published_or_final_version / abstract / Education / Doctoral / Doctor of Philosophy

A renormalization analysis is presented for a generalized Harper equation (1 + α cos(2π(ω(i + 1/2) + φ)))ψi+1 + (1 + α cos(2π(ω(i − 1/2) + φ)))ψi−1 +2λ cos(2π(iω + φ))ψi = Eψi. (0.1) For values of the parameter ω having periodic continued-fraction expansion, we construct the periodic orbits of the renormalization strange sets in function space that govern the wave function fluctuations of the solutions of the generalized Harper equation in the strong-coupling limit λ→∞. For values of ω with non-periodic continued fraction expansions, we make some conjectures based on work of Mestel and Osbaldestin on the likely structure of the renormalization strange set.

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 definition of semi-hyperbolic dynamical systems generated by Lipschitz continuous and not necessarily invertible mappings in Banach spaces is presented in this thesis. Like hyperbolic mappings, they involve a splitting into stable and unstable spaces, but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional. Bi-shadowing is a combination of the concepts of shadowing and inverse shadowing and is usually used to compare pseudo-trajectories calculated by a computer with the true trajectories. In this thesis, the concept of bi-shadowing in a Banach space is defined and proved for semi-hyperbolic dynamical systems generated by Lipschitz mappings. As an application to the concept of bishadowing, linear delay differential equations are shown to be bi-shadowing with respect to pseudo-trajectories generated by nonlinear small perturbations of the linear delay equation. This shows robustness of solutions of the linear delay equation with respect to small nonlinear perturbations. Complicated dynamical behaviour is often a consequence of the expansivity of a dynamical system. Semi-hyperbolic dynamical systems generated by Lipschitz mappings on a Banach space are shown to be exponentially expansive, and explicit rates of expansion are determined. The result is applied to a nonsmooth noninvertible system generated by delay differential equation. It is shown that semi-hyperbolic mappings are locally φ-contracting, where -0 is the Hausdorff measure of noncompactness, and that a linear operator is semi-hyperbolic if and only if it is φ-contracting and has no spectral values on the unit circle. The definition of φ-bi-shadowing is given and it is shown that semi-hyperbolic mappings in Banach spaces are φ-bi-shadowing with respect to locally condensing continuous comparison mappings. The result is applied to linear delay differential equations of neutral type with nonsmooth perturbations. Finally, it is shown that a small delay perturbation of an ordinary differential equation with a homoclinic trajectory is ‘chaotic’.

Thesis (Ph. D.)--University of Hong Kong, 2007. / Title proper from title frame. Also available in printed format.

This dissertation provides the conceptual development, modeling and simulation, physical implementation, and measured hardware results for a practicable digital coherent chaotic communication system. Such systems are highly desirable for robust communications due to the maximal entropy signal characteristics that satisfy Shannon's ideal noise-like waveform and provide optimal data transmission across a flat communications channel. At the core of the coherent chaotic communications system is a fully digital chaotic circuit, providing an efficiently controllable mechanism that overcomes the traditional bottleneck of chaotic circuit state synchronization. The analytical, simulation, and hardware results yield a generalization of direct sequence spread spectrum waveforms, that can be further extended to create a new class of maximal entropy waveforms suitable for optimized channel performance, maximal entropy transmission of chaotically spread amplitude modulated data constellations, and permission-based multiple access systems.

published_or_final_version / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

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