The purpose of this paper is to clarify the problems of exponents by introducing notation not customarily used and by demonstrating certain theorems in regard to the properties of the exponential functions.

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Tsang, Chi Shing Sidney. / "October 2010." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 58-59). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- The main results --- p.6 / Chapter 1.2 --- Structure of the thesis --- p.8 / Chapter 2 --- The Upper Lyapunov Exponent --- p.10 / Chapter 2.1 --- Notation --- p.10 / Chapter 2.2 --- The upper Lyapunov exponent --- p.11 / Chapter 2.3 --- Cocycles --- p.12 / Chapter 2.4 --- The Theorem of Furstenberg and Kesten --- p.14 / Chapter 3 --- Contraction Properties --- p.19 / Chapter 3.1 --- Two basic lemmas --- p.20 / Chapter 3.2 --- Contracting sets --- p.25 / Chapter 3.3 --- Strong irreducibility --- p.29 / Chapter 3.4 --- A key property --- p.30 / Chapter 3.5 --- Contracting action on P(Rd) and converges in direction --- p.36 / Chapter 3.6 --- Lyapunov exponents --- p.39 / Chapter 3.7 --- Comparison of the top Lyapunov exponents and Fursten- berg's theorem --- p.43 / Chapter 4 --- Analytic Dependence of Lyapunov Exponents on The Probabilities --- p.48 / Chapter 4.1 --- Continuity and analyticity properties for i.i.d. products --- p.49 / Chapter 4.2 --- The proof of the main result --- p.50 / Chapter 5 --- The Expression of The Upper Lyapunov Exponent in Complex Functions --- p.54 / Chapter 5.1 --- The set-up --- p.54 / Chapter 5.2 --- The main result --- p.56 / Bibliography --- p.58

A new method for evaluating the Lyapunov exponent for a Hamiltonian system involves a spatial evaluation, rather than a numerical time integration. The introduction of a novel vector field to the phase space allows the Lyapunov exponent to be expressed in a form that does not involve time. The Lyapunov exponent is seen to be a property of the geometry and topology of ergodic regions of phase space. As a result it has a more regular behavior than previously thought. The Lyapunov exponent is found to be a differentiable function of the perturbation coupling in regions where it was previously thought to be discontinuous. Properties of the Lyapunov function once taken for granted are shown to be artifacts of the traditional computation methods. The technique is discussed with examples from a system of coupled quartic oscillators. / Graduation date: 1996

The purpose of this work is to present a methodology to predict vibrations of drilllstrings for oil recovery service. The work extends a previous model of the drill collar between two stabilizers in the literature to include drill collar flexibility utilizing a modal coordinate condensed, finite element approach. The stiffness due to the gravitational forces along the drillstring axis is included. The model also includes the nonlinear effects of drillstring-wellbore contact, friction and quadratic damping. Bifurcation diagrams are presented to illustrate the effects of speed, friction at wellbore, stabilizer clearance and drill collar length on chaotic vibration response. Their effects shifts resonance peaks away from the linear natural frequency values and influences the onset speed for chaos. A study is conducted on factors for improving the accuracy of Lyapunov Exponents to predict the presence of chaos. This study considers the length of time to steady state, the number and duration of linearization sub-intervals, the presence of rigid body modes and the number of finite elements and modal coordinates. The Poincare map and frequency spectrum are utilized to confirm the prediction of Lyapunov exponent analysis. The results may be helpful for computing Lyapunov exponents of other types of nonlinear vibrating systems with many degrees of freedom. Vibration response predictions may assist drilling rig operators in changing a variety of controlled parameters to improve operation procedures and/or equipment.

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 223-231).

Increased attention has been paid in recent years to the means in which the body maintains stability and the subtleties of the neurocontroller. Variability of kinematic data has been used as a measure of stability but these analyses are not appropriate for quantifying stability of dynamic systems. Response of biological control systems depend on both temporal and spatial inputs, so means of quantifying stability should account for both. These studies utilized tools developed for the analysis of deterministic chaos to quantify local dynamic stability of musculoskeletal systems. The initial study aimed to answer the oft assumed conjecture that reduced gait speeds in people with neuromuscular impairments lead to improved stability. Healthy subjects walked on a motorized treadmill at an array of speeds ranging from slow to fast while kinematic joint angle data were recorded. Significant (p < 0.001) trends showed that stability monotonically decreased with increasing walking speeds. A second study was performed to investigate dynamic stability of the trunk. Healthy subjects went through a variety of motions exhibiting either symmetric flexion in the sagittal plane or asymmetric flexion including twisting at both low and high cycle frequencies. Faster cycle frequencies led to significantly (p<0.001) greater instability than slower frequencies. Motions that were hybrids of flexion and rotation were significantly (p<0.001) more stable than motions of pure rotation or flexion. Finding means of increasing dynamic stability may provide great understanding of the neurocontroller as well as decrease instances of injury related to repetitive tasks. Future studies should look in greater detail at the relationships between dynamic instability and injury and between local dynamic stability and global dynamic stability. / Master of Science

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Earthquakes are amongst the most destructive of natural phenomena and have been the subject of significant research effort over many decades, to predict the onset of seismic events. Electromagnetic emissions detected prior to earthquakes provide a potential data source for seismic predictions and research suggests that specific pre-seismic electromagnetic activity can be directly related to specific earthquakes although it is still an open issue as to the precise links between these electromagnetic emissions and subsequent earthquakes. In this research, findings of the long memory or the self-organization of several pre-earthquake MHz electromagnetic time-series provide significant outcomes regarding the earthquake prediction. It is also recognised that enhanced radon gas emission has an equally long history as being associated with seismic activity. In general, several anomalous soil radon emissions have been observed prior to earthquakes and this has been recorded all over the world. The abnormal soil radon exhalation from the interior of the earth has been associated with earthquakes and is considered as an important field of research. The research reported in this thesis compared and contrasted the merits of combining electromagnetic emission data and radon exhalation data as precursors of earthquakes with the aim of enhancing earthquake prediction methodology. The findings from the long-memory analysis of radon disturbances in the soil indicated a very significant issue: the radon disturbances in the soil prior to earthquakes exhibit similar behaviour as the MHz RF disturbances of general failure. So, the radon precursors and the MHz electromagnetic correspond to the same pre-earthquake phase. Geological explanations were proposed in view of the asperity model. Persistent and anti-persistent MHz anomalies were due to the micro-cracking of the heterogeneous medium of the earth's crust which may have led the system's evolution towards the global failure. Fractal methods have been used on historical data, to investigate MHz electromagnetic time-series spectra on emissions preceding major earthquakes over the period 2007 to 2014 and the characteristics of enhanced radon emissions have been studied over the period 2008 to 2015 for seismic events occurring in the Aegean Region. It has been found that both the electromagnetic emissions and the radon exhalation data exhibit similar fractal behaviour and are associated with impending seismic activity. Hence both phenomena are relevant to earthquake predictions and should both be employed in any systematic approach to this problem as the varying geological and geographic conditions under which earthquakes can occur, might preclude one or other data from being measurable. According to the several techniques applied in this thesis, all should be employed in sequential steps, albeit the power-law spectral fractal analysis is the most significant to trace long-memory patterns of 1/f processes as those of the processes of earthquakes.

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