of seldom studied super-lattice and quasi-patterns. The second and third project areas will also contribute in several ways to the recently flourishing field of dynamic bifurcation in PDEs. The second area will investigate how slow spatial ramps select and control patterns, while the third will reveal new front, pattern, and defect formation phenomena through such slowly-varying temporal quenches. Both will study new types of dynamic bifurcation in PDEs, and develop new tools in infinite-dimensional geometric singular perturbation theory, such as slow invariant manifolds, and geometric blow-up for systems in the presence of neutral continuous spectrum.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.