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P.E. McSharry (1999) Innovations in Consistent Nonlinear Deterministic Prediction DPhil thesis, University of Oxford

Abstract

Observations arising from `real world' systems are invariably noisy. The absence of structurally perfect equations with which to describe these systems further complicates attempts at formulating a coherent modelling framework. In this thesis, both the construction and application of models within the paradigm of nonlinear determinism are explored. Nonlinear determinism may offer an intuitive explanation for irregular behaviour in systems which do not seem to be inherently stochastic. This approach is particularly valuable when a system has to be analysed from a time series of observations. In practice, the reconstruction of model-state spaces from delays of observations may be guided by embedding theorems, however, faced with many sources of uncertainty, caution should be exercised when choosing important parameters such as the reconstruction dimension. A new method which investigates the quality of such reconstructions is introduced, providing a means of tailoring the model-state space and approximated dynamics to achieve accurate predictions. This method is illustrated by calculating the minimum reconstruction dimension which provides a delay embedding of the Ikeda map. Further applications show that two infinite dimensional delay-differential equation systems can be successfully reconstructed in low dimensions. A novel technique which aims at identifying structural error in a particular model is presented; this technique identifies regions of model-state space which are inconsistent with the observational uncertainty. Realising that all models will be structurally inept to some extent, an ensemble over different model structures is considered in an attempt to account for this structural uncertainty. The characterisation of dynamical systems through Lyapunov exponents is fundamental to nonlinear analysis. A thorough investigation of the statistical properties of Lyapunov exponent estimates for a number of theoretical systems are presented. A method which aims to improve these Lyapunov exponent estimates is introduced. In addition to identifying non-convergence in these estimates, the method is used to explore a system's parameter space.

Keywords

Nonlinear dynamics, time series, uncertainty, prediction, chaos, Lyapunov exponents.