Dynamic bifurcation of patterns through spatio-temporal heterogeneity

通过时空异质性动态分叉模式

• 批准号: 2307650
• 负责人: Ryan Goh
• 金额: $ 30万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2307650&HistoricalAwards=false
• 关键词: Dynamic; bifurcation; patterns; through; spatio

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.

该项目的重点是自然发生的和人造的空间模式如何与时空异质性相互作用。在这种情况下，模式指的是重复出现的几何空间结构，并出现在各种物理领域中。 在自然和实验环境中，时空异质性（例如杂质、外力、动态淬火和系统参数的缓慢演化）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。该项目由三个示例推动和组织：光敏化学反应系统中的定向淬火、流体对流辊中的缓慢空间斜坡以及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且相关的数学模型中的模式的相互作用，并将重点关注异质性如何诱发在空间同质环境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，包括动物手指的形成、皮肤图案、组织形成、半干旱气候下的植被图案、早期宇宙的结构形成以及晶体的缓慢冷却功能材料中的相。 该项目的结果还可以帮助设计和组装不同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来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。它重点关注这种异质性如何在空间延伸的系统中引起动态分叉。它将在三个项目领域开发和应用有限和无限维动力系统理论、泛函分析和数值计算的新技术，研究淬火系统中的模式和准模式、存在缓慢变化的空间坡道的前沿和模式，以及通过缓慢的时间淬灭来动态分叉模式。第一个区域将证明在实验相关的反应扩散系统中使用淬火图案的模空间。它还寻求将空间动力学技术扩展到没有单一可区分的无界方向的 2 维和 3 维域。除了深入了解猝灭和其他异质性对此类域中图案的影响之外，这些技术还将广泛用于非线性波和相干结构领域的许多不同设置。它还将研究异质性如何影响很少研究的超晶格和准图案的形成。第二和第三个项目领域也将以多种方式为最近蓬勃发展的偏微分方程动态分岔领域做出贡献。第二个领域将研究缓慢的空间斜坡如何选择和控制模式，而第三个领域将通过这种缓慢变化的时间淬灭揭示新的锋面、模式和缺陷形成现象。两者都将研究偏微分方程中的新型动态分岔，并开发无限维几何奇异摄动理论的新工具，例如慢不变流形和存在中性连续谱的系统的几何爆炸。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。