Neste trabalho estudamos relações entre expoente de Lyapunov e continuidade absoluta da folheação central para difeomorfismos parcialmente hiperbólicos conservativos de \'T POT. 3\'. Sobre tal tema, provamos que tipicamente (\'C POT. 1\' aberto e \'C POT. 2\' denso) os difeomorfismos parcialmente hiperbólicos, conservativos de classe \'C POT. 2\' , do toro \'T POT. 3\', apresentam folheação central não absolutamente contínua. Desta maneira, respondemos positivamente uma pergunta proposta em [20]. Também neste trabalho, estudamos entropia topológica para Sistema Iterado de Funções. Neste contexto, damos uma nova demonstração para uma conjectura proposta em [14] e provada primeiramente em [15]. Apresentamos um método geométrico que nos permite calcular entropia para transformações de \'S POT. 1\', como em [15]. Além de disso o método apresentado se verifica para casos mais gerais, como por exemplo: transformações não comutativas / In this work we study relations between Lyapunov exponents, absolute continuity of center foliation for conservative partially hyperbolic diffeomorphisms of \'T POT. 3\'. About this theme, (on a \'C POT. 1\' open and \'C POT. 2\'dense set) of conservative partially hyperbolic \'C POT. 2\' diffeomorphisms of the 3-torus presents non absolutely continuous center foliation. So, we answer positively a question proposed in [20]. Also in this work, we study topological entropy for Iterated Functions Systems. In this setting, we give a proof for a conjecture proposed in [14] and firstly proved in [15]. We present a geometrical method that allows us to calcule the entropy for transformations of \'S POT. 1\', like in [15]. Furthermore this method holds for more general cases, for example: non commutative transformations

La dynamique d'un difféomorphisme d'une variété compacte est essentiellement concentrée sur l'ensemble récurrent par chaînes, qui est partitionné en classes de récurrence par chaînes, disjointes et indécomposables. Le travail de Bonatti et Crovisier [BC] montre que, pour les difféomorphismes C¹-génériques, une classe de récurrence par chaînes ou bien est une classe homocline, ou bien ne contient pas de point périodique. Une classe de récurrence par chaînes sans point périodique est appelée classe apériodique.Il est clair qu'une classe homocline hyperbolique ni contient d'orbite périodique faible ni supporte de mesure non hyperbolique.Cette thèse tente de donner une caractérisation des classes homoclines non hyperboliques en montrant qu'elles contiennent des orbites périodiques faibles ou des mesures ergodiques non hyperboliques. Cette thèse décrit également les décompositions dominées sur les classes apériodiques.Le premier résultat de cette thèse montre que, pour les difféomorphismes C¹-génériques, si les orbites périodiques contenues dans une classe homocline H(p) ont tous leurs exposants de Lyapunov bornés loin de zéro, alors H(p) doit être (uniformément) hyperbolique. Ceci est dans l'esprit des travaux sur la conjecture de stabilité, mais il y a une différence importante lorsque la classe homocline H(p) n'est pas isolée. Par conséquent, nous devons garantir que des orbites périodiques "faibles'', crées par perturbations au voisinage de la classe homocline, sont contenues dans la classe. En ce sens, le problème est de nature "intrinsèque'', et l'argument classique de la conjecture de stabilité est impraticable.Le deuxième résultat de cette thèse prouve une conjecture de Díaz et Gorodetski [DG]: pour les difféomorphismes C¹-génériques, si une classe homocline n'est pas hyperbolique, alors elle porte une mesure ergodique non hyperbolique. C'est un travail en collaboration avec C. Cheng, S. Crovisier, S. Gan et D. Yang. Dans la démonstration, nous devons appliquer une technique introduité dans [DG], et qui améliore la méthode de [GIKN], pour obtenir une mesure ergodique comme limite d'une suite de mesures périodiques.Le troisième résultat de cette thèse énonce que, génériquement, une décomposition dominée non-triviale sur une classe apériodique stable au sens de Lyapunov est en fait une décomposition partiellement hyperbolique. Plus précisément, pour les difféomorphismes C¹-génériques, si une classe apériodique stable au sens de Lyapunov a une décomposition dominée non-triviale Eoplus F, alors, l'un des deux fibrés est hyperbolique: soit E contracté, soit F dilaté.Dans les démonstrations des résultats principaux, nous construisons des perturbations qui ne sont pas obtenues directement à partir des lemmes de connexion classiques. En fait, il faut appliquer le lemme de connexion un grand nombre (et même un nombre infini) de fois. Nous expliquons les méthodes de connexions multiples dans le Chapitre 3. / The dynamics of a diffeomorphism of a compact manifold concentrates essentially on the chain recurrent set, which splits into disjoint indecomposable chain recurrence classes. By the work of Bonatti and Crovisier [BC], for C¹-generic diffeomorphisms, a chain recurrence class either is a homoclinic class or contains no periodic point. A chain recurrence class without a periodic point is called an aperiodic class.Obviously, a hyperbolic homoclinic class can neither contain weak periodic orbit or support non-hyperbolic ergodic measure.This thesis attempts to give a characterization of non-hyperbolic homoclinic classes via weak periodic orbits inside or non-hyperbolic ergodic measures supported on it. Also, this thesis gives a description of the dominated splitting on Lyapunov stable aperiodic classes.The first result of this thesis shows that for C¹-generic diffeomorphisms, if the periodic orbits contained in a homoclinic class H(p) have all their Lyapunov exponents bounded away from 0, then H(p) must be (uniformly) hyperbolic. This is in spirit of the works of the stability conjecture, but with a significant difference that the homoclinic class H(p) is not known isolated in advance. Hence the "weak'' periodic orbits created by perturbations near the homoclinic class have to be guaranteed strictly inside the homoclinic class. In this sense the problem is of an "intrinsic" nature, and the classical argument of the stability conjecture does not pass through.The second result of this thesis proves a conjecture by Díaz and Gorodetski [DG]: for C¹-generic diffeomorphisms, if a homoclinic class is not hyperbolic, then there is a non-hyperbolic ergodic measure supported on it. This is a joint work with C. Cheng, S. Crovisier, S. Gan and D. Yang. In the proof, we have to use a technic introduced in [DG], which developed the method of [GIKN], to get an ergodic measure by taking the limit of a sequence of periodic measures.The third result of this thesis states that, generically, a non-trivial dominated splitting over a Lyapunov stable aperiodic class is in fact a partially hyperbolic splitting. To be precise, for C¹-generic diffeomorphisms, if a Lyapunov stable aperiodic class admits a non-trivial dominated splitting Eoplus F, then one of the two bundles is hyperbolic: either E is contracted or F is expanded.In the proofs of the main results, we construct several perturbations which are not simple applications of the connecting lemmas. In fact, one has to apply the connecting lemma several (even infinitely many) times. We will give the detailed explanations of the multi-connecting processes in Chapter 3.

We study one-parameter families of quasi-periodically forced monotone interval maps and provide sufficient conditions for the existence of a parameter at which the respective system possesses a non-uniformly hyperbolic attractor. This is equivalent to the existence of a sink-source orbit, that is, an orbit with positive Lyapunov exponent both forwards and backwards in time. The attractor itself is a non-continuous invariant graph with negative Lyapunov exponent, often referred to as ‘SNA’. In contrast to former results in this direction, our conditions are C² -open in the fibre maps. By applying a general result about saddle-node bifurcations in skew-products, we obtain a conclusion on the occurrence of non-smooth bifurcations in the respective families. Explicit examples show the applicability of the derived statements.

This thesis tests the ability of the Lorenz's (2005) chaotic model to simulate predictability curve of the ECMWF model calculated from data over the 1986 to 2011 period and demonstrates similarity of the predictability curves for the Lorenz's model with N = 90 variables. This thesis also tests approximations of predictability curves and their differentials, aiming to correct the ECMWF model estimated parameters and thus allow for estimation of the largest Lyapunov exponent, model error and limit value of the predictability curve. The correction is based on comparing the parameters estimated for the Lorenz's and ECMWF and on comparison with the largest Lyapunov exponent (λ=0,35 day-1 ) and limit value of the predictability curve (E∞=8,2) of the Lorenz's model. Parameters are calculated from approximations made by the Quadratic hypothesis with and without model error, as well as by Logarithmic and General hypotheses and by hyperbolic tangent employing corrections with and without model error. Average value of the largest Lyapunov exponent is estimated to be λ=0,37 day-1 for the ECMWF model, limit values of the predictability curves are estimated with lower theoretically derived values and new approach of calculation of model error based on comparison of models is presented.

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In this dissertation, we consider two different strategies for improving the projection-based reduced order model (ROM) accuracy: (I) adding closure terms to the standard ROM; (II) using Lagrangian data to improve the ROM basis. Following strategy (I), we propose a new data-driven reduced order model (ROM) framework that centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. The VMS methodology is a natural fit for the hierarchical structure of the ROM basis: In the first step, we use the ROM projection to separate the scales into three categories: (i) resolved large scales, (ii) resolved small scales, and (iii) unresolved scales. In the second step, we explicitly identify the VMS-ROM closure terms, i.e., the terms representing the interactions among the three types of scales. In the third step, we use available data to model the VMS-ROM closure terms. Thus, instead of phenomenological models used in VMS for standard numerical discretizations (e.g., eddy viscosity models), we utilize available data to construct new structural VMS-ROM closure models. Specifically, we build ROM operators (vectors, matrices, and tensors) that are closest to the true ROM closure terms evaluated with the available data. We test the new data-driven VMS-ROM in the numerical simulation of four test cases: (i) the 1D Burgers equation with viscosity coefficient $nu = 10^{-3}$; (ii) a 2D flow past a circular cylinder at Reynolds numbers $Re=100$, $Re=500$, and $Re=1000$; (iii) the quasi-geostrophic equations at Reynolds number $Re=450$ and Rossby number $Ro=0.0036$; and (iv) a 2D flow over a backward facing step at Reynolds number $Re=1000$. The numerical results show that the data-driven VMS-ROM is significantly more accurate than standard ROMs. Furthermore, we propose a new hybrid ROM framework for the numerical simulation of fluid flows. This hybrid framework incorporates two closure modeling strategies: (i) A structural closure modeling component that involves the recently proposed data-driven variational multiscale ROM approach, and (ii) A functional closure modeling component that introduces an artificial viscosity term. We also utilize physical constraints for the structural ROM operators in order to add robustness to the hybrid ROM. We perform a numerical investigation of the hybrid ROM for the three-dimensional turbulent channel flow at a Reynolds number $Re = 13,750$. In addition, we focus on the mathematical foundations of ROM closures. First, we extend the verifiability concept from large eddy simulation to the ROM setting. Specifically, we call a ROM closure model verifiable if a small ROM closure model error (i.e., a small difference between the true ROM closure and the modeled ROM closure) implies a small ROM error. Second, we prove that a data-driven ROM closure (i.e., the data-driven variational multiscale ROM) is verifiable. For strategy (II), we propose new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct new Lagrangian ROMs. We show that the new Lagrangian ROMs are orders of magnitude more accurate than the standard Eulerian ROMs, i.e., ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs' accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis. / Doctor of Philosophy / Reduced order models (ROMs) are popular in physical and engineering applications: for example, ROMs are widely used in aircraft designing as it can greatly reduce computational cost for the aircraft's aeroelastic predictions while retaining good accuracy. However, for high Reynolds number turbulent flows, such as blood flows in arteries, oil transport in pipelines, and ocean currents, the standard ROMs may yield inaccurate results. In this dissertation, to improve ROM's accuracy for turbulent flows, we investigate three different types of ROMs. In this dissertation, both numerical and theoretical results show that the proposed new ROMs yield more accurate results than the standard ROM and thus can be more useful.

Exploring the concepts of long range aerial transport of microorganisms is the main motivation of this study. For this purpose we use theories and concepts of dynamical systems in the context of geophysical fluid systems. We apply powerful notions such as finite-time Lyapunov exponent (FTLE) and the associated Lagrangian coherent structures (LCS) and we attempt to provide mathematical explanations and frameworks for some applied questions which are based on realistic concerns of atmospheric transport phenomena. Accordingly, we quantify the accuracy of prediction of FTLE-LCS features and we determine the sensitivity of such predictions to forecasting parameters. In addition, we consider the spatiotemporal resolution of the operational data sets and we propose the concept of probabilistic source and destination regions which leads to the definition of stochastic FTLE fields. Moreover, we put forward the idea of using ensemble forecasting to quantify the uncertainty of the forecast results. Finally, we investigate the statistical properties of localized measurements of atmospheric microbial structure and their connections to the concept of local FTLE time-series. Results of this study would pave the way for more efficient models and management strategies for the spread of infectious diseases affecting plants, domestic animals, and humans. / Ph. D.

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