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.

本研究项目包含一系列关键的数学和科学问题，这些问题涉及图案形成、化学和燃烧、神经科学、电子工程和多粒子系统等领域的多尺度问题。第一个项目是关于模式的形成，并分析范式反应扩散系统中锋面、脉冲和斑点的动力学和稳定性。第二个项目研究复杂的多尺度化学反应、生化网络和燃烧中使用的模型还原方法，并结合扩散的影响。第三个项目涉及一种全新的解决方案，被称为环面鸭式，它是在神经科学模型中发现的。这些解决方案有助于理解周期性尖峰和爆发之间的转变。第四个项目将侧重于如何模拟和分析切断对前线动态的影响。该项目涉及研究生和博士后研究人员，并与国家实验室的科学家合作。本研究项目涉及图案形成、化学与燃烧、神经科学、电子工程和多粒子系统等领域的多尺度问题。在模式形成项目中，将研究范式反应-扩散系统。目标是发展新的分析技术和数学理论，以确定稳定模式形成区域的边界，分析半强脉冲相互作用的稳定性，模拟一维系统中的脉冲散射和二维系统中的点，扩展调制脉冲稳定性的重整化群方法，并预测脉冲和锋面的动态分岔。第二个项目集中在精确的模型还原方法大规模燃烧，化学和生化系统显示多个时间尺度。目标是分析，开发和改进尖端的模型简化方法，以寻找在扩散存在时控制有效系统动力学的低维流形。在第三个项目中，将研究环面鸭形和偏微分方程中的鸭形新现象。针对具有多维快慢变量的快慢系统，提出了一种通用环面鸭形理论。众所周知，环面鸭翼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系统和强迫范德波尔方程，环面鸭翼在强音尖峰和爆发之间的过渡机制中至关重要。还将对称为调幅爆破节奏的新型爆破节奏进行详细的研究。第四个项目将研究截断对反应项的影响，引入反应项以精确模拟低粒子密度区域，对传播锋的速度、形状和稳定性的影响。将研究与四阶模型、二维空间动力学、前沿起始和前沿前体有关的一系列重要问题。