Topics in Computational Dynamics

复杂系统的模型通常涉及非线性微分方程，必须用数值近似来获得详细信息。为了了解这种近似的行为，本项目探讨了原始模型的动态行为的异同，以及通过数值模拟得到的结果。了解原始模型和数值模拟何时产生相似的动态行为，可以在根据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复杂生物和物理现象的鲁棒性。本项目特别强调的是将这些结果应用于气候动力学模式的动力学行为。待开发的技术包括基于李亚普诺夫指数理论的降维方法和不确定性量化方法。生物和物理过程的理解将增加，由于计算技术的发展，以解决详细的动态行为，微观模型。在科学和工程的许多领域，动力系统被用作复杂现象的模型。这个项目的重点是非线性动力系统的近似解。特别是，研究者和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联。主要研究方向为随时间变化的正交变量变换在离散化行波动力学中的应用。具体兴趣包括使用Lyapunov向量（时变稳定性分析中特征向量的类似物）对耗散非线性微分方程进行降维，时变线性哈密顿系统的鲁棒稳定性，以及分析数值时间步进技术时变稳定性的技术，周期介质中双稳态反应扩散方程平面波解的稳定性。离散介质中时空离散化对行波和非平面行波的影响。研究这些系统的技术结合了数值分析和动力系统的思想，以更好地理解定性和定量的计算动力学。

Models of complex systems often involve nonlinear differential equations that must be approximated numerically to gain detailed information. To understand the behavior of such approximations, this project explores similarities and differences in the dynamic behavior of the original model and what is obtained through numerical simulations. Understanding when the original model and numerical simulations yield similar dynamic behavior leads to greater confidence when making inferences from simulation results. In addition, stability analysis is useful in understanding the robustness of complex biological and physical phenomena. Of particular emphasis in this project is the application of these results to the dynamic behavior of models of climate dynamics. The techniques to be developed include approaches to dimension reduction and uncertainty quantification based upon Lyapunov exponent theory. The understanding of biological and physical processes will increase due to the computational techniques developed to address the dynamic behavior of detailed, microscopic models. In many areas of science and engineering dynamical systems are employed as models of complex phenomena. The focus of this project is on approximation of solutions of nonlinear dynamical systems. In particular, the investigator and 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 main research topics to be investigated are related to the application of time dependent orthogonal change of variables and in the dynamics of traveling waves under discretization. Specific interests include the use of Lyapunov vectors (analogues of eigenvectors in time dependent stability analysis) for dimensional reduction of dissipative nonlinear differential equations, robust stability for time dependent linear Hamiltonian systems, and techniques for analyzing time dependent stability of numerical time stepping techniques, stability of plane wave solutions to bistable reaction-diffusion equations in periodic media, the impact of time and space discretization on traveling waves, and non-planar traveling waves in discrete media. Techniques to investigate these systems combine numerical analysis and dynamical systems ideas to gain a better understanding of computational dynamics both qualitatively and quantitatively.