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The dynamics of two three-degree-of-freedom systems with quadratic nonlinearities are studied. The first system has two simultaneous two-to-one internal resonances. The second has a combination internal resonance. In both cases the response to a primary resonant excitation of the third mode is studied. The method of multiple time scales is used to obtain the equations that govern the amplitudes and phases of the first system. Then the fixed points of these equations are obtained and their stability is determined. The fixed points undergo Hopf bifurcations, and the overall system response can be periodic or periodically, quasiperiodically, or chaotically modulated. The method of the time-averaged Lagrangian is used to obtain the equations that govern the amplitudes and phases of the second system. The fixed points of these equations are obtained and their stability is determined. These fixed points undergo Hopf bifurcations, and the overall system response can be periodic or a two- or three-torus. / Master of Science

The modified Riks/Wempner method was examined for elastic, geometrically nonlinear space trusses. A computer program, RWCNR, was developed using the Riks/Wempner algorithm. The study includes an examination of three trusses, of which two exhibit bifurcation. A method of branching onto secondary equilibrium paths was studied, and the computer program incorporates this concept. This method of branching was found to work well provided the intersection of the fundamental and secondary equilibrium paths is good. The computer program was found to be very useful and reliable. It successfully traced fundamental equilibrium paths, located critical points (load levels), and branched onto secondary equilibrium paths. The modified Riks/Wempner method was found to be a very reliable method of obtaining equilibrium paths up to, and beyond, limit points, and it could branch onto secondary equilibrium paths. / Master of Science

Thesis (MSc)--Stellenbosch University, 2000. / ENGLISH ABSTRACT: This study examined nonlinear modelling techniques as applied to dynamic systems, paying specific attention to the Method of Surrogate Data and its possibilities. Within the field of nonlinear modelling, we examined the following areas of study: attractor reconstruction, general model building techniques, cost functions, description length, and a specific modelling methodology. The Method of Surrogate Data was initially applied in a more conventional application, i.e. testing a time series for nonlinear, dynamic structure. Thereafter, it was used in a less conventional application; i.e. testing the residual vectors of a nonlinear model for membership of identically and independently distributed (i.i.d) noise. The importance of the initial surrogate analysis of a time series (determining whether the apparent structure of the time series is due to nonlinear, possibly chaotic behaviour) was illustrated. This study confrrmed that omitting this crucial step could lead to a flawed conclusion. If evidence of nonlinear structure in the time series was identified, a radial basis model was constructed, using sophisticated software based on a specific modelling methodology. The model is an iterative algorithm using minimum description length as the stop criterion. The residual vectors of the models generated by the algorithm, were tested for membership of the dynamic class described as i.i.d noise. The results of this surrogate analysis illustrated that, as the model captures more of the underlying dynamics of the system (description length decreases), the residual vector resembles Li.d noise. It also verified that the minimum description length criterion leads to models that capture the underlying dynamics of the time series, with the residual vector resembling Li.d noise. In the case of the "worst" model (largest description length), the residual vector could be distinguished from Li.d noise, confirming that it is not the "best" model. The residual vector of the "best" model (smallest description length), resembled Li.d noise, confirming that the minimum description length criterion selects a model that captures the underlying dynamics of the time series. These applications were illustrated through analysis and modelling of three time series: a time series generated by the Lorenz equations, a time series generated by electroencephalograhpic signal (EEG), and a series representing the percentage change in the daily closing price of the S&P500 index. / AFRIKAANSE OPSOMMING: In hierdie studie ondersoek ons nie-lineere modelleringstegnieke soos toegepas op dinamiese sisteme. Spesifieke aandag word geskenk aan die Metode van Surrogaat Data en die moontlikhede van hierdie metode. Binne die veld van nie-lineere modellering het ons die volgende terreine ondersoek: attraktor rekonstruksie, algemene modelleringstegnieke, kostefunksies, beskrywingslengte, en 'n spesifieke modelleringsalgoritme. Die Metode and Surrogaat Data is eerstens vir 'n meer algemene toepassing gebruik wat die gekose tydsreeks vir aanduidings van nie-lineere, dimanise struktuur toets. Tweedens, is dit vir 'n minder algemene toepassing gebruik wat die residuvektore van 'n nie-lineere model toets vir lidmaatskap van identiese en onafhanlike verspreide geraas. Die studie illustreer die noodsaaklikheid van die aanvanklike surrogaat analise van 'n tydsreeks, wat bepaal of die struktuur van die tydsreeks toegeskryf kan word aan nie-lineere, dalk chaotiese gedrag. Ons bevesting dat die weglating van hierdie analise tot foutiewelike resultate kan lei. Indien bewyse van nie-lineere gedrag in die tydsreeks gevind is, is 'n model van radiale basisfunksies gebou, deur gebruik te maak van gesofistikeerde programmatuur gebaseer op 'n spesifieke modelleringsmetodologie. Dit is 'n iteratiewe algoritme wat minimum beskrywingslengte as die termineringsmaatstaf gebruik. Die model se residuvektore is getoets vir lidmaatskap van die dinamiese klas wat as identiese en onafhanlike verspreide geraas bekend staan. Die studie verifieer dat die minimum beskrywingslengte as termineringsmaatstaf weI aanleiding tot modelle wat die onderliggende dinamika van die tydsreeks vasvang, met die ooreenstemmende residuvektor wat nie onderskei kan word van indentiese en onafhanklike verspreide geraas nie. In die geval van die "swakste" model (grootse beskrywingslengte), het die surrogaat analise gefaal omrede die residuvektor van indentiese en onafhanklike verspreide geraas onderskei kon word. Die residuvektor van die "beste" model (kleinste beskrywingslengte), kon nie van indentiese en onafhanklike verspreide geraas onderskei word nie en bevestig ons aanname. Hierdie toepassings is aan die hand van drie tydsreekse geillustreer: 'n tydsreeks wat deur die Lorenz vergelykings gegenereer is, 'n tydsreeks wat 'n elektroenkefalogram voorstel en derdens, 'n tydsreeks wat die persentasie verandering van die S&P500 indeks se daaglikse sluitingsprys voorstel.

published_or_final_version / Civil Engineering / Doctoral / Doctor of Philosophy

The Hamiltonian of a stably stratified incompressible fluid in an internal water wave in a deep ocean is constructed. Studying the ocean internal wave field with its full dynamics is formidable (or unsolvable) so we consider a test-wave Hamiltonian to study the dynamical and statistical properties of the internal water wave field in a deep ocean. Chaos is present in the internal test-wave dynamics using actual coupling coefficients. Moreover, there exists a certain separatrix net that fills the phase space and is covered by a thin stochastic layer for a two-triad pure resonant interaction. The stochastic web implies the existence of diffusion of the Arnold type for the minimum dimension of a non-integrable autonomous system. For non-resonant case, stochastic layer is formed where the separatrix from KAM theory is disrupted. However, the stochasticity does not increase monotonically with increasing energy. Also, the problem of relaxation process is studied via microscopic Hamiltonian model of the test-wave interacting nonlinearly with ambient waves. Using the Mori projection technique, the projected trajectory of the test-wave is transformed to a form which corresponds to a generalized Langevin equation. The mean action of the test-wave grows ballistically for a short time regime, and quenches back to the normal diffusion for a intermediate time regime and regresses linearly to a state of statistical equilibrium. Applying the Nakajima-Zwanzig technique on the test-wave system, we get the generalized master equation on the test-wave system which is non-Markovian in nature. From our numerical study, the distribution of the test-wave has non-Gaussian statistics.

Sustained resonance in a linear oscillator is achievable with a drive whose constant frequency matches the resonant frequency of the oscillator. In oscillators with nonlinear restoring forces, i.e., Dung-type oscillators, resonant frequency changes with amplitude, so a constant frequency drive generates a beat oscillation instead of sustained resonance. Dung-type oscillators can be driven into sustained resonance, called autoresonance (AR), when drive frequency is swept in time to match the changing resonant frequency of the oscillator. It is found that near-optimal drive linear sweep rates for autoresonance can be estimated from the beat oscillation resulting from constant frequency excitation. Specically, a least squares estimate of the slope of the Teager-Kaiser instantaneous frequency versus time plot for the rising half-cycle of the beat response to a stationary drive provides a near-optimal estimate of the linear drive sweep rate that sustains resonance in the pendulum, Dung and Dung-Van der Pol oscillators. These predictions are confirmed with model-based numerical simulations. A closed-form approximation to the AM-FM nonlinear resonance beat response of a Dung oscillator driven at its low-amplitude oscillator frequency is obtained from a solution to an associated Mathieu equation. AR time responses are found to evolve along a Mathieu equation primary resonance stability boundary. AR breakdown occurs at sweep rates just past optimal and map to a single stable point just off the Mathieu equation primary resonance stability boundary. Optimal AR sweep rates produce oscillating phase dierences with extrema near 90 degrees, allowing extended time in resonance. AR breakdown occurs when phase difference equals 180 degrees. Nonlinear resonance of the van der Pol type may play a role in the extraordinary sensitivity of the human ear. / The mechanism for maintaining the cochlear amplifier at its critical point is currently unknown. The possibility of open-loop control of cochlear operating point, maintaining criticality on average through periodically varying damping (super-regeneration) motivates a study of spontaneous otoacoustic emission (SOAE) amplitude modulation on a short (msec) time scale. An example of periodic amplitude modulation within a wide filter bandwidth is found that appears to be a beat oscillation of two SOAEs. / by Carey Witkov. / Thesis (Ph.D.)--Florida Atlantic University, 2011. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2011. Mode of access: World Wide Web.

The synchronization of coupled semiconductor lasers with delay is investigated by numerical simulations of the nonlinear dynamic models complemented by a stability analysis of the linearized system. The equations used in the dissertation are based on the well known "Lang-Kobayashi" model modified to include unidirectional and bidirectional coupling. Stability diagrams are calculated and supplemented by numerically integrated time series. Synchronization is determined and quantified by computing the cross-correlation function. It is found that synchronized states are achievable for a wide range of coupling constants and delay times. These findings have implications for experiment and technological applications, notably in cryptography. / by Michael S. London. / Thesis (Ph.D.)--Florida Atlantic University, 2011. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2011. Mode of access: World Wide Web.