Dynamics of Coupled Cell Systems

该项目旨在促进对相互作用的微分方程和映射网络动力学的理解。该项目特别关注振荡和其他复杂动态行为的方式，如异斜周期和混沌动力学，可以由网络架构强制。许多物理、电子和生物系统可以通过相互作用的微分方程和地图网络来建模。这种类型的模型可能涉及连续和离散动力系统的混合，可能带有时滞和滤波器，称为混合系统。该项目的一个高优先级是发展混合系统的数学理论，特别关注工程和计算机应用。该项目将使用最初为研究对称动力系统而开发的数学技术，研究动力系统的统计和随机特性，并开发用于构建满足特定动力和结构特性的相对简单的动力系统（模块）的新方法。耦合微分方程系统在广泛的物理应用中被用作模型，例如神经网络、物种形成、基因动力学、约瑟夫森结阵列以及机械、电气和计算机系统。该项目旨在发展对这些系统动力学的改进的数学理解。这是过去研究的自然发展和延续，特别是在美国国家科学基金会资助的“耦合细胞系统中的同步和结构”项目中。部分研究将与英国（曼彻斯特和埃克塞特）和葡萄牙（波尔图）的小组合作进行。曼彻斯特大学的研究小组非常关注电子、计算机系统和生物学领域的应用，这些应用将有助于为网络理论方面的发展提供一个框架。至少两名研究生将直接获得资助。更一般地说，这项研究将通过研讨会、微分方程领域研究生课程的发展和网络领域本科生研究项目，直接影响休斯顿大学研究生和本科生的数学教育。研究人员长期致力于数学和科学教师专业发展的拓展项目，特别是通过休斯顿教师协会和耶鲁国家倡议。拟议的研究将影响这些计划的发展和内容。研究者将继续通过公开讲座、教育出版物和科普著作的方式，努力提高大众对数学及其应用的理解。

The project aims to advance the understanding of the dynamics of networks of interacting differential equations and maps. The project focuses in particular on the ways oscillatory and other complex dynamical behavior, such as heteroclinic cycles and chaotic dynamics, can be forced by the network architecture. Many physical, electrical and biological systems can be modeled by networks of interacting differential equations and maps. Models of this type, which can involve a mix of continuous and discrete dynamical systems, possibly with time delays and filters, are called hybrid systems. A high priority of the project is the development of a mathematical theory of hybrid systems, with a special focus on engineering and computer applications. The project will use mathematical techniques originally developed for the study of symmetric dynamical systems, work on statistical and stochastic properties of dynamical systems, and develop new methods for the construction of relatively simple dynamical systems (modules) which satisfy specified dynamical and structural properties.Systems of coupled differential equations are used as models in a wide range of physical applications such as neuronal networks, speciation, gene dynamics, arrays of Josephson junctions, and mechanical, electrical and computer systems. The project aims to develop an improved mathematical understanding of the dynamics of these systems. It is a natural development and continuation of past research undertaken, notably in an NSF-sponsored project on "Synchrony and Structure in Coupled Cell Systems.? Part of the research will be undertaken in collaboration with groups in the United Kingdom (Manchester and Exeter) and Portugal (Porto). The Manchester group has a strong focus on applications in the areas of electrical and computer systems and biology, and these applications will help give a framework for the development of the theoretical aspects of networks. At least two graduate students will be directly supported by the grant. More generally, the research will directly impact the mathematical education of graduate and undergraduate students at the University of Houston through seminars, the development of graduate course in the area of differential equations, and undergraduate research projects in the area of networks. The investigator has shown a long-term commitment to the development of outreach programs for the professional development of mathematics and science teachers, notably through the Houston Teachers Institute and the Yale National Initiative. The proposed research will impact the development and content of these programs. The investigator will continue to work towards improving the understanding of mathematics and its applications among the general population by means of public lectures, educational publications, and popular scientific writings.