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

Thesis (Ph. D.)--University of Washington, 2006. / Vita. Includes bibliographical references (leaves 186-200).

Thesis (Ph.D.)--University of Missouri-Columbia, 2006. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (February 28, 2007) Vita. Includes bibliographical references.

Nonlinear optical phenomena associated with high-order soliton breakup in photonic crystal fibres and squeezed state generation in three dimensional photonic crystal microcavities are investigated. In both cases, the properties of periodically patterned, high-index contrast dielectric structures are engineered to control the dispersion and local field enhancements of the electromagnetic field. Ultra-short pulse propagation in a polarization-maintaining microstructured fibre (with 1 um core diameter and 1.1 m length) is investigated experimentally and theoretically. For an 80 MHz train of 130 fs pulses with average propagating powers in the fibre up to 13.8 mW, the output spectra consist of multiple discrete solitons that shift continuously to lower energies as they propagate in the lowest transverse mode of the fibre. The number of solitons and the amount that they shift both increase with the launched power. All of the data is quantitatively consistent with solutions of the nonlinear Schrodinger equation, but only when the Raman nonlinearity is treated without approximation, and self-steepening is included. The feasibility of using a parametric down-conversion process to generate squeezed electromagnetic states in 3D photonic crystal microcavity structures is investigated for the first time. The spectrum of the squeezed light is theoretically calculated by using an open cavity quantum mechanical formalism. The cavity communicates with two main channels, which model vertical radiation losses and coupling into a single-mode waveguide respectively. The amount of squeezing is determined by the correlation functions relating the field quadratures of light coupled into the waveguide. All of the relevant model parameters are realistically estimated using 3D finite-difference time-domain (FDTD) simulations. Squeezing up to ~20% below the shot noise level is predicted for reasonable optical excitation levels. To preserve the squeezed nature of the light generated in the microcavity, a unidirectional coupling geometry from the microcavity to a ridge waveguide in a slab photonic crystal structure is studied. The structure was successfully fabricated in a silicon membrane, and experimental measurements of the efficiency for the signal coupled out of the structure are in good agreement with the result of FDTD simulations. The coupling efficiency of the cavity mode to the output channel is ~60%.

Epilepsy is a nervous system disorder which affects approximately 1% of the world's population. Nearly 25% of people who have epilepsy are resistant to traditional treatments such as medication and are not candidates for surgery [32]. A new form of treatment has emerged that attempts to disrupt epileptic activity in the brain by electrically stimulating neural tissue. However, the nature of this treatment requires that it is able to accurately predict the onset of a seizure in order to time the intervention correctly. Recent studies suggest that EEG recordings may be generated by a low dimensional nonlinear process [35] [36] [6]. This paper will investigate nonlinearity tests, as well as the use of methods from the theory of nonlinear dynamical systems in the prediction of seizures or seizure like events (SLEs) from complex time series. To do this data is generated from a nonlinear dynamical system with a stochastic time dependent parameter, which attempts to emulate the different states of an epileptic brain. Two kinds of nonlinearity tests were used in simulations, one which specifies a model in the alternative hypothesis (Keenans test) and one which simply states that the process is `not linear' (Surrogate data test). The tests were applied to the generated data, as well as a short EEG recording from a person with epilepsy and a simple nonstationary example. Both tests were able to correctly identify the model as nonlinear, neither test identified the EEG data as nonlinear and there were contradicting results when the tests were applied to nonstationary data. Estimates of the correlation dimension and Lyapunov exponent were then used to classify the preictal state of the model data. Correlation dimensions showed the best ability to classify states, so they were used in the prediction algorithm. The results of the simulation was that the correlation dimension was able to successfully predict half of the SLEs, however there was an alarmingly high false prediction rate. These results suggest that even though a complicated model may fit the data better, when dealing with prediction it is usually best to use a simple model. A simpler approach with better understood statistical properties may be able to improve on the prediction of SLEs as well as reduce the computational cost of performing them.

Epilepsy is a nervous system disorder which affects approximately 1% of the world's population. Nearly 25% of people who have epilepsy are resistant to traditional treatments such as medication and are not candidates for surgery [32]. A new form of treatment has emerged that attempts to disrupt epileptic activity in the brain by electrically stimulating neural tissue. However, the nature of this treatment requires that it is able to accurately predict the onset of a seizure in order to time the intervention correctly. Recent studies suggest that EEG recordings may be generated by a low dimensional nonlinear process [35] [36] [6]. This paper will investigate nonlinearity tests, as well as the use of methods from the theory of nonlinear dynamical systems in the prediction of seizures or seizure like events (SLEs) from complex time series. To do this data is generated from a nonlinear dynamical system with a stochastic time dependent parameter, which attempts to emulate the different states of an epileptic brain. Two kinds of nonlinearity tests were used in simulations, one which specifies a model in the alternative hypothesis (Keenans test) and one which simply states that the process is `not linear' (Surrogate data test). The tests were applied to the generated data, as well as a short EEG recording from a person with epilepsy and a simple nonstationary example. Both tests were able to correctly identify the model as nonlinear, neither test identified the EEG data as nonlinear and there were contradicting results when the tests were applied to nonstationary data. Estimates of the correlation dimension and Lyapunov exponent were then used to classify the preictal state of the model data. Correlation dimensions showed the best ability to classify states, so they were used in the prediction algorithm. The results of the simulation was that the correlation dimension was able to successfully predict half of the SLEs, however there was an alarmingly high false prediction rate. These results suggest that even though a complicated model may fit the data better, when dealing with prediction it is usually best to use a simple model. A simpler approach with better understood statistical properties may be able to improve on the prediction of SLEs as well as reduce the computational cost of performing them.

Nonlinear acoustic media for implementations of parametric generation of low frequencies has so far been restricted to small values of the parameter B/A, typically between 3 and 13. Parametric amplification, defined as the generation of a low difference frequency signal resulting from the nonlinear interactions of two higher frequency fundamentals is enhanced by medium with a large coefficient of nonlinearity and low sound speed. The acoustic properties of a highly nonlinear medium were estimated and introduced in a numerical model, to evaluate the parametric amplification induced by a thin layer of such material in contact with a spherical transducer. The numerical model predicted a significant enhancement of the sound pressure level for the difference frequency component relative to that obtained when the transducer is driven linearly at the difference frequency. A source was then constructed to compare the theoretical predictions with experimental values and an enhancement of 17dB compared to the linear operation of the transducer was measured. The difference between the parametric amplification achieved with the nonlinear medium and the parametric amplification that would be obtained in water is 73dB.

Resonant or near resonant optical nonlinearities in semiconductor coupled quantum-well systems are discussed. Quantum engineered coupled or cascade quantumwell structures can provide giant nonlinear susceptibilities for various optical nonlinear processes. Nonlinearities integrated within quantum cascade lasers (QCL) showed great potential in various applications in the infrared range. Several schemes of nonlinearities are proposed and discussed in this work. Integrating difference frequency generation (DFG) with QCL can yield long wavelength radiation, such as terahertz light. The DFG process does not require population inversion at a transition associated with low photon energy; however, this requirement is necessary to lasers, such as QCL, and is hard to meet, because of the thermal backfilling and inefficient injection or pumping at room temperature. Therefore terahertz radiation due to DFG QCL for room temperature is proposed. On the other hand, the second harmonic generation can double laser frequency, and then push radiation frequency of AlInAs/GaInAs/InP based QCL to short wavelengths such as 3 μm and shorter. Optical nonlinearities can extend working frequencies of light sources, and also can help to improve light detection. For example, a sum frequency generation can upconvert mid/far-IR signal into near-IR signal with strong near-IR pump light, namely high efficient near-IR photon detector could be employed to detect mid/far-IR light. A specific designed quantum well structure of this frequency up-conversion scheme is discussed. A scheme of monolithic in-plane integration of the optical nonlinearities with QCL is also proposed. In this scheme, an optical nonlinear section is made from the same quantum well structure of a QCL, and is under an independent applied bias. Due to the independence of the applied bias, the nonlinearities can be tuned flexibly. In particular, a widely tunable Raman laser based on this scheme could be achieved. A frequency up-conversion based on sum frequency generation process in coupled quantum-well structure is also proposed for mid-infrared detection. By converting mid-IR signal to near-IR, superior near-IR detector such as silicon avalanche photo diode (APD) can be employed. The scheme can provide lower noise equivalent power (NEP) or higher detectivity compared with regular semiconductor photo detectors. A scheme of lasing without inversion (LWI) based on QCL for THz radiation is proposed. A ladder type three-level system for LWI process is integrated into a boundto- continue high power QCL at 10 μm. The proposed LWI generates THz signal at 69 μm. An optical gain about 80 cm-1 is achieved, against a waveguide loss about 30 cm-1 in a semi insulator (SI) surface plasmon waveguide.

This paper present three kinds of nonlinear numerical methods to search for perfect complementary codes, include Newtonian Methods¡BLevenberg-Marquardt Algorithm and Trust-Regions. By searching for the solution of theses nonlinear equations, we can get complementary codes when setting for the length of element codes and the flock size. These search results is very generous. Complete complementary codes¡Bsuper complementary code and poly-phase complementary code are subsets of these searching results¡C These nonlinear equations are set to have ideal auto-correlation and cross-correlation properties, so the searching results of these nonlinear equations are still have perfect orthogonal complementary properties. Because the orthogonal complementary code is obtained via these nonlinear equations, the results are the most generous. So nonlinear numerical method is a good choice to search for another complementary code we don¡¦t know.

Abstract The Purchasing power parity ¡]PPP¡^ theory was originally developed by a Swidish economist, Gustav Cassel, in 1916. It is a method using the long-run equilibrium exchange rate of two currencies to measure the currencies' purchasing power. It is based on the law of one price, the idea that, in an efficient market, identical goods must have only one price internationally. This parity is a central building block of many theoretical and empirical models of exchange rate determination, since most are relied on PPP as the basis for long-run real exchange rates. While the literature on the PPP hypothesis is voluminous today and still growing, the doctrine has not found well. The validity of PPP can be examined by testing the stationary of real exchange rates. Most of the empirical evidences relied mainly on using linear structure to explore PPP in the past. By using traditional unit root test, the PPP is hard to hold in the long run. There is a growing consensus that previous empirical research reflects the poor power of the tests rather necessarily against PPP. Therefore, the use of more powerful tests is needed. Recently, an alternative point of view based on the presence of market frictions that impede commodity trade has arisen. The adjustment of real exchange rates is perhaps described more appropriately as a nonlinear process once market frictions are taken into account. There are several reasons that theoretically explain why the adjustment process of deviations from PPP is nonlinear, such as transactions and transportation costs and tariffs and non-tariff barriers to international trade. Therefore, the analysis of real exchange rate should be conducted under the nonlinear structure. This study uses the STAR methodology proposed by Granger & Teräsvirta (1993) and Teräsvirta (1994) to examine whether the deviation of PPP is a nonlinear dynamic adjustment among the following countries: Australia, Denmark, Italy, Japan, Luxembourg, Norway, Spain, Sweden and the United Kingdom. If the linear hypothesis was rejected, then to distinguish if the model of STAR is LSTAR or ESTAR. This study finds that the deviations from equilibrium exchange rates show strong evidence of nonlinear properties. The deviations of exchange rates for all countries can be explained by the LSTAR model. In conclusion, this study finds the real exchange rates exhibit the property of nonlinear mean reversion for most countries.