Orstavik, Odd-Halvdan Sakse; (1999) Analysis of chaotic multi-variate time-series from spatio-temporal dynamical systems. Doctoral thesis (Ph.D.), University College London (United Kingdom).

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Abstract

This work concerns the analysis of chaotic multi-variate time-series from spatio-temporal dynamical systems (STS). Such systems can be thought of as consisting of a collection of sub-systems at different spatial locations coupled together into one large system. These arise in many applications throughout science and engineering including most types of fluid flow, pattern formation in chemical and biological systems, dynamics of ecosystems, road traffic, vibration of structures such as beams, plates and shells, and many others. In many situations there is a desire to analyse data from STS in situations where little is known about the system generating the data. In particular one may have no idea of the system's structure, or even its state space. It has until now been an open question how to characterise, control and predict the future evolution of STS in these circumstances. To answer these questions this thesis builds on the chaotic time-series analysis framework that has been successfully developed for the analysis of lowdimensional systems. Coupled map lattices (CML) are used as model systems since these feature many of the characteristics of STS. Several new results that apply to spatio-temporal systems are presented and can be summarised as follows. By using a mix of temporal and spatial embedding techniques one is able to carry out reconstruction and cross-prediction on a time-series generated by a CML and the results show that spatio-temporal delay reconstructions give better predictability than standard methods using either time delays or spatial delays only. A framework for embedding spatio-temporal systems is proposed. Results also show that by using spatio-temporal embedding techniques with local observations one cannot detect the presence of spatial extent in CML's thus suggesting the impossibility of reconstructing the whole system from localised information. New methods for calculating Lyapunov spectra for STS, and for extracting related quantities such as KS entropy density and Lyapunov dimension density, have been developed both for the case where the underlying dynamics is known and directly from time-series.

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