Lattice Differential Equations and the Computation of Stability Spectra

研究者和他的同事考虑微分方程解的近似问题。 微分方程通常用于在科学和工程的许多领域中模拟物理和生物现象。 微分方程是一种规则，是解与其导数之间的关系，它决定了初始配置如何演变为未来配置。 该项目的重点是李雅普诺夫指数和相关量的近似，这些指数和量提供了关于稳定性、附近构型在附近演化的趋势和不稳定性、附近构型在远离的趋势的信息，以及关于晶格微分方程的分析和计算，即，空间离散时间连续的微分方程。 这项工作的大部分强调严格的分析与算法的实际实现相结合。 这项研究的中心主题是：将动力系统和数值分析思想与微分方程的建模和分析相结合。 离散模型在物理和生物系统的建模中起着重要的作用。 特别感兴趣的是行波解的格子微分方程。 Sacker-Sell和Lyapunov谱区间是特征值的真实的部分的自然类似物，为时变微分方程提供稳定性信息。 研究人员开发，分析，并证明使用数值技术的近似这些频谱间隔。 一套计算模块的稳定性信息的计算和功能行波正在开发中。 它是由分析的数值技术的形式，应该证明有用的工作科学家和工程师。

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.