| The theory and applications of deterministic chaos have received a great deal of attention during the last decade, with several new and valuable approaches introduced that can be used to obtain a clearer understanding of the origins of such signals and the nature of the systems responsible for their presence. Mutual information theory, for example, a concept introduced by A. Fraser (Physical Review A, 1986), can be used to address the choice of an optimal embedding time step in order to avoid oversampling experimental data. For the most part, however, current tools for the analysis of apparently chaotic signals lack in their ability to adequately address the significance of time evolution within their methodology.; This dissertation introduces a new method for probing whether a signal has a deterministic or purely random origin. The approach employs a time-dependent clustering quantizer (TBC) to transform the original waveform data into a symbol train, which can then be analyzed for excluded symbol combinations. A hypothesis test is used to bound the likelihood of randomness of a complex time series, using Markoff chains to calculate the probability of missing and existing symbol combinations. Finally, J. Theiler's technique of surrogate data (Physica D, 1992) is employed to strengthen these quantitative results. It is shown that the new TBC quantizer unifies the concepts of mutual information theory with attractor reconstruction time-embedding, as a means of obtaining dynamically optimal signal coarsening.; Future chaotic system research and directions for applications of the TBC method include possible new attractor reconstructions with a generalization of the underlying time-dependent clustering method quantizer, development of cluster-based models for complex dynamical systems such as weather and communication phenomena, as well as the fundamental problem of controlling the behavior of systems subject to chaotic behavior. |
