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

这项研究项目包括一系列关键的数学和科学问题，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等领域中出现的多尺度问题。第一个项目是关于图案形成的，并分析了范例反应扩散系统中前沿、脉冲和斑点的动力学和稳定性。第二个项目研究了在复杂的多尺度化学反应、生化网络和燃烧中使用的模型简化方法，通过纳入扩散的影响。第三个项目涉及一类全新的解决方案，被称为环鸭，它是在神经科学的模型中发现的。这些解决方案有助于理解周期性尖峰和突增之间的转换。第四个项目将侧重于模拟和分析切断对锋面动态的影响的方法。该项目涉及研究生和博士后研究员参与研究，以及与国家实验室的科学家合作。这项研究项目解决了图案形成、化学和燃烧、神经科学、电气工程和多粒子系统中的一系列多尺度问题。在图案形成项目中，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来确定稳定图案形成区域的边界，分析半强脉冲相互作用的稳定性，模拟一维系统中脉冲的散射和二维系统中的光斑，扩展用于调制脉冲稳定性的重整化群方法，以及预测脉冲和前沿的动态分叉。第二个项目集中于显示多个时间尺度的大规模燃烧、化学和生化系统的精确模型降阶方法。我们的目标是分析、发展和改进尖端的模型降阶方法，以寻找在扩散存在的情况下管理有效系统动力学的低维流形。在第三个项目中，我们将研究偏微分方程环冠状和冠状的新现象。对于具有多维快变量和慢变量的快慢系统，将发展一种通用环面鸭牌理论。已知存在于许多神经科学模型中，如Hindmarsh-Rose方程、Morris-LeCar-Terman模型、Wilson-Cowan-Izhikevich系统和强迫van der Pol方程，环鸭在紧张性放电和爆发之间的转变过程中起着关键作用。还将对被称为调幅爆发节律的新爆发节律进行详细研究。第四个项目将研究截断对反应项的影响，反应项被引入以准确地模拟低粒子密度的区域，对传播锋面的速度、形状和稳定性的影响。我们将研究与四阶模式、二维空间动力学、锋面启动和锋面前兆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.