Dynamics of Coupled Cell Systems

该项目旨在促进对相互作用的微分方程和地图网络动态的理解。该项目特别关注振荡和其他复杂的动力学行为，如异宿周期和混沌动力学，可以通过网络架构来实现。许多物理、电学和生物系统可以通过相互作用的微分方程和映射网络来建模。这种类型的模型，它可以涉及连续和离散动态系统的混合，可能有时间延迟和过滤器，被称为混合系统。该项目的一个高度优先事项是混合系统的数学理论的发展，特别注重工程和计算机应用。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，并开发新方法来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）。耦合微分方程系统被用作广泛物理应用的模型，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的动力学的更好的数学理解。这是过去研究的自然发展和延续，特别是在NSF赞助的“耦合细胞系统的同步性和结构”项目中。部分研究将与联合王国（曼彻斯特和埃克塞特）和葡萄牙（波尔图）的团体合作进行。曼彻斯特小组非常关注电气和计算机系统以及生物学领域的应用，这些应用将有助于为网络理论方面的发展提供一个框架。至少有两名研究生将直接得到赠款的支持。更一般地说，研究将直接影响休斯顿大学研究生和本科生的数学教育，通过研讨会，研究生课程的发展在微分方程领域，并在网络领域的本科生研究项目。研究人员已经表现出长期致力于数学和科学教师专业发展的推广计划的发展，特别是通过休斯顿教师研究所和耶鲁国家倡议。拟议的研究将影响这些方案的发展和内容。研究人员将继续致力于通过公开讲座，教育出版物和科普著作来提高公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.