Mathematical Sciences: Interpretation of Data from Chaotic Processes

9305659 Sauer本研究的重点是对从混沌过程和模拟中收集的数据进行解释。利用确定性物理过程产生的单变量或多变量时间序列，可以忠实地重构该过程的相空间。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。在这些数学基础的许多应用中包括用于降噪(滤波)和时间序列预测的计算技术，以增强和/或取代来自信号处理和时间序列分析领域的传统方法，这些方法对于来自混沌过程的数据的适用性有限。这项研究的目标是开发可靠的诊断学，根据从实验或环境测量的数据，将线性和非线性噪声过程与非线性动力学区分开来。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。还与合作者一起研究了通过跟踪方法对混沌过程进行计算机模拟的有效性。这项研究的中心目标是通过测量和分析系统产生的定量信号来识别和分析系统中的非线性行为。新的技术允许利用确定性混沌的范例，对一些早先被认为是噪音或太难研究的系统进行可利用的确定性描述。在物理、化学、工程和生物/医学环境中发现了这类复杂的现象。例如，研究人员正在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。这项建议中所追求的重建技术的第一个数学基础是在1980年传播的，这几乎立即导致了几个科学学科的密集研究，其影响尚未完全感受到。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，在这一提议下，将扩大它们的有效性领域，并增加它们在自然、实验和工程相关背景下研究复杂系统的能力。

9305659 Sauer The focus of this research is the interpretation of data collected from chaotic processes and simulations. Using univariate or multivariate time series produced by a deterministic physical process, the phase space of the process can be faithfully reconstructed. 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. Among the many applications of these mathematical foundations are computational techniques for noise reduction (filtering) and time series prediction, to enhance and/or replace conventional methods from the fields of signal processing and time series analysis, which are of limited applicability for data from chaotic processes. A related task that is targeted by this research is the development of reliable diagnostics with which to distinguish linear and nonlinear noise processes from nonlinear dynamics, on the basis of data measured from experiment or the environment. New techniques for these applications are developed in this project, involving methods from numerical analysis used in conjunction with existing signal processing methods. The study of the validity of computer simulations of chaotic processes by means of shadowing is also pursued in conjunction with collaborators. The central goal of this research is to identify and analyze nonlinear behaviors in systems, by measurement and analysis of quantitative signals produced by that system. New techniques allow exploitable deterministic descriptions, using the paradigms of deterministic chaos, for some systems written off earlier as noise, or as too difficult to study. Complex phenomena of this type are being identified in physical, chemical, engineering and biological/medical settings. As an example, hippocampal slices from mammals and other small neural systems are being studied by the investigator, in conjunction with a g roup of medical researchers headed by a neurosurgeon specializing in epilepsy, with the purpose of detecting deterministic information processing in the brain. The first mathematical foundations of the reconstruction techniques pursued in this proposal were disseminated in 1980, which led almost immediately to intensive research across several scientific disciplines, the impact of which has not yet been fully felt. The investigator has done previous work on expanding these conceptual foundations and developing related computational implementations, and under this proposal will widen their areas of validity and increase their power for the study of complex systems in natural, experimental, and engineering-related contexts.