Numerical Spectral Analysis and Approximation of Functional Traveling Waves

在科学和工程的许多领域，微分方程被用作物理和生物现象的模型。研究者和他的同事们在这个建议中的重点是微分方程的近似解。研究者对稳定性谱（微分算子和差分算子的点谱、Sacker-Sell谱和Lyapunov指数）的分析和计算以及类似于时变偏微分方程的离散模型的分析和逼近技术感兴趣，但使用差分算子而不是空间微分算子。离散模型在物理和生物系统的建模中起着重要的作用。特别有趣的是格微分方程的行波解。所采取的方法是将动力系统和数值分析思想与常微分方程、偏微分方程和格微分方程的建模和分析相结合。这个项目是关于发展和分析有效的，准确的数值技术，是有用的计算和分析动力系统。Sacker-Sell和Lyapunov谱区间是特征值实部的自然类似物，为时变微分方程提供稳定性信息。研究者开发，分析，并证明使用数值技术来近似这些光谱间隔。目前正在开发一套用于稳定性信息计算和功能行波计算的计算模块。它是由一种形式的数值技术分析支持的，这种形式应该证明对工作中的科学家和工程师有用。研究者和他的同事们考虑微分方程解的近似和计算问题。在科学和工程的许多领域中，微分方程通常用于模拟物理和生物现象。微分方程是一种规则，一种解与解的变化率之间的关系，它决定了初始构型如何演变成未来构型。这个项目的重点是李雅普诺夫指数的近似和相关的量，这些量提供了关于稳定性的信息，即附近构型进化和保持在附近的趋势，以及不稳定性，即附近构型分离的趋势。这种类型的分析对于理解环境中发生的复杂生物现象和识别不稳定性（例如，天气预报模型）非常有用。晶格微分方程的分析和计算，即在空间上离散而在时间上连续的微分方程，在物理和生物系统的建模中是重要的，其中空间成分是自然离散的，特别是对于材料和生理学的微观模型。

Differential equations are used as models of physical and biological phenomena in many areas of science and engineering. The focus of the investigator and his colleagues in this proposal is on the approximation of solutions of differential equations. The investigator is interested in the analysis and computation of stability spectra (point spectrum of differential and difference operators, Sacker-Sell spectrum, and Lyapunov exponents) and techniques for analysis and approximation of discrete models similar to time dependent partial differential equations but with a difference operator instead of a spatial differential operator. Discrete models play a prominent role in the modeling of physical and biological systems. Of particular interest are traveling wave solutions of lattice differential equations. The approach taken is to combine dynamical systems and numerical analysis ideas with the modeling and analysis of ordinary, partial, and lattice differential equations. This project is concerned with the development and analysis of efficient, accurate numerical techniques that are useful for the computation and analysis of dynamical systems. Sacker-Sell and Lyapunov spectral intervals are natural analogues of the real parts of the eigenvalues that provide stability information for time varying differential equations. The investigator develops, analyzes, and justifies the use of numerical techniques for the approximation of these spectral intervals. A suite of computational modules for the computation of stability information and for functional traveling waves is being developed. It is backed by analysis of the numerical techniques in a form that should prove useful to working scientists and engineers.The investigator and his colleagues consider issues in the approximation and computation of solutions of differential equations. Differential equations are commonly used to model physical and biological phenomena in many areas of science and engineering. A differential equation is a rule, a relationship between the solution and the rate of change of the solution, that determines how an initial configuration evolves into future configurations. The focus of this project is on the approximation of Lyapunov exponents and related quantities that provide information on stability, the tendency for nearby configurations to evolve and stay nearby, and instability, the tendency for nearby configurations to move apart. This type of analysis is useful in understanding complex biological phenomena that occur in the environment and in identifying instabilities in, for example, models of weather prediction. The analysis and computation of lattice differential equations, i.e., differential equations that are discrete in space and continuous in time, are important in the modeling of physical and biological systems in which the spatial component is naturally discrete, in particular for microscopic models in materials and physiology.