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.

This thesis discusses a development of the exponential and logarithmic functions.

No description available.

Thesis (Ph. D.)--Illinois State University, 2005. / Title from title page screen, viewed on April 13, 2007. Dissertation Committee: Norma C. Presmeg (chair), Beverly S. Rich, Nerida F. Ellerton, Sherry L. Meier. Includes bibliographical references (leaves 207-217) and abstract. Also available in print.

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.

In this work we study the Generalized Lane-Emden equation and the interplay between the exponents involved and their consequences on the existence and non existence of radial solutions on a unit ball in n dimensions. We extend the analysis to the phase plane for a clear understanding of the behavior of solutions and the relationship between their existence and the growth of nonlinear terms, where we investigate the critical exponent p and a sub-critical exponent, which we refer to as ^p. We discover a structural change of solutions due the existence of this sub-critical exponent which we relate to the same change in behavior of the Lane- Emden equation solutions, for ; = 0; andp = 2, due to the same sub-critical exponent. We hypothesize that this sub-critical exponent may be related to a weighted trace embedding.

The Wiggins-Holmes extension of the generalized Melnikov method (GMM) is applied to weakly damped parametrically excited cross waves with surface tension in a long rectangular wave channel in order to determine if these cross waves are chaotic. The Lagrangian density function for surface waves with surface tension is simplified by transforming the volume integrals to surface integrals and by subtracting the zero variation integrals. The Lagrangian is written in terms of the three generalized coordinates (or, equivalently the three degrees of freedom) that are the time-dependent components of the velocity potential. A generalized dissipation function is assumed to be proportional to the Stokes material derivative of the free surface. The generalized momenta are calculated from the Lagrangian and the Hamiltonian is determined from a Legendre transformation of the Lagrangian. The first order ordinary differential equations derived from the Hamiltonian are usually suitable for the application of the GMM. However, the cross wave equations of motion must be transformed in order to obtain a suspended system for the application of the GMM. Only three canonical transformations that preserve the dynamics of the cross wave equations of motion are made because of an extension of the Herglotz algorithm to nonautonomous systems. This extension includes two distinct types of the generalized Herglotz algorithm (GHA). The system of nonlinear nonautonomous evolution equations determined from Hamilton's equations of motion of the second kind are averaged in order to obtain an autonomous system. The unperturbed system is analyzed to determine hyperbolic saddle points that are connected by heteroclinic orbits The perturbed Hamiltonian system that includes surface tension satisfies the KAM nondegeneracy requirements; and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the Melnikov integral is identically zero implying that a higher dimensional GMM is necessary in order to demonstrate by the GMM that the motion is chaotic. However, numerical calculations of the largest Liapunov characteristic exponent demonstrate that the perturbed dissipative system with surface tension is also chaotic. A chaos diagram is computed in order to search for possible regions of the damping parameter and the Floquet parametric forcing parameter where chaotic motions may exist. / Graduation date: 1999

In order to understand water quality in the coastal ocean and its effects on human health, the necessity arises to locate the sources of contaminants and track their transport throughout the ocean. Dynamical systems methods are applied to the study of transport of enterococci as an indicator of microbial concentration in the vicinity of Hobie Beach, an urban, subtropical beach in Miami, FL that is used for recreation and bathing on a daily basis. Previous studies on water quality have shown that Hobie Beach has high microbial levels despite having no known point source. To investigate the cause of these high microbial levels, a combination of measured surface drifter trajectories and numerically simulated flows in the vicinity of Hobie Beach is used. The numerically simulated flows are used to identify Lagrangian Coherent Structures (LCSs), which provide a template for transport in the study area. Surface drifter trajectories are shown to be consistent with the simulated flows and the LCS structure. LCSs are then used to explain the persistent water contamination and unusually high concentrations of microbes in the water off of this beach as compared with its neighboring beaches. From the drifter simulations, as well as field experiments, one can see that passive tracers are trapped in the area along the coastline by LCS. The Lagrangian circulation of Hobie Beach, influenced primarily by tide and land geometry causes a high retention rate of water near the shore, and can be used to explain the elevated levels of enterococci in the water.