Dynamical aspects in nonautonomous and random differential equations and applications

这个项目是研究由各种应用问题引起的非自治和随机微分方程的动力学。具体地说，研究人员将探索(1)在相变、神经脉冲传播、种群遗传学和细胞神经网络中出现的扩散随机和非均匀介质中的波前动力学；(2)非自治和随机线性抛物型方程的谱和Lyapunov指数，以及在竞争系统中的一致持久性和共存的应用；以及(3)随机强迫振子中的全局动力学，包括超导电路中的约瑟夫森结。该项目包括推广微分方程和动力系统中的经典概念，以及为研究该项目中的问题开发和实施新的工具和技术。该项目的结果将加深对所研究系统动力学的了解，将为进一步研究这些系统及其相关系统提供理论和方法论基础，并将使三个相互独立但相互关联的数学分支更加紧密地联系在一起：微分方程、拓扑动力系统和度规动力系统，丰富了它们各自的内容。现实的物理和生物系统受到外部环境变化的影响，通常位于各向异性或非均匀介质中。因此，通过涉及非自治或随机或随机微分方程的模型来研究这类系统得到了越来越多的关注。由于缺乏一般的方法论，以及推广经典概念和概念的困难，人们对这些方程中的许多仍然知之甚少。研究人员将扩展相关的经典概念和概念，并开发新的工具和技术来研究这些方程的动力学。该项目的结果将深入了解随机外部影响和介质的不均匀对物理和生物问题模型中动力学的影响，包括相变、神经脉冲传播、种群动力学、细胞神经网络、图案形成和超导电路。该项目将为该领域的研究生提供培训，并将为学生创造与其他学科的科学家互动的机会。

This project is to investigate the dynamics of nonautonomous and random differential equations arising from a variety of applied problems. In particular, the investigator will explore (1) wave front dynamics in diffusive random and inhomogeneous media arising in phase transitions, nerve pulse propagation, population genetics, and cellular neural networks; (2) spectra and Lyapunov exponents for nonautonomous and random linear parabolic equations, with applications to uniform persistence and coexistence in competitive systems; and (3) global dynamics in stochastically forced oscillators, including Josephson junctions in superconducting circuits. The project involves extension of classical notions in differential equations and dynamical systems as well as the development and implementation of new tools and techniques for the study of the problems in the project. The results of the project will enhance the understanding of the dynamics of the systems under investigation, will provide theoretical and methodological foundations for the further study of these systems and related ones, and will bring closer together three separate, but related, branches of mathematics: differential equations, topological dynamical systems, and metric dynamical systems, enriching each of them.Realistic physical and biological systems are influenced by variations in the external environment, and are often situated in anisotropic or inhomogeneous media. For this reason, the study of such systems via models involving nonautonomous or random or stochastic differential equations has been gaining more and more attention. Due to a lack of general methodology and difficulties in generalizing classical concepts and notions, there is still little understanding of many of these equations. The investigator will extend relevant classical concepts and notions and develop new tools and techniques to study the dynamics of these equations. The results of the project will provide deep insight into the effects of random external influences and inhomogeneity of media on the dynamics in models of physical and biological problems including phase transitions, nerve pulse propagation, population dynamics, cellular neural networks, pattern formation, and superconducting circuits. The project will provide graduate students training in the area and will create opportunities for students to interact with scientists from other disciplines.