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Thesis (M.S.)--Rochester Institute of Technology, 1991. / Typescript. Includes bibliographical references.

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (leaves [117]-122).

Thesis (M.S. in Applied Physics)--Naval Postgraduate School, June 2002. / Thesis advisor(s): Bruce Denardo, Andres Larraza. Includes bibliographical references (p. 55). Also available online.

In previous papers by Awrejcewicz in 1986 and Narayanan and Jayaraman in 1991, it was claimed that the nonlinear oscillator with dry friction exhibited chaos for several forcing frequencies. The chaos determination was achieved using the characteristic exponent of Lyapunov which requires the right-hand side of the differential equation to be differentiable. With the addition of the dry friction term, the right-hand side of the equation of motion is not continuous and therefore not differentiable. Thus this approach cannot be used. The Filippov definition must be employed to handle the discontinuity in the spatial variable. The behavior of the nonlinear oscillator with dry friction is studied using a numerical solver which produces the Filippov solution. The results show that the system is not chaotic; rather it has a stable periodic limit cycle for at least one forcing frequency. Other forcing frequencies produce results that do not clearly indicate the presence of chaotic motion. / Graduation date: 2002

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The topic of this dissertation, modeling and identification of nonlinear oscillation, represents an area of mathematical systems theory that has received little attention in the past. Primarily, the types of oscillation of interest are those found in biological systems where theoretical foundations for mathematical models are insufficient. These oscillations are also observed in other systems including electrical, mechanical, and chemical. The contributions of this dissertation are a generalized class of autonomous differential equations that are found to exhibit stable limit cycles, and an investigation of a method of system identification that can be used to estimate the model parameters. Here the observed signal is modeled as the response of a nonlinear system that can be described by differential equations. Modeling the signal in this way shifts the emphasis from signal characteristics, such as spectral content, to system characteristics, such as parameter values and system structure. This shift in emphasis may provide a better method for monitoring complex systems that exhibit periodic behavior such as patients under anesthesia. A class of autonomous differential equations, called the generalized oscillator models, are presented as one nᵗʰ-order differential equations with nonlinear coefficients. The coefficients are chosen to change sign depending on the magnitude of the phase variables. The coefficients are negative near the origin and positive away from the origin. Motivated by the generalized Routh-Hurwitz criterion, this coefficient sign changing produces the desired oscillation. Properties of the generalized oscillator model are investigated using the describing function method of analysis and numerical simulation. Several descriptive examples are presented. Based on the generalized oscillator model as a set of candidate models, the system identification problem is formed as a mathematical programming problem. The method of quasilinearization is investigated as method of solving the identification problem. Two examples are presented that demonstrate the method. It is shown that in general, the method of quasilinearization as a solution to the system identification problem will not converge regardless of the initial starting point. This result indicates that although the quasilinearization method is useful for solving two-point boundary value problems, it is not useful (in its present form) for solving the system identification problem.

This work investigates the response of the solar atmosphere to mechanical and thermal driving due to global solar oscillations. It was discovered that the coupling of thermal and mechanical modes was very important in reconciling theoretical predictions of the expected change in the solar limb due to solar oscillations and experimental observations of the variability in the solar limb darkening function undertaken at SCLERA (Santa Catalina Laboratory for Experimental Relativity). The coupling between the thermal and mechanical modes occur mainly due to the nonlocal nature of the radiation field. Previous theoretical calculations that used approximations for the radiative transfer that ignored the nonlocal nature of the radiation field predicted expected temperature perturbations (compared to the fluid displacement) that were much too small to be observed. Much larger ratios were found when the radiative transfer was treated properly. A particular solar oscillation can be influenced by the presence of a large number of other modes, if these modes can change the average properties of the medium. If the basic nonlinear equations are statistically averaged, the influence of the "mean field" can be investigated. This nonlinear effect can become important in the analysis for single modes in the upper photosphere.