Topics in Computational Dynamics

复杂系统的模型通常涉及到非线性微分方程组，必须对其进行数值近似才能获得详细的信息。为了了解这种近似的行为，本项目探索了原始模型的动态行为与通过数值模拟获得的行为的异同。当了解原始模型和数值模拟何时产生类似的动态行为时，可以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复杂生物和物理现象的稳健性。在这个项目中特别强调的是将这些结果应用于气候动力学模型的动态行为。有待开发的技术包括基于Lyapunov指数理论的降维方法和不确定性量化方法。由于开发了处理详细微观模型的动态行为的计算技术，对生物和物理过程的理解将会增加。在科学和工程的许多领域中，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解的逼近。特别是，研究人员和同事对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感兴趣。拟研究的主要课题是变量的含时正交变换在离散化下的行波动力学中的应用。具体内容包括利用Lyapunov向量(类似于时间相关稳定性分析中的特征向量)降维的耗散非线性微分方程组，含时线性哈密顿系统的鲁棒稳定性，以及数值时间推进技术的时间相关稳定性，周期介质中双稳态反应扩散方程的平面波解的稳定性，时间和空间离散化对行波的影响，以及离散介质中的非平面行波。研究这些系统的技术结合了数值分析和动力系统的思想，以在定性和定量上更好地理解计算动力学。

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.