Approximation of Infinite Dimensional Dynamics

在科学和工程的许多领域中，微分方程式被用作复杂现象的模型。这一建议的重点是无穷维动力系统解的逼近。特别是，这位研究人员和他的同事们感兴趣的是了解有限维近似的动力学以及它们如何与原始动力系统的动力学相关。需要研究的具体问题包括：空间离散反应扩散方程解的稳定性，所谓的反扩散格点微分方程解的动力学，中立型混合型泛函微分方程行波解的动力学和计算，时滞时滞方程周期轨道的严格计算，含时线性哈密顿系统的鲁棒稳定性，基于非线性流的Lyapunov类指数的数值计算技术，以及时滞方程和偏微分方程稳定谱的逼近技术。研究这些系统的技术结合了数值分析和动力系统的思想，以更好地理解近似动力学。在复杂系统中，用于建模的微分方程通常是无限维的，但为了执行数值模拟，这些模型必然要用有限维系统来近似。为了理解这些模型的行为，研究人员和他的同事们对原始无限维系统和有限维近似的行为的异同感兴趣，例如通过离散化获得的那些。了解离散化的影响有助于在从模拟结果进行推断时获得更大的信心。此外，稳定性分析有助于理解环境中发生的复杂生物现象的稳健性，并有助于识别不稳定因素，例如在天气预报模型中。生物和物理过程的离散模型正变得越来越重要，因为对详细的微观模型的需求增加，并应受益于改进的分析和计算能力。

In many areas of science and engineering differential equations are employed asmodels of complex phenomena. The focus in this proposal is on the approximationof solutions of infinite dimensional dynamical systems. In particular, the investigatorand his colleagues are interested in understanding the dynamics of the finite dimensional approximations and how they relate to the dynamics of the original dynamical system. The specific issues to be investigated are related to stability of solutions of spatially discrete reaction-diffusion equations, the dynamics of so-called anti-diffusion lattice differential equations, dynamics and computation of traveling waves for neutral mixed type functional differential equations and rigorous computation of periodic orbits for retarded delay equations, robust stability for time dependent linear Hamiltonian systems, numerical techniques for efficient computation of Lyapunov exponent like quantities based upon nonlinear flows, and approximation techniques for stability spectra of delay equations and partial differential equations. Techniques to investigate these systems combine numerical analysis and dynamical systems ideas to gain a better understanding of approximation dynamics. Often in complex systems the differential equations used in modeling are infinite dimensional, but necessarily these models are approximated by finite dimensional systems in order to perform numerical simulations. In order to understand the behavior of these models, the investigator and his colleagues are interested in similarities and differences in the behavior of the original infinite dimensional system and finite dimensional approximations, for example those obtained through discretization. Understanding the impact of discretization leads to greater confidence when making inferences from simulation results. In addition, stability analysis is useful in understanding the robustness of complex biological phenomena that occur in the environment and in identifying instabilities, for example, in models of weather prediction. Discrete models of both biological and physical processes are becoming more important as the need for detailed, microscopic models increases and should benefit from improved analysis and computational capabilities.