Dynamical Systems and Singular Perturbation Theory for Multiscale Reaction-Diffusion Systems

该研究项目包括一系列关键的数学和科学问题，用于图案形成，化学和燃烧，神经科学，电气工程和多粒子系统领域中出现的多尺度问题。第一个项目是关于模式的形成，并分析了范式反应扩散系统中前沿，脉冲和点的动态和稳定性。第二个项目研究复杂的多尺度化学反应，生物化学网络和燃烧中使用的模型简化方法，也包括扩散的影响。第三个项目涉及一种全新的解决方案，称为环面鸭，在神经科学模型中发现。这些解决方案有助于理解周期性尖峰和爆发之间的转换。第四个项目将侧重于如何模拟和分析截断对前沿动态的影响。该项目涉及研究生和博士后研究员，以及与国家实验室的科学家合作。该研究项目涉及图案形成，化学和燃烧，神经科学，电气工程和多粒子系统中的多尺度问题。在模式形成计画中，将研究范例反应扩散系统。目标是发展新的分析技术和数学理论，用于确定稳定模式形成制度的边界，分析半强脉冲相互作用的稳定性，模拟1-D系统中的脉冲散射和2-D系统中的斑点，扩展重整化群方法用于调制脉冲的稳定性，以及预测脉冲和前沿的动态分叉。第二个项目的中心是针对具有多个时间尺度的大规模燃烧、化学和生化系统的精确模型简化方法。目标是分析，开发和改进尖端的模型简化方法，以找到在扩散存在下控制有效系统动力学的低维流形。在第三个计画中，我们将研究偏微分方程式中的环面鸭翼与鸭翼的新现象。一般环面鸭翼的理论将开发的快，慢系统与多维快，慢变量。已知存在于许多神经科学模型中，例如Hindmarsh-Rose方程、Morris-Lecar-Terman模型、Wilson-Cowan-Izhikevich系统和强迫货车der Pol方程，环面鸭翼在紧张性尖峰和爆发之间的过渡状态中是关键的。还将对称为调幅爆发节律的新爆发节律进行详细研究。第四个项目将研究反应项的截止值对传播前沿的速度、形状和稳定性的影响，这些截止值是为了精确模拟低粒子密度区域而引入的。将研究与四阶模型、二维空间动力学、锋面引发和锋面前驱有关的一系列重要问题。

This research project encompasses a series of critical mathematical and scientific questions for multiscale problems arising in the fields of pattern formation, chemistry and combustion, neuroscience, electrical engineering, and multi-particle systems. The first project is on pattern formation and analyzes the dynamics and stability of fronts, pulses, and spots in paradigm reaction-diffusion systems. The second project studies model reduction methods used in complex multiscale chemical reactions, biochemical networks, and combustion by incorporating also the effects of diffusion. The third project involves a completely new class of solutions, known as torus canards, found in models from neuroscience. These solutions help understand the transitions between periodic spiking and bursting. The fourth project will focus on ways to model and analyze the impacts of cut-offs on the dynamics of fronts. The project involves graduate students and postdoctoral fellows in the research, as well as collaborations with scientists at national laboratories. This research project addresses a series of questions concerning multiscale problems in pattern formation, chemistry and combustion, neuroscience, electrical engineering, and multi-particle systems. In the pattern formation project, paradigm reaction-diffusion systems will be studied. The goals are to develop new analytical techniques and mathematical theory for determining the boundaries of the stable pattern-forming regimes, analyzing the stability of semi-strong pulse interactions, modeling the scattering of pulses in 1-D systems and spots in 2-D systems, extending renormalization group methods for stability of modulating pulses, and predicting the dynamic bifurcations of pulses and fronts. The second project centers on accurate model reduction methods for large-scale combustion, chemical, and biochemical systems exhibiting multiple time scales. The goals are to analyze, develop, and improve cutting-edge model reduction methods for finding the low-dimensional manifolds that govern the effective system dynamics in the presence of diffusion. In the third project, the new phenomena of torus canards and canards in partial differential equations will be investigated. A theory of generic torus canards will be developed for fast-slow systems with multi-dimensional fast and slow variables. Known to exist in many neuroscience models, such as the Hindmarsh-Rose equations, the Morris-Lecar-Terman model, the Wilson-Cowan-Izhikevich system, and the forced van der Pol equation, torus canards are critical in the transition regimes between tonic spiking and bursting. A detailed study will also be carried out of the new bursting rhythms known as amplitude-modulated bursting rhythms. The fourth project will study the impacts of cut-offs on the reaction terms, introduced to accurately model regions of low particle densities, on the speeds, shapes, and stability of propagating fronts. A series of important problems related to fourth-order models, two-dimensional space dynamics, front initiation, and front pre-cursors will be studied.