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.

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

No description available.

In this thesis I make extensive use of the Lyapunov analysis formalism to unravel fundamental mechanisms of instability in two different systems : the Kuramoto model of globally coupled phase-oscillators and the Lorenz 96 (L96) atmospheric "toy" model, portraying the evolution of a physical quantity along a latitude circle. I start by introducing the relevant theoretical background, with special attention on the main tools I have been using throughout this work : Lyapunov Exponents (LEs), which quantify the asymptotic growth rates of infinitesimal perturbations in a system, and by extension, its degree of chaoticity, and Covariant Lyapunov Vectors (CLVs), which indicate the phase space direction (or the geometry) associated with these growth rates. The Kuramoto model is central in the study of synchronization among oscillatory units characterized by their various natural frequencies, but little is known on its chaotic dynamics in the unsynchronized state. I thus investigate the scaling behavior of the first LE, upon different assumptions on the natural frequencies, and make use of educated structural simplifications to analyze the origin of chaos in the finite size model. On the other hand, the L96 model has been devised to gather the main dynamical ingredients of atmospheric dynamics, namely advection, damping, external (solar) forcing and transfers across different scales of motion, in a minimalist and functional way. It features two coupled dynamical layers : the large scale variables, representing synoptic scale atmospheric dynamics, and the small scale variables, faster and more numerous, associated with convective scale dynamics. The core of the study revolves around geometrical properties of CLVs, in the aim of understanding the processes underlying the observed multiscale chaoticity, and an exhaustive study of a non-trivial ensemble of CLVs featuring relevant projection on the slow subspace.

Thesis (doctoral)--Universität Augsburg, 2000. / Includes bibliographical references (p. 177-179) and index.

Στην παρούσα διπλωματική εργασία, μελετάμε την έννοια και σημασία των εκθετών Lyapunov μέσω μεθόδων ανάλυσης πειραματικών δεδομένων που εφαρμόζονται στην φυσική, στην γεωλογία, στην αστρονομία, στην νευροβιολογία, στην οικολογία και στα οικονομικά. Οι εκθέτες Lyapunov παίζουν πολύ σημαντικό ρόλο στην ανίχνευση χάους, το οποίο εμφανίζεται σε πολλούς τομείς της επιστήμης και της τεχνολογίας. Άρα, το θέμα τους ανήκει στην θεωρία των χαοτικών δυναμικών συστημάτων αλλά και γενικότερα όλων των δυναμικών συστημάτων, τα οποία πρέπει να αναλυθούν σωστά και με ακρίβεια για να πάρουμε τα σωστά συμπεράσματα όσον αφορά τους εκθέτες Lyapunov. Σκοπός της μελέτης είναι η εύρεση των εκθετών Lyapunov για διάφορα δυναμικά συστήματα και η εξήγηση των αποτελεσμάτων όσον αφορά την δυναμική συμπεριφορά του κάθε συστήματος. Επίσης, παρουσιάζονται εφαρμογές στην επιστήμη όπου οι εκθέτες Lyapunov παίζουν σημαντικό ρόλο και εξηγούνται οι κυριότεροι αλγόριθμοι υπολογισμού αυτών των εκθετών υπό διαφορετική υλοποίηση και σε διαφορετικά υπολογιστικά πακέτα, όπως το Matlab, το Mathematica και ακόμα σε γλώσα προγραμματισμού C με σκοπό την εύρεση του καλύτερου και πιο ακριβή αλγόριθμου. Επιπρόσθετα, παρουσιάζονται τα συμπεράσματα μετά την ανάλυση όλων των αλγορίθμων και των αποτελεσμάτων και προτείνεται ο καλύτερος και αποτελεσματικότερος αλγόριθμος όσον αφορά την απόδοση, τον χρόνο εκτέλεσης, αλλά και το μέγεθος των σφαλμάτων. Στο τέλος, υπάρχει παράρτημα με επιμέρους κώδικες που χρησιμοποιούνται, όπως ακόμα και η βιβλιογραφία από την οποία αντλήθηκαν πολύ σημαντικές πληροφορίες. / In this paper, we study the meaning and importance of Lyapunov exponents through experimental data analysis methods applied in physics, geology, astronomy, neurobiology, ecology and economics. The Lyapunov exponents play an important role in the detection of chaos, which occurs in many areas of science and technology. So, their issue concerns the theory of chaotic dynamical systems and generally all dynamical systems, which must be analyzed properly and accurately to get the right conclusions for the Lyapunov exponents. The purpose of this paper is to find the Lyapunov exponents for various dynamical systems and the explanation of the results concerning the dynamic behavior of each system. Also, several applications in science are presented where Lyapunov exponents play an important role and the main algorithms, which calculate these exponents under different implementation and in different computer packages such as Matlab, Mathematica, and even in programming language C, are explained to find the best and most accurate algorithm. Additionally, conclusions are drawn after analyzing all the algorithms and the results and it is suggested the best and most efficient algorithm regarding the performance, the execution time and also the magnitude of errors. In the end, there is an appendix with individual codes which are used, as even the bibliography from which very important information are derived.

No description available.

We consider diffeomorphisms of a compact Riemann Surface. A development of Oseledec's Multiplicative Ergodic Theorem is given, along with a development of measure theoretic entropy and dimension. The main result, due to L.S. Young, is that for certain diffeomorphisms of a surface, there is a beautiful relationship between these three concepts; namely that the entropy equals dimension times expansion.