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

Global temperature variations between 1861 and 1984 are forecast usingsregularization networks, multilayer perceptrons and linearsautoregression. The regularization network, optimized by stochasticsgradient descent associated with colored noise, gives the bestsforecasts. For all the models, prediction errors noticeably increasesafter 1965. These results are consistent with the hypothesis that thesclimate dynamics is characterized by low-dimensional chaos and thatsthe it may have changed at some point after 1965, which is alsosconsistent with the recent idea of climate change.s

The failure of rolling element bearings is one of the foremost causes of breakdown in rotary machinery. So far, a variety of vibration-based techniques have been developed to monitor the condition of bearings; however, the role of vibration behavior is rarely considered in the proposed techniques. This thesis presents an analytical study of a healthy rotor-bearing system to gain an understanding of the different categories of bearing vibration. In this study, a two degree-of-freedom model is employed, where the contacts between the rolling elements and races are considered to be nonlinear springs. The analytical investigations confirm that the nature of the inner ring oscillation depends on the internal clearance. A fault-free bearing with a small backlash exhibits periodic behavior; however, bearings categorized as having normal clearance oscillate chaotically. The results from the numerical simulations agree with those from the experiments confirming bearing’s chaotic response at various rotational speeds. Bearing faults generate periodic impacts which affect the chaotic behavior. This effect manifests itself in the phase plane, Poincare map, and chaotic quantifiers such as the Lyapunov exponent, correlation dimension, and information entropy. These quantifiers serve as useful indices for detecting bearing defects. To compare the sensitivity and robustness of chaotic indices with those of well-accepted fault detection techniques, a comprehensive investigation is conducted. The test results demonstrate that the Correlation Dimension (CD), Normalized Information Entropy (NIE), and a proposed time-frequency index, the Maximum Approximate Coefficient of Wavelet transform (MACW), are the most reliable fault indicators. A neuro-fuzzy diagnosis system is then developed, where the strength of the aforementioned indices are integrated to provide a more robust assessment of a bearing’s health condition. Moreover, a prognosis scheme, based on the Adaptive Neuro Fuzzy Inference System (ANFIS), in combination with a set of logical rules, is proposed for estimating the next state of a bearing’s condition. Experimental results confirm the viability of forecasting health condition under different speeds and loads.

The failure of rolling element bearings is one of the foremost causes of breakdown in rotary machinery. So far, a variety of vibration-based techniques have been developed to monitor the condition of bearings; however, the role of vibration behavior is rarely considered in the proposed techniques. This thesis presents an analytical study of a healthy rotor-bearing system to gain an understanding of the different categories of bearing vibration. In this study, a two degree-of-freedom model is employed, where the contacts between the rolling elements and races are considered to be nonlinear springs. The analytical investigations confirm that the nature of the inner ring oscillation depends on the internal clearance. A fault-free bearing with a small backlash exhibits periodic behavior; however, bearings categorized as having normal clearance oscillate chaotically. The results from the numerical simulations agree with those from the experiments confirming bearing’s chaotic response at various rotational speeds. Bearing faults generate periodic impacts which affect the chaotic behavior. This effect manifests itself in the phase plane, Poincare map, and chaotic quantifiers such as the Lyapunov exponent, correlation dimension, and information entropy. These quantifiers serve as useful indices for detecting bearing defects. To compare the sensitivity and robustness of chaotic indices with those of well-accepted fault detection techniques, a comprehensive investigation is conducted. The test results demonstrate that the Correlation Dimension (CD), Normalized Information Entropy (NIE), and a proposed time-frequency index, the Maximum Approximate Coefficient of Wavelet transform (MACW), are the most reliable fault indicators. A neuro-fuzzy diagnosis system is then developed, where the strength of the aforementioned indices are integrated to provide a more robust assessment of a bearing’s health condition. Moreover, a prognosis scheme, based on the Adaptive Neuro Fuzzy Inference System (ANFIS), in combination with a set of logical rules, is proposed for estimating the next state of a bearing’s condition. Experimental results confirm the viability of forecasting health condition under different speeds and loads.

The detection for phase locking in a forced oscillator with dual attractors and ill-defined phase structure is hard until a quantitative approach was constructed for detecting phase locking via stroboscopic method. We study the route to weak phase locking in a chaotic system ¡§Chua oscillator¡¨ with complex attractor structure by analyzing the stroboscopic points. The onset of weak phase locking detected by using this statistical approach and the critical coupling strength calculated by conditional Lyapunov exponent are matched well. Detailed structure of phase locking intensity is described by the Arnold tongue diagram. Moreover, we apply this approach on three hyper-chaotic systems with multi-scroll attractor, including hyper-chaotic Rössler system, hyper-chaotic Lorenz system, and modified MCK oscillator. The weak phase locking between hyper-chaotic system and a periodic or a chaotic driving force is observable following the condition of stroboscopic point concentration.

Synchronization of two identical chaotic systems with matched and mismatched perturbations by utilizing adaptive sliding mode control (ASMC) technique is presented in this thesis. The sliding surface function is designed based on Lyapunov stability theorem and linear matrix inequality (LMI) optimization technique. Adaptive mechanisms embedded in the proposed control scheme are used to adapt the unknown upper bounds of the perturbations. The designed tracking controller can not only suppress the mismatched perturbations when the controlled dynamics (master-slave) are in the sliding mode, but also drive the trajectories of synchronization errors into a small bounded region whose size can be adjusted through the designed parameters. The stability of overall controlled synchronization systems is guaranteed. Application of proposed chaotic synchronization technique to secure communication as well as several numerical examples are given to demonstrate the feasibility of the proposed design technique.

Theoretical and numerical studies of chaotic mixing are performed to circumvent the difficulties of efficient mixing, which come from the lack of turbulence in microfluidic devices. In order to carry out efficient and accurate parametric studies and to identify a fully chaotic state, a spectral element algorithm for solution of the incompressible Navier-Stokes and species transport equations is developed. Using Taylor series expansions in time marching, the new algorithm employs an algebraic factorization scheme on multi-dimensional staggered spectral element grids, and extends classical conforming Galerkin formulations to nonconforming spectral elements. Lagrangian particle tracking methods are utilized to study particle dispersion in the mixing device using spectral element and fourth order Runge-Kutta discretizations in space and time, respectively. Comparative studies of five different techniques commonly employed to identify the chaotic strength and mixing efficiency in microfluidic systems are presented to demonstrate the competitive advantages and shortcomings of each method. These are the stirring index based on the box counting method, Poincare sections, finite time Lyapunov exponents, the probability density function of the stretching field, and mixing index inverse, based on the standard deviation of scalar species distribution. Series of numerical simulations are performed by varying the Peclet number (Pe) at fixed kinematic conditions. The mixing length (lm) is characterized as function of the Pe number, and lm ∝ ln(Pe) scaling is demonstrated for fully chaotic cases. Employing the aforementioned techniques, optimum kinematic conditions and the actuation frequency of the stirrer that result in the highest mixing/stirring efficiency are identified in a zeta potential patterned straight micro channel, where a continuous flow is generated by superposition of a steady pressure driven flow and time periodic electroosmotic flow induced by a stream-wise AC electric field. Finally, it is shown that the invariant manifold of hyperbolic periodic point determines the geometry of fast mixing zones in oscillatory flows in two-dimensional cavity.

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

In this dissertation, we devise a quantization scheme to construct eigenfunctions by classical periodic orbits in both regular systems as well as chaotic systems. Our method is based on the principle that eigenfunctions can be resolved from a time-dependent wavefunction. This is different from the classical (or EBK) quantization scheme that constructs eigenfunction in the energy-domain. The advantage of our method is that it can be applied to more varieties of systems, including some chaotic systems. Three systems, the simple harmonic oscillator, the x⁴-potential oscillator, and the x²y² quartic-oscillator, are used as examples for our eigenfunction construction. The key to the constructions is a family (or families) of periodic orbits with a newly defined quantization rule, the resolving quantization rule. The eigenspectrum for the x⁴-potential oscillator is also computed. Furthermore, the classical Green's function is used to explain the relation between the resolving quantization rule and the classical quantization rule. This dissertation begins with an introduction in Chapter 1. The semiclassical theory for the eigenfunction construction by periodic orbits is developed in Chapter 2. In Chapter 3 and Chapter 4, eigenfunctions are constructed for the simple harmonic oscillator, the x⁴-potential oscillator, and the x²y² quartic-oscillator. The eigenspectrum for the x⁴-potential oscillator is computed in Chapter 5. Chapter 6 is devoted to discussions including the interpretation of the resolving quantization rule from the classical Green's function, the interpretation of the photoabsorption spectrum for a Rydberg atom in a magnetic field, and the comparison of our method with the EBK quantization scheme. Conclusions are made in Chapter 7. / text

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