Chaotic Systems: Reliability of Simulations and Interpretation of Data

9626197绍尔本研究侧重于混沌动力学的计算方面，重点是在科学和工程领域具有影响的问题。第一个主要领域涉及的问题是，即使在原则上，典型的非双曲混沌系统的长期计算机模拟是否可能近似地匹配真实的系统行为。这项研究的一部分涉及在计算机模拟轨迹和计算机模型轨迹之间开发故障或不匹配之间预期等待时间的定量定律。第二个主要领域是研究者正在进行的关于产生非周期时间序列的物理和生物实验的解释的工作。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，可以忠实地重建过程的相空间，作为系统识别、滤波、预测和控制等应用的基础。在各种物理可验证的背景下保证这一点的拓扑和微分同构嵌入定理是研究者和同事以前和正在进行的研究的结果。计算机模拟是现代科学的重要组成部分。正在制定的国家政策决定部分依赖于对非线性模型的长期计算机模拟的解释。本文探讨的问题对非线性过程的模拟分析至关重要。本提案的一个目标是在物理相关模型中探索这些问题，特别是隔离和量化这些表示的局限性，特别是为了长期建模的目的。本研究的第二个重点是解释从实验和自然界的混沌系统中收集的数据。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。作为一个例子，研究人员与一组医学研究人员一起研究了哺乳动物的海马切片和其他小的神经系统，这些研究人员由一名专门研究癫痫的神经外科医生领导，目的是检测大脑中的确定性信息处理。研究者已经完成了扩展这些概念基础和开发相关计算实现的先前工作，并致力于扩大其有效性领域，并增加其在自然，实验和工程相关环境中复杂系统研究的能力。本项目开发了这些应用的新技术，包括与现有信号处理方法结合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.