Date of Graduation
Eberly College of Arts and Sciences
Practical methods, based upon linear systems theory, are explored for applications to nonlinear phenomena and are extended to a larger class of problems. An algorithm for stabilizing, characterizing, and tracking unstable steady states and periodic orbits in multidimensional dynamical systems is developed and applied to stabilize and characterize an unstable four-cell flame front of the Kuramoto-Sivashinsky equation with six unstable degrees of freedom. A new method is presented for probing chemical reaction mechanisms experimentally with perturbations and measurements of the response. Time series analysis and the methods of linear control theory are used to determine the Jacobian matrix of a reaction at a stable stationary state subjected to random perturbations. The method is demonstrated with time series of a model system, and its performance in the presence of noise is examined. A new theory based on the construction of a multitude of linear models, each serving to represent one small region of the phase space, is presented together. Details of its implementation are presented in predicting chaotic Kuramoto-Sivashinsky wave fronts, demonstrating how it overcomes some of the problems associated with high dimensionality phase spaces. Motivated by the relationship between nonlinear prediction methods and the capabilities of neural systems, we demonstrate the possible role of nonlinear phenomena in the morphogenesis of neural tracts.
Mihaliuk, Eugene Robert, "Identification and control of dynamical systems" (1999). Graduate Theses, Dissertations, and Problem Reports. 3166.