Lattice Differential Equations and the Computation of Stability Spectra

这位研究人员和他的同事们研究了微分方程解的近似问题。在科学和工程的许多领域中，常用微分方程来模拟物理和生物现象。微分方程是一条规则，是解与其导数之间的关系，它决定了初始构型如何演变为未来的构型。这个项目的重点是关于Lyapunov指数和相关量的近似，它提供了关于稳定性的信息，附近组态在附近演化的趋势，以及不稳定性，附近组态移动的趋势，以及格点微分方程的分析和计算，即在空间上离散的和在时间上连续的微分方程。这项工作的大部分都强调将严格的分析与算法的实际实现相结合。本研究以动力系统与数值分析思想相结合以及微分方程的建模与分析为中心主题。离散模型在物理和生物系统的建模中扮演着重要的角色。格点微分方程行波解是特别有趣的。Sacker-Sell和Lyapunov谱区间是为时变微分方程提供稳定性信息的特征值实部的自然类似。研究人员开发、分析并证明使用数值技术来近似这些光谱间隔是合理的。一套用于计算稳定性信息和函数行波的计算模块正在开发中。它以对数值技术的分析为后盾，这种形式应该被证明对工作的科学家和工程师有用。

The investigator and his colleagues consider issues in the approximation 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 its derivatives, 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 nearby, and instability, the tendency for nearby configurations to move apart, and on the analysis and computation of lattice differential equations, i.e., differential equations that are discrete in space and continuous in time. Much of this work emphasizes the blending of rigorous analysis with practical implementation of algorithms. This research has as a central theme: the combination of dynamical systems and numerical analysis ideas together with the modeling and analysis of differential equations. 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. 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.