The aim of this thesis is to seek a low-dimensional description of baroclinic instability in general, and of the Martian atmosphere in particular, where both forcing and spatial resonance are relevant to the dynamics of the system being analysed. The Proper Orthogonal Decomposition (POD) is used to determine a basis for the modal decomposition of climatic simulations of Mars, obtained by using two General Circulation Models (GCMs): (a) a simple GCM, which is an idealised model in which the meteorological primitive equations are solved on a sphere with simplified physical parameters and (b) the Martian GCM, a more realistic model in which a comprehensive range of the relevant Martian physical parameters and topography are represented. Results of these analyses are presented for a range of Martian seasons and climatic conditions. The effects of using different forms of energy norm in performing the analysis is considered, with the objective of providing analyses which represents the physically most significant components of the circulation, with optimal efficiency. Reduced low-dimensional models that replicated the full simple GCM streamfunction simulations are formulated by projecting the spherical quasi-geostrophic equations onto the PODs of the large-scale calculations. The resulting models are analysed by using a combination of solution continuation and numerical integration methods. A thorough analysis of the models reveals that a 6-D POD model is capable of reproducing the amplitude, frequency and behaviour of the leading oscillatory structures of the simple GCM, to within a 1% error. Such an excellent reproduction of the original system is shown to be due to (1) an accurate vertical formulation scheme, (2) the use of the correct norm, (3) a sufficiently high level of truncation and (4) the fact that the original system is a steady wave flow. The behaviour of the various regimes observed in the low-order models are comparable with observations from studies of large-scale waves and instabilities in planetary atmospheres, including a range of hydrodynamical experiments on baroclinic wave interactions of a stratified fluid in cylindrical containers.

We report progress made on an analytic investigation of low-frequency cardiorespiratory variability in humans. The work is based on an existing physiological model of chemically-mediated blood-gas control via the central and peripheral chemoreceptors, that of Grodins, Buell & Bart (1967). Scaling and simplification of the Grodins model yields a rich variety of dynamical subsets; the thesis focusses on the dynamics obtained under the normoxic assumption (i.e., when oxygen is decoupled from the system). In general, the method of asymptotic reduction yields submodels that validate or invalidate numerous (and more heuristic) extant efforts in the literature. Some of the physiologically-relevant behaviour obtained here has therefore been reported before, but a large number of features are reported for the first time. A particular novelty is the explicit demonstration of cardiorespiratory coupling via chemosensory control. The physiology and literature reviewed in Chapters 1 and 2 set the stage for the investigation. Chapter 3 scales and simplifies the Grodins model; Chapters 4, 5, 6 consider carbon dioxide dynamics at the central chemoreceptor. Chapter 7 begins analysis of the dynamics mediated by the peripheral receptor. Essentially all of the dynamical behaviour is due to the effect of time delays occurring within the conservation relations (which are ordinary differential equations). The pathophysiology highlighted by the analysis is considerable, and includes central nervous system disorders, heart failure, metabolic diseases, lung disorders, vascular pathologies, physiological changes during sleep, and ascent to high altitude. Chapter 8 concludes the thesis with a summary of achievements and directions for further work.

In this thesis we consider mathematical problems related to different aspects of hard sphere systems. In the first part we study planar billiards, which arise in the context of hard sphere systems when only one or two spheres are present. In particular we investigate the possibility of elliptic periodic orbits in the general construction of hyperbolic billiards. We show that if non-absolutely focusing components are present there can be elliptic periodic orbits with arbitrarily long free paths. Furthermore, we show that smooth stadium like billiards have elliptic periodic orbits for a large range of separation distances. In the second part we consider hard sphere systems with a large number of particles, which we model by the Boltzmann equation. We develop a new approach to derive hydrodynamic limits, which is based on classical methods of geometric singular perturbation theory of ordinary differential equations. This method provides new geometric and dynamical interpretations of hydrodynamic limits, in particular, for the of the dissipative Boltzmann equation.

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (leaves [51]-52).

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-02-18Bitstream added on 2014-06-13T20:16:03Z : No. of bitstreams: 1 alves_ff_me_sjrp.pdf: 362207 bytes, checksum: 9a797ca400dea6e139af98c5a9f10378 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Este trablaho apresenta os conceitos de Lyapounov e de espaços próprios e fornece um resultado devido a Oseledets, o qual trata da existência desses expoentes (e, consequentemente, dos espaços próprios) do ponto de vista da teoria da medida. A prova do teorema que nós fornecemos foi dada originalmente por Mañe e posteriormente melhorada por Viana. / This work presents the concepts of Lyapounov exponents and of proper spaces and provides a result due to Oseledets, wich deals with the existence of these exponents (and consequently, of the proper spaces) from a measure-theoretical point of view. The proof of the theorem which we provide was originally given by Mañe later improved by Viana.

We review the concept of invariant measure and study conditions under which linear combinations of averages along periodic orbits are dense in the space of invariant measures. / Revisamos el concepto de medida invariante y estudiamos condiciones bajo las cuales combinaciones lineales de promedios a lo largo de órbitas periódicas son densas en el espacio de medidas invariantes.

Submitted by Nilton Cesar Moretti Junior null (niiilton@hotmail.com) on 2018-07-31T04:58:47Z No. of bitstreams: 1 Dissertação-Final.pdf: 1149444 bytes, checksum: 6c44dc0b9f2462ee08c23da4a240fa0a (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2018-07-31T18:13:18Z (GMT) No. of bitstreams: 1 morettijunior_nc_me_sjrp.pdf: 1461434 bytes, checksum: 7a6418b1192346448fc927ec6c6650dc (MD5) / Made available in DSpace on 2018-07-31T18:13:18Z (GMT). No. of bitstreams: 1 morettijunior_nc_me_sjrp.pdf: 1461434 bytes, checksum: 7a6418b1192346448fc927ec6c6650dc (MD5) Previous issue date: 2017-08-30 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho estudamos vários resultados relacionados com sistemas dinâmicos, teoria dos números e combinatória. Em particular, provamos os teoremas de Van Der Waerden, Szemeredi, Koksma e Weyl. / In this work we study several results connected with dynamical systems, number thoery and combinatorics. In particular, we prove Van Der Waerden, Szemer edi, Koksma and Weyl’s theorems.

Made available in DSpace on 2014-06-11T19:26:15Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-03-02Bitstream added on 2014-06-13T19:47:09Z : No. of bitstreams: 1 andrade_rmd_me_sjrp.pdf: 453172 bytes, checksum: 949b0dbffc53bf3b227c07bef2ee3856 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo principal deste trabalho e estudar a dinâmica de uma partícula pontual no interior de subconjuntos do plano. Tais sistemas são conhecidos na literatura como bilhares. Apresentaremos os principais conceitos desses sistemas e veremos que tais sistemas deixam invariante uma medida de probabilidade, o que nos permite aplicar a Teoria Ergódica ao problema do bilhar / The main goal of this work is to study the dynamical behavior of a point-like (dimensionless) particle in the interior of planar regions. Such systems are known in the literature as billiards. We're going to present the principal concepts of those systems and we'll see that such system turns the probability measure invariant, which allows us to apply the Ergodic Theory to billiard problems

Orientador: Vanderlei Minori Horita / Banca: Daniel Smania Brandão / Banca: Ali Messaoudi / Resumo: Este trablaho apresenta os conceitos de Lyapounov e de espaços próprios e fornece um resultado devido a Oseledets, o qual trata da existência desses expoentes (e, consequentemente, dos espaços próprios) do ponto de vista da teoria da medida. A prova do teorema que nós fornecemos foi dada originalmente por Mañe e posteriormente melhorada por Viana. / Abstract: This work presents the concepts of Lyapounov exponents and of proper spaces and provides a result due to Oseledets, wich deals with the existence of these exponents (and consequently, of the proper spaces) from a measure-theoretical point of view. The proof of the theorem which we provide was originally given by Mañe later improved by Viana. / Mestre

Cette thèse porte sur le comportement asymptotique des systèmes dynamiques et contient cinq chapitres indépendants.Nous considérons dans la première partie de la thèse trois systèmes dynamiques concrets. Les deux premiers chapitres présentent deux modèles de systèmes physiques : dans le premier, nous étudions la structure géométrique des langues d'Arnold de l'équation modélisant le contact de Josephson; dans le deuxième, nous nous intéressons au problème de Lagrange de recherche de la vitesse angulaire asymptotique d'un bras articulé sur une surface. Dans le troisième chapitre nous étudions la géométrie plane du billard elliptique avec des méthodes de la géométrie complexe.Les quatrième et cinquième chapitres sont dédiés aux méthodes générales d'étude asymptotique des systèmes dynamiques. Dans le quatrième chapitre nous prouvons la convergence des moyennes sphériques pour des actions du groupe libre sur un espace mesuré. Dans le cinquième chapitre nous fournissons une forme normale pour un produit croisé qui peut s'avérer utile dans l'étude des attracteurs étranges de systèmes dynamiques. / This thesis deals with the questions of asymptotic behavior of dynamical systems and consists of six independent chapters. In the first part of this thesis we consider three particular dynamical systems. The first two chapters deal with the models of two physical systems: in the first chapter, we study the geometric structure and limit behavior of Arnold tongues of the equation modeling a Josephson contact; in the second chapter, we are interested in the Lagrange problem of establishing the asymptotic angular velocity of the swiveling arm on the surface. The third chapter deals with planar geometry of an elliptic billiard.The forth and fifth chapters are devoted to general methods of studying the asymptotic behavior of dynamical systems. In the forth chapter we prove the convergence of markovian spherical averages for free group actions on a probablility space. In the fifth chapter we provide a normal form for skew-product diffeomorphisms that can be useful in the study of strange attractors of dynamical systems.