Ce travail de thèse porte sur la dynamique du flot linéaire des surfaces de translation et de sa renormalisation par le flot de Teichmüller introduite par H. Masur et W. Veech en 1982. Une version combinatoire de cette renormalisation, l'induction de Rauzy sur les échanges d'intervalles, fût introduite auparavant par G. Rauzy en 1979. D'une part, nous faisons une étude combinatoire des classes de Rauzy qui forment une partition de l'ensemble des permutations irréductibles et interviennent dans l'algorithme d'induction de Rauzy. Nous donnons une formule pour la cardinalité de chaque classe. D'autre part, nous étudions un modèle de billard infini $\ZZ^2$-périodique dans le plan appelé le \og vent dans les arbres \fg introduit dans une version stochastique par P.~et T. Ehrenfest en 1912 et par J. Hardy et J. Weber en 1980 dans la version périodique. Nous construisons une famille de directions pour lesquelles le flot du billard est divergent donnant ainsi des exemples de $\ZZ^2$-cocycles divergents au-dessus d'échanges d'intervalles. De plus, nous démontrons que le taux polynomial de diffusion générique est $2/3$ autrement dit que la distance maximale atteinte par une particule au temps $t$ est de l'ordre de $t^{2/3}$.

Thesis (M.S.)--Baylor University, 2005. / Includes bibliographical references (p. 120-121).

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Nonlinear dynamical systems theory has inspired a new set of useful tools to be applied in climate studies. In this work we presented specific examples where information has been gained by the application of methods from nonlinear dynamical systems theory. The main goal is to understand the relative importance of stochastic forcing versus deterministic coupling within the context of Coupled General Circulation Models. This work address this important subject by approaching this goal through the development of a hierarchy of models with increasing complexity that we assert contain the essential dynamics of ENSO. We examined the effect of noise in a low order model and found that it is not restricted to blurring the attractor trajectories in phase space, but includes important changes in the dynamics of the system. The main results indicate that the presence of noise in a nonlinear system has two different effects. The presence of noise acts to increase the maximum Lyapunov exponent and can result in noise induced chaos if the system was originally stable. However, the same arguments are not valid if the original system is already in the chaotic regime, where the noise inclusion acts to decrease the maximum Lyapunov exponent, therefore increasing the system stability. The system of interest includes coupled ocean-atmosphere interactions and here we mimic this interaction by coupling two low order models with very different dominant time scales. These subsystems interact in a complex, nonlinear way and the behavior of the whole system cannot be explained by a linear summation of dynamics of the system parts. We used information theory concepts to detect the influence of the slow system dynamics in synchronizing the fast system in coupled models. We introduced a fast-slow coupled system, where both the slowness of the ocean model and the intensity of the boundary forcing anomalies contribute to the asymmetry and phase locking of both subsystems. The mechanisms controlling the fast modelspread were uncovered revealing uncertainty dynamics depending on the location of ensemble members in the model’s phase space. As an intermediate step between low order models and CGCMs we study the effect of noise on an intermediate complexity model. The addition of gaussian noise to the Zebiak-Cane model in order to understand the effects of noise on its attractor led to a way of estimating the noise level based on the effects of noise on the correlation dimension curves. We investigate the intrinsic predictability of the coupled models used here, and the different time scales associated with fast and slow modes were detected using the Finite Size Lyapunov Exponents. We found new estimates for the prediction horizon of ENSO for the Zebiak-Cane model as well as for the NCAR CCSM3 model and observations. The whole analysis of observations and CCSM3 was possible after applying noise reduction techniques. We also improved our understanding of three different noise reduction techniques by comparing the Local Projective Noise Reduction, the Interactive Ensemble strategy, and a Random Interactive Ensemble applied to CCSM3. The main difference between these two noise reduction techniques is when the process is applied. The Local Projective Noise Reduction can be applied to both model and observations, and it is done a posteriori in phase space, therefore the trajectories to be adjusted already posses the physical mechanisms embedded in them. The Interactive Ensemble approach can only be applied to model simulations and has shown to be a very useful technique for noise reduction since its done a priori while the system evolves instead of a posteriori, besides the fact that it allows to retrieve the spatial distribution of the noise level in physical space.

In this work, some new results to exploit the recurrence properties of quasiperiodic dynamical systems are presented by means of a two dimensional visualization technique, Recurrence Plots(RPs). Quasiperiodicity is the simplest form of dynamics exhibiting nontrivial recurrences, which are common in many nonlinear systems. The concept of recurrence was introduced to study the restricted three body problem and it is very useful for the characterization of nonlinear systems. I have analyzed in detail the recurrence patterns of systems with quasiperiodic dynamics both analytically and numerically. Based on a theoretical analysis, I have proposed a new procedure to distinguish quasiperiodic dynamics from chaos. This algorithm is particular useful in the analysis of short time series. Furthermore, this approach demonstrates to be efficient in recognizing regular and chaotic trajectories of dynamical systems with mixed phase space. Regarding the application to real situations, I have shown the capability and validity of this method by analyzing time series from fluid experiments. / In dieser Arbeit stelle ich neue Resultate vor, welche zeigen, wie man Rekurrenzeigenschaften quasiperiodischer, dynamischer Systeme für eine Datenanalyse ausnutzen kann. Die vorgestellten Algorithmen basieren auf einer zweidimensionalen Darstellungsmethode, den Rekurrenz-Darstellungen. Quasiperiodizität ist die einfachste Dynamik, die nicht-triviale Rekurrenzen zeigt und tritt häufig in nichtlinearen Systemen auf. Nicht-triviale Rekurrenzen wurden im Zusammenhang mit dem eingeschränkten Dreikörper-problem eingeführt. In dieser Arbeit, habe ich mehrere Systeme mit quasiperiodischem Verhalten analytisch untersucht. Die erhaltenen Ergebnisse helfen die Wiederkehreigenschaften dieser Systeme im Detail zu verstehen. Basierend auf den analytischen Resultaten, schlage ich einen neuen Algorithmus vor, mit dessen Hilfe selbst in kurzen Zeitreihen zwischen chaotischem und quasiperiodischem Verhalten unterschieden werden kann. Die vorgeschlagene Methode ist besonders effizient zur Unterscheidung regulärer und chaotischer Trajektorien mischender dynamischer Systeme.Die praktische Anwendbarkeit der vorgeschlagenen Analyseverfahren auf Messdaten, habe ich gezeigt, indem ich erfolgreich Zeitreihen aus fluid-dynamischen Experimenten untersucht habe.

Eine klassische Art, die Dynamik nichtlinearer Systeme zu beschreiben, besteht in der Analyse ihrer Fourierspektren. Für periodische und quasiperiodische Prozesse besteht das Fourierspektrum nur aus diskreten Deltafunktionen. Das Spektrum einer chaotischen Bewegung ist hingegen durch das Vorhandensein einer stetigen Komponente gekennzeichnet. In der Arbeit geht es um einen eigenartigen, weder regulären noch vollständig chaotischen Zustand mit sogenanntem singulärstetigen Leistungsspektrum. <br /> Unsere Analyse ergab verschiedene Fälle aus weit auseinanderliegenden Gebieten, in denen singulär stetige (fraktale) Spektren auftreten. Die Beispiele betreffen sowohl physikalische Prozesse, die auf iterierte diskrete Abbildungen oder gar symbolische Sequenzen reduzierbar sind, wie auch Prozesse, deren Beschreibung auf den gewöhnlichen oder partiellen Differentialgleichungen basiert. / One of the classical ways to describe the dynamics of nonlinear systems is to analyze theur Fourier spectra. For periodic and quasiperiodic processes the Fourier spectrum consists purely of discrete delta-functions. On the contrary, the spectrum of a chaotic motion is marked by the presence of the continuous component. In this work, we describe the peculiar, neither regular nor completely chaotic state with so called singular-continuous power spectrum. <br /> Our investigations concern various cases from most different fields, where one meets the singular continuous (fractal) spectra. The examples include both the physical processes which can be reduced to iterated discrete mappings or even symbolic sequences, and the processes whose description is based on the ordinary or partial differential equations.

Cette thèse traite des systèmes paramétrés. Ils modélisent des applications dans divers domaines, comme la robotique ou la calibration. Soit S un système paramétré. Nous cherchons à décrire les ouverts connexes U de l'espace des paramètres tels que S restreint à U admet un nombre constant de solutions réelles. En robotique, nous détectons les positions cuspidales des robots plan 3-RPR. En calibration photographique, nous décrivons le nombre de solutions réalisables du problème Perspective-3- Points. D'un point de vue théorique, nous montrons que sous certaines hypothèses, le calcul de la variété discriminante d'un système paramétré peut se réduire à un calcul de projection. Dans le cas des systèmes quelconques, nous introduisons la décomposition équidimensionnelle régulière. Notre algorithme possède de bonnes performances en pratique et nous permet par ailleurs de déduire un nouvel algorithme pour le calcul du radical d'un idéal.

University of Macau / Faculty of Science and Technology / Department of Electromechanical Engineering

This thesis is concerned with the modeling and numerical study of nonsmooth dynamical systems (NSDS). The first part of the thesis deals with the modeling of some DC-DC power converters using the complementarity formalism. This mathematical theoretical framework allows us to ensure existence and uniqueness of solutions in a "natural" and synthetic way. Specifically, it works pretty well in power electronic converters because it incorporates generalized discontinuous conduction modes (GDCM), characterized by a reduction of the dimension of the effective dynamics. For systems with a single diode, analytical state-space conditions for the presence of a GDCM are stated and simulation results, showing a variety of behaviours, such as persistent or re-entering GDCM, are also presented. Furthermore, the analysis and simulation of a parallel resonant converter (PRC), which has four diodes, illustrate the convenience of the complementarity formalism to simulate electrical systems with a large number of ideal diodes. We also present the simulation of a boost converter with a sliding mode control, even though a general control theory for complementarity systems is not still developed.In the second part of the thesis we focus on the bifurcation analysis in NSDS, and in particular, we have studied different mechanical systems which involve impacts and dry-friction. It is known that nonsmooth or discontinuous dynamical systems can exhibit the bifurcations also exhibited by smooth systems. In addition to these, there are also some novel transitions so-called discontinuity-induced bifurcations (DIBs) which are unique to these systems. We have investigated the complex behaviour occurring in an impacting mechanical system. DIBs such as corner impact bifurcations and transitions from complete to uncomplete chattering motions have been analysed in detail. Another type of DIBs recently classified are the so-called sliding bifurcations. Such bifurcations are a characteristic feature of so-called Filippov systems. We present detailed examples of all the different sliding bifurcation scenarios in a dry friction oscillator using a measured friction characteristic firstly introduced by Popp. Furthermore, a codimension-two degenerate switching-sliding bifurcation is displayed. In this case of degenerate switching-sliding bifurcation two curves of codimension-one sliding bifurcations, crossing-sliding and adding-sliding, branch out from the codimension-two point. Also, a cusp smooth codimension-two bifurcation is shown and coexistence of periodic orbits in the region between both fold codimension-one curves is studied.We have also investigated the dynamic behaviour of the two-block Burridge model for earthquake simulations. Previous numerical studies investigated by Ruina verified that, with a friction force of Coulomb type, the system presents only periodic behaviour. We show that chaotic regions can be observed in a symmetric configuration even if a Coulomb friction is considered with the relaxation of one of the assumptions assumed in the seismological literature. Furthermore, we have studied the behaviour of the system with asymmetric configuration. Different periodic solutions and regions of chaos can be observed varying the asymmetry of the system. With respect to the bifurcation point of view, we have analysed several smooth bifurcations (smooth and DIBs) observed in this system.Chapter 6 of this thesis presents the SICONOS software platform dedicated to simulation of NSDS. We give an overview of the SICONOS software and the way NSDS are modeled and simulated within the platform. Routines for analysis (stability, bifurcations, invariant manifolds,.) of NSDS implemented in the platform are explained in detail. To conclude this part, several representative samples are shown in order to illustrate the SICONOS platform abilities.Conclusion and some open problems are presented in the last chapter.

Aquesta tesi doctoral està emmarcada en el camp de l'astrodinàmica. Presenta solucions a problemes identificats en el disseny de missions que utilitzen òrbites entorn dels punts de libració, fent servir la teoria de sistemes dinàmics.El problema restringit de tres cossos és un model per estudiar el moviment d'un cos de massa infinitessimal sota l'atracció gravitatòria de dos cossos molt massius. Els cinc punts d'equilibri d'aquest model, en especial L1 i L2, han estat motiu de nombrosos estudis per aplicacions pràctiques en les últimes dècades (SOHO, Genesis...). Genèricament, qualsevol missió en òrbita al voltant del punt L2 del sistema Terra-Sol es veu afectat per ocultacions degudes a l'ombra de la Terra. Si l'òrbita és al voltant de L1, els eclipsis són deguts a la forta influència electromagnètica del Sol. D'entre els diferents tipus d'òrbites de libració, les òrbites de Lissajous resulten de la combinació de dues oscil.lacions perpendiculars. El seu principal avantatge és que les amplituds de les oscil.lacions poden ser escollides independentment i això les fa adapatables als requeriments de cada missió. La necessitat d'estratègies per evitar eclipsis en òrbites de Lissajous entorn dels punts L1 i L2 motivaren la primera part de la tesi. En aquesta part es presenta una eina per la planificació de maniobres en òrbites de Lissajous que no només serveix per solucionar el problema d'evitar els eclipsis, sinó també per trobar trajectòries de transferència entre òrbites d'amplituds diferents i planificar rendez-vous. Per altra banda, existeixen canals de baix cost que uneixen els punts L1 i L2 d'un sistema donat i representen una manera natural de transferir d'una regió de libració a l'altra. Gràcies al seu caràcter hiperbòlic, una òrbita de libració té uns objectes invariants associats: les varietats estable i inestable. Si tenim present que la varietat estable està formada per trajectòries que tendeixen cap a l'òrbita a la qual estan associades quan el temps avança, i que la varietat inestable fa el mateix però enrera en el temps, una intersecció entre una varietat estable i una d'inestable proporciona un camí asimptòtic entre les òrbites corresponents. Un mètode per trobar connexions d'aquest tipus entre òrbites planes entorn de L1 i L2 es presenta a la segona part de la tesi, i s'hi inclouen els resultats d'aplicar aquest mètode als casos dels problemes restringits Sol Terra i Terra-Lluna.La idea d'intersecar varietats hiperbòliques es pot aplicar també en la cerca de camins de baix cost entre les regions de libració del sistema Sol-Terra i Terra-Lluna. Si existissin camins naturals de les òrbites de libració solars cap a les lunars, s'obtindria una manera barata d'anar a la Lluna fent servir varietats invariants, cosa que no es pot fer de manera directa. I a l'inversa, un camí de les regions de libració lunars cap a les solars permetria, per exemple, que una estació fos col.locada en òrbita entorn del punt L2 lunar i servís com a base per donar servei a les missions que operen en òrbites de libració del sistema Sol-Terra. A la tercera part de la tesi es presenten mètodes per trobar trajectòries de baix cost que uneixen la regió L2 del sistema Terra-Lluna amb la regió L2 del sistema Sol-Terra, primer per òrbites planes i més endavant per òrbites de Lissajous, fent servir dos problemes de tres cossos acoblats. Un cop trobades les trajectòries en aquest model simplificat, convé refinar-les per fer-les més realistes. Una metodologia per obtenir trajectòries en efemèrides reals JPL a partir de les trobades entre òrbites de Lissajous en el model acoblat es presenta a la part final de la tesi. Aquestes trajectòries necessiten una maniobra en el punt d'acoblament, que és reduïda en el procés de refinat, arribant a obtenir trajectòries de cost zero quan això és possible. / This PhD. thesis lies within the field of astrodynamics. It provides solutions to problems which have been identified in mission design near libration points, by using dynamical systems theory. The restricted three body problem is a well known model to study the motion of an infinitesimal mass under the gravitational attraction of two massive bodies. Its five equilibrium points, specially L1 and L2, have been the object of several studies aimed at practical applications in the last decades (SOHO, Genesis...). In general, any mission in orbit around L2 of the Sun-Earth system is affected by occultations due to the shadow of the Earth. When the orbit is around L1, the eclipses are caused by the strong electromagnetic influence of the Sun. Among all different types of libration orbits, Lissajous type ones are the combination of two perpendicular oscillations. Its main advantage is that the amplitudes of the oscillations can be chosen independently and this fact makes Lissajous orbits more adaptable to the requirements of each particular mission than other kinds of libration motions. The need for eclipse avoidance strategies in Lissajous orbits around L1 and L2 motivated the first part of the thesis. It is in this part where a tool for planning maneuvers in Lissajous orbits is presented, which not only solves the eclipse avoidance problem, but can also be used for transferring between orbits having different amplitudes and for planning rendez-vous strategies.On the other hand, there exist low cost channels joining the L1 and L2 points of a given sistem, which represent a natural way of transferring from one libration region to the other one. Furthermore, there exist hyperbolic invariant objects, called stable and unstable manifolds, which are associated with libration orbits due to their hyperbolic character. If we bear in mind that the stable manifold of a libration orbit consists of trajectories which tend to the orbit as time goes by, and that the unstable manifold does so but backwards in time, any intersection between a stable and an unstable manifold will provide an asymptotic path between the corresponding libration orbits. A methodology for finding such asymptotic connecting paths between planar orbits around L1 and L2 is presented in the second part of the dissertation, including results for the particular cases of the Sun-Earth and Earth-Moon problems. Moreover, the idea of intersecting hyperbolic manifolds can be applied in the search for low cost paths joining the libration regions of different problems, such as the Sun-Earth and the Earth-Moon ones. If natural paths from the solar libration regions to the lunar ones was found, it would provide a cheap way of transferring to the Moon from the vicinity of the Earth, which is not possible in a direct way using invariant manifolds. And the other way round, paths from the lunar libration regions to the solar ones would allow for the placement of a station in orbit around the lunar L2, providing services to solar libration missions, for instance. In the third part of the thesis, a methodology for finding low cost trajectories joining the lunar L2 region and the solar L2 region is presented. This methodology was developed in a first step for planar orbits and in a further step for Lissajous type orbits, using in both cases two coupled restricted three body problems to model the Sun-Earth-Moon spacecraft four body problem. Once trajectories have been found in this simplified model, it is convenient to refine them to more realistic models. A methodology for obtaining JPL real ephemeris trajectories from the initial ones found in the coupled models is presented in the last part of the dissertation. These trajectories need a maneuver at the coupling point, which can be reduced in the refinement process until low cost connecting trajectories in real ephemeris are obtained (even zero cost, when possible).