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

多尺度反应扩散系统的动力系统和奇异摄动理论

• 批准号: 1616064
• 负责人: Tasso Kaper
• 金额: $ 54.28万
• 依托单位: Trustees of Boston University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1616064&HistoricalAwards=false
• 关键词: Dynamical; Systems; Singular; Perturbation; Theory

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.

该研究项目涵盖了一系列关键的数学和科学问题，涉及模式形成、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形成，并分析范式反应扩散系统中前沿、脉冲和斑点的动态和稳定性。第二个项目研究了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，还纳入了扩散效应。第三个项目涉及一类全新的解决方案，称为环鸭翼，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突发之间的转变。第四个项目将重点关注如何建模和分析切断对锋面动态的影响。该项目涉及研究生和博士后研究人员，以及与国家实验室科学家的合作。该研究项目解决了有关模式形成、化学和燃烧、神经科学、电气工程和多粒子系统中多尺度问题的一系列问题。在模式形成项目中，将研究范式反应扩散系统。目标是开发新的分析技术和数学理论，以确定稳定图案形成范围的边界，分析半强脉冲相互作用的稳定性，对一维系统中的脉冲散射和二维系统中的斑点进行建模，扩展重整化群方法以实现调制脉冲的稳定性，并预测脉冲和前沿的动态分叉。第二个项目的重点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、开发和改进尖端模型简化方法，以找到在扩散情况下控制有效系统动力学的低维流形。在第三个项目中，将研究偏微分方程中的环鸭翼和鸭翼的新现象。将为具有多维快慢变量的快慢系统开发通用环面鸭翼理论。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 系统和强制 van der Pol 方程，环面鸭翼在强直之间的过渡机制中至关重要尖峰和爆裂。还将对称为调幅突发节奏的新突发节奏进行详细研究。第四个项目将研究截止值对反应项的影响，引入这些反应项是为了精确模拟低粒子密度区域，以及传播前沿的速度、形状和稳定性。将研究与四阶模型、二维空间动力学、锋起始和锋前兆有关的一系列重要问题。