Neural networks for the characterization of nonlinear deterministic systems

This dissertation addresses the identification and characterization of the long term behavior of nonlinear systems from experimental data. Nonlinear signal processing techniques, based on artificial neural networks (ANNs), are used to construct empirical models from the experimental observations. These ad hoc models allow us to reconstruct the experimental dynamics and explore the associated bifurcation structure in phase and parameter space. We show that the usual discrete delay-based approaches are often incapable of reproducing observed bifurcation sequences. Poincare map representations extracted from the experimental time series can, sometimes, be used to circumvent this problem. A complete dynamic picture including bifurcations of steady states can, however, only be captured by a continuous-time model. We develop a novel ANN configuration for the construction of continuous-time models of systems. This configuration couples a "principal component" approach with a composite ANN based on an integrator scheme. The resulting ANN is able to correctly reproduce nontrivial behaviors of experimentally observed dynamic transitions. The approaches are demonstrated using data from several experimental systems: spatiotemporal variations from CO oxidation on Pt(110), period doublings and mixed-mode oscillations from Cu electrodissolution in phosphoric acid, and quasiperiodic behavior from Rayleigh-Binard convection in small aspect-ratio cells.; The breakdown in the predictive capabilities of discrete delay-based approximations can be partially attributed to the noninvertible character of such approximations. Noninvertible dynamics concepts can be used to validate the quality of the approximations of the neural network mapping and probe its extrapolation/interpolation range.; The approaches presented here lead to relatively simple models describing the dynamics of the systems, but they certainly do not substitute for a fundamental understanding of the underlying physical or chemical mechanism. Nevertheless these approaches provide a powerful tool for the understanding of the instabilities and for real-time prediction and control applications of systems for which fundamental models are not available. The availability of fully nonlinear models may allow us to design controllers using powerful nonlinear control techniques with potentially better results than the traditional schemes based on linearizations around set points.