Computational Methods in Applications of Nonlinear Dynamics

近年来，工程和科学中动态现象的建模，从高能激光到天气预报，由于对非线性效应的更复杂的处理，其精度已经显著提高。专注于如何进行改进的非线性动力学领域也经历了同样重要的增长。然而，更精确的建模会导致更精细的计算模拟。这个项目的一个重点是研究计算机模拟的有效性问题。我们计划研究非线性动力学和计算机算法之间的相互作用，这些相互作用令人惊讶地经常被证明对模拟结果有重大影响。该项目的第二个重点是研究实验多变量时间序列和棘波训练数据解释中的基本问题，包括来自物理实验、癫痫发作和信号处理的数据。在物理、化学、工程和生物/医学环境中，正在确定复杂的确定性时间序列。将开发技术来重建过程的动态行为，有可能预测或控制过程。该项目的更广泛影响来自于PI本科生和研究生教学中研究主题的整合，对研究生研究的支持，以及通过与非数学家合作者的合作来加强研究基础设施。提案中提出的基本问题与本科生和研究生水平的数值分析和动力系统导论之间有着密切的联系；PI已经共同撰写了一篇关于后一主题的文本，并正在完成一篇关于前者的本科生文本(预计将于2005年12月出版)，这表明了将研究的好处转化为教学的持续承诺。

In recent years, modeling of dynamic phenomena in engineering and science, from high-energy lasers to weather prediction, has dramatically increased in accuracy due to more sophisticated treatment of nonlinear effects. The field of nonlinear dynamics, which is focused on exactly how to carry out the improvements, has experienced a parallel increase in importance. However, more accurate modeling leads to more delicate computational simulations. One focus of this project is to investigate issues of validity of computer simulations. We plan to study the interaction between nonlinear dynamics and computer arithmetic, that surprisingly often proves to have a major impact on simulation outcomes.The second focus of the project is research on fundamental questions in the interpretation of experimental multivariate time series and spike train data, including data from physical experiments, epileptic seizures, and signal processing. Complex deterministic time series are being identified in physical, chemical, engineering and biological/medical settings. Techniques will be developed to reconstruct the dynamical behavior of the process, with potential to predict or control the process.The broader impact of the project derives from integration of the research topics in the PI's undergraduate and graduate teaching, the support of graduate student research, and the enhancement of research infrastructure through collaboration with non-mathematician collaborators. There are close connections between the fundamental questions posed in the proposal and introductory numerical analysis and dynamical systems at the undergraduate and graduate levels; the PI has co-authored a text on the latter topic and is finishing an undergraduate text on the former (publication expected Dec. 2005), demonstrating an ongoing commitment to transferring benefits of research into teaching.