If an empirically derived dynamical model adequately reproduces the observed dynamic behaviour of the system it represents, then the structure and the parameters of that model must encode the observed behaviours. Signature Characterisation is a method for seeking and extracting those components from the empirical model that are precursors to the changes in the system behaviours. The empirical model must therefore be directed purely by the observed data. The Volterra Series is a non-parametric modelling method that allows empirical models of a large class of non-anticipative, nonlinear, time invariant, fading memory systems. Fortunately, this is not a strong discriminator because most physical systems belong to this class. In this work a framework for identifying arbitrary degree truncated Volterra Series Expansion is developed, implemented and tested. The main drawback of the Volterra Series is an explosion of the number of parameters needed to represent the high-degree Volterra Kernels. I have developed a tensorial representation of the Volterra Kernels, that simplifies both the notational and the computer representation of the high-degree kernels. This tensorial formulation has the advantage of scalability in both the memory length and the number of terms. The limits are set purely by the capacity of the computational device. I have used this formulation to test its correct functionality by numerically identifying the kernels of two nonlinear models through only their input- output data; one an analytical model the other a simulation model. The identified kernels are symmetric in the permutation of their indices; thereby, the candidate signatures are the distinct parameters of the kernels. Extracting these signatures for the analytical model, I have shown that at present a multi-layered neural network can predict the onset of amplitude distortion and frequency distortion with around 60% certainty. The labelling for supervised training was conducted manually and was the main source of the uncertainty. For the simulation model, I have shown that using the same extraction method produces vectors of the distinct Volterra Kernels that for the candidate signatures. However, the number of candidate signatures become difficult label; the usefulness of creating image graphs of the candidate signatures allows the analyst to discover the patterns of changes at a glance. The key conclusions are that, provided the system under observation conforms to the theories of the Volterra Series Expansion, then the tensorial identification of the kernels can be obtained to any degree. The identified kernels are symmetric, therefore the distinct subset of the kernel forms the candidate signatures. Using either a neural network or by manual means, the changes in the candidate signatures can be easily related to the changes in the simple models. The work of developing the Signature Characterisation Method using the Volterra Series Expansion has produced the set of tools for identify an arbitrary degree Volterra Kernels. Initial trails have shown that the candidate signatures do encode the system behaviour. One one hand, the ability to draw inferences from the image graphs needs to be developed further; on the other hand, the neural network approach for signature characterisation is suitable for progressing to more industrial trials.

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A renormalization analysis is presented for a generalized Harper equation (1 + α cos(2π(ω(i + 1/2) + φ)))ψi+1 + (1 + α cos(2π(ω(i − 1/2) + φ)))ψi−1 +2λ cos(2π(iω + φ))ψi = Eψi. (0.1) For values of the parameter ω having periodic continued-fraction expansion, we construct the periodic orbits of the renormalization strange sets in function space that govern the wave function fluctuations of the solutions of the generalized Harper equation in the strong-coupling limit λ→∞. For values of ω with non-periodic continued fraction expansions, we make some conjectures based on work of Mestel and Osbaldestin on the likely structure of the renormalization strange set.

Bibliography: leaves 95-97 / iv, 97 leaves : ill ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1986