Chaotic Systems: Reliability of Simulations and Interpretation of Data

9626197鲍尔这项研究集中在混沌动力学的计算方面，重点放在涉及科学和工程的问题上。第一个主要领域涉及的问题是，即使在原则上，是否有可能对典型的非双曲混沌系统进行长期的计算机模拟，以近似匹配真实的系统行为。这项研究的一部分涉及开发计算机模拟轨迹和计算机模型轨迹之间的故障或不匹配之间的预期等待时间的量化规律。第二个主要领域是调查员正在进行的关于解释产生非周期性时间序列的物理和生物实验的工作。利用确定性物理过程产生的单变量或多变量时间序列或事件间时间间隔，可以忠实地重构过程的相空间，作为系统辨识、滤波、预测和控制等应用的基础。在各种物理可验证的上下文中保证这一点的拓扑和微分同胚嵌入定理已经是研究者和同事先前和正在进行的研究的结果。计算机模拟是现代科学的主要内容。国家决策的制定在一定程度上依赖于对非线性模型的长期计算机模拟的解释。提案中探讨的问题对于分析非线性过程的模拟是至关重要的。这项建议的一个目标是在物理相关的模型中探索这些问题，特别是分离和量化这些表示的限制，特别是为了长期建模的目的。这项研究的第二个重点是对从实验和自然中的混沌系统收集的数据的解释。复杂的确定性时间序列在物理、化学、工程和生物/医学环境中被识别。例如，研究人员与一名专门研究癫痫的神经外科医生领导的一组医学研究人员一起研究了哺乳动物和其他小型神经系统的海马片，目的是检测大脑中确定性的信息处理。研究人员已经在扩展这些概念基础和开发相关计算实现方面做了以前的工作，并努力扩大它们的有效性领域，增加它们在自然、实验和工程相关背景下研究复杂系统的能力。在这个项目中为这些应用开发了新的技术，涉及到结合现有信号处理方法使用的数值分析方法。

9626197 Sauer This research focuses on computational aspects of chaotic dynamics, with emphasis on questions that have implications across the sciences and engineering. The first main area involves the question of whether it is possible, even in principle, for long-term computer simulations of typical nonhyperbolic chaotic systems to approximately match true system behavior. Part of this research involves developing quantitative laws of the expected waiting time between breakdowns, or mismatches, between computer simulation trajectories and the trajectories of computer models. The second major area is ongoing work by the investigator on the interpretation of physical and biological experiments that generate aperiodic time series. Using univariate or multivariate time series or inter-event time intervals produced by a deterministic physical process, the phase space of the process can be faithfully reconstructed as the basis for applications such as system identification, filtering, prediction and control. Topological and diffeomorphic embedding theorems guaranteeing this in various physically verifiable contexts have been the results of previous and ongoing research by the investigator and coworkers. Computer simulations are a staple of modern science. National policy decisions are being made which rely partially on the interpretation of long-term computer simulations of nonlinear models. The questions explored in the proposal are critical to the analysis of simulations of nonlinear processes. One goal of this proposal is to explore these questions in physically relevant models, and in particular to isolate and quantify the limitations of these representations, especially for the purpose of long-term modeling. The second focus of this research is the interpretation of data collected from chaotic systems in experiments and nature. Complex deterministic time series are being identified in physical, chemical, engineering and biological/medical setti ngs. As an example, hippocampal slices from mammals and other small neural systems are studied by the investigator, in conjunction with a group of medical researchers headed by a neurosurgeon specializing in epilepsy, with the purpose of detecting deterministic information processing in the brain. The investigator has done previous work on expanding these conceptual foundations and developing related computational implementations, and works to widen their areas of validity and increase their power for the study of complex systems in natural, experimental, and engineering-related contexts. New techniques for these applications are developed in this project, involving methods from numerical analysis used in conjunction with existing signal processing methods.