Dynamic bifurcation of patterns through spatio-temporal heterogeneity

Trustees Of Boston University, Boston

Investigators

Abstract

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.