This thesis proposes a novel method for locating a p-dimensional invariant torus of an n-dimensional map. A set of non-linear equations is formulated and solved using the Newton-Raphson scheme. The method requires a set of sampled points on a guess invariant torus. An interpolant is passed through these points to compute the pointwise shift on the invariant torus, which is used to formulate the equation of invariance for the torus under the given map. The principal application of this method is to locate invariant tori of continuous systems. These tori occur for continuous dynamical systems having quasiperiodic orbits in state space. The discretization of the continuous system in terms of a map is accomplished in terms of its flow function. Results for one-dimensional invariant tori in two and three-dimensional state space and for two-dimensional invariant tori in three and four-dimensional maps are presented. / Master of Science

Slender structures in fluid flow exhibit a variety of rich behaviors. Here we study the equilibrium shapes of perfectly flexible strings that are moving with a uniform velocity and axial flow in viscous fluid. The string is acted upon by local, anisotropic, linear drag forces and a uniform body force. Generically, the configurations of the string are planar, and we provide analytical expressions for the equilibrium shapes of the string as a first order five parameter dynamical system for the tangential angle of the body ($theta$). Phase portraits in the angle-curvature ($theta,partial_s theta$) plane are generated, that can be shown to be $pi$ periodic after appropriate scaling and reflection operations. The rich parameter space allows for different kinds of phase portraits that give rise to a variety of curve geometries. Some of these solutions are unstable due to the presence of compressive stresses. Special cases of the problem include sedimenting filaments, dynamic catenaries, and towed strings. We also discuss equilibrium configurations of towed cables and other relevant problems with fixed boundary conditions. Special cases of the boundary value problem involve towing of neutrally buoyant cables and strings with pure axial flow between two fixed points. / Master of Science

A low-order discrete dynamical system (DDS) for finite-rate chemistry of H2-air combustion is derived in 3D. Fourier series with a single wavevector are employed to represent dependent variables of subgrid-scale (SGS) behaviors for applications to large-eddy simulation (LES). A Galerkin approximation is applied to the governing equations for comprising the DDS. Regime maps are employed to aid qualitative determination of useful values for bifurcation parameters of the DDS. Both isotropic and anisotropic assumptions are employed when constructing regime maps and studying bifurcation parameters sequences. For H2-air reactions, two reduced chemical mechanisms are studied via the DDS. As input to the DDS, physical quantities from experimental turbulent flow are used. Numerical solutions consisting of time series of velocities, species mass fractions, temperature, and the sum of mass fractions are analyzed. Numerical solutions are compared with experimental data at selected spatial locations within the experimental flame to check whether this model is suitable for an entire flame field. The comparisons show the DDS can mimic turbulent combustion behaviors in a qualitative sense, and the time-averaged computed results of some species are quantitatively close to experimental data.

This thesis investigates the possibility of approximating stationary solutions of stochastic differential equations using numerical methods. We consider a particular class of stochastic differential equations, which are known to generate random dynamical systems. The existence of stochastic stationary solution is proved using global attractor approach. Euler's numerical method, applied to the stochastic differential equation, is proved to generate a discrete random dynamical system. The existence of stationary solution is proved again using global attractor approach. At last we prove that the approximate stationary point converges in mean-square sense to the exact one as the time step of the numerical scheme diminishes.

On sait que la matrice d'incidence associée à une substitution ne suffit pas pour déterminer complètement le système dynamique associé, même dans des cas très simples, il existe plusieurs substitutions associées à une même matrice car il existe de nombreux mots ayant le même abélianisé. Dans cette thèse, on étudie les points communs de deux lignes brisées associées à deux substitutions $sigma_1$ et $sigma_2$ irréductibles unimodulaires de type Pisot qui ont la même matrice d'incidence. On identifie les points communs de ces deux lignes brisées à partir d'un algorithme. On montre ainsi que l'intersection de ces deux lignes brisées est aussi une ligne brisée associée au point fixe d'une nouvelle substitution. On montre plus précisément que si $sigma_1$ vérifie la conjecture Pisot et $0$ est un point intérieur à son fractal de Rauzy alors ces points communs peuvent être engendrés par une substitution définie sur un alphabet appelé alphabet des paires équilibrées. Cette substitution est obtenue à partir d'un algorithme, l'algorithme des paires équilibrées. On obtient ainsi l'intersection des intérieurs des deux fractals de Rauzy. En prenant la clôture de cet ensemble on obtient un ensemble substitutif. La condition que $0$ est un point intérieur au fractal de Rauzy associé à la substitution $sigma_1$ nous permet de montrer que l'intersection des deux fractals de Rauzy est de mesure positive. Dans une deuxième partie du travail on s'intéresse à l'étude de la frontière du fractal de Rauzy. Le fractal de Rauzy est dit fractal mais c'est en fait sa frontière qui est fractale. / The matrix of a substitution is not sufficient to completely determine the dynamics associated with it, even in the simplest cases since there are many words with the same abelianization. In this paper we study the common points of the canonical broken lines associated with two different irreducible Pisot unimodular substitutions σ1 and σ2 having the same incidence matrix. We prove that if σ1 verifies the Pisot conjecture and 0 is an inner point to the Rauzy fractal associated with the substitution σ1 then these common points can be generated with a substitution on an alphabet of so-called balanced pairs, and we obtain in this way the intersection of the interior of two Rauzy fractals.

In the presented thesis, we study an application of nonstandard analysis to dynamical systems, in particular to ω-limit set, stability and global attractor. We recall the definition and properties of elementary embedding, in detail ex- plore the introduction of infinitesimals to the real line and study metric spaces using nonstandard methods, in particular continuity and compactness which are closely related to the theory of dynamical systems. Last we attend to dynamical systems and present nonstandard characterizations of some of its properties such as asymptotic compactness and dissipativity and using these characterizations we prove one of the basic results of this theory - existence of a global attractor. 1

Modelos preditivos de mercado são ferramentas importantes para tomadores de decisões no âmbito público e privado. Devido à complexidade dinâmica do mercado imobiliário, composta pela interação de dois submercados distintos (mercado de ativos imobiliários e mercado de consumo de espaço) e pela limitação de dados disponíveis, o estudo analítico de mercados imobiliários requer a modelagem paramétrica de um sistema de equações que os descrevam, seguido pela identificação dos parâmetros deste sistema utilizando dados reais de uma região. Neste trabalho, estudamos o modelo dinâmico de mercado imobiliário proposto por Wheaton (1999), criado a partir do popular modelo de quatro quadrantes de autoria de DiPasquale e Wheaton (1996). Utilizamos técnicas de identificação de sistemas para elaborar um modelo de aprendizado para o estoque imobiliário, e o implementamos em Matlab. Aplicamos o método elaborado em dados simulados, para validá-lo, e então aplicamos o mesmo método, com adaptações, em dados reais do mercado imobiliário canadense. Os resultados obtidos validam o método de identificação de sistema dinâmico quando testado em dados simulados, e corroboram o modelo de Wheaton (1999) como modelo preditivo em dados reais. Ademais, os resultados indicam que um modelo que seja capaz de entender a evolução dinâmica dos parâmetros estáticos do modelo de Wheaton (1999), poderia melhorar os resultados deste como ferramenta preditiva. / Predictive market models are important tools for decision-makers in the public and private spheres. Due to the dynamic complexity of the real estate market, consisting of the interaction of two distinct submarkets (real estate asset market and space consumption market) and the lack of real estate data, the analytical study of real estate markets requires the parametric modeling of a system of equations describing them, followed by the identification of the parameters of this system using real data from a region. In this work, we study the dynamic real estate market model proposed by Wheaton (1999), created from the popular four-quadrant model of DiPasquale e Wheaton (1996). We use system identification techniques to develop a learning model for real estate inventory data, and implement it in Matlab. We apply the method devised in simulated data to validate it, and then apply the same method with adaptations in real data of the Canadian real estate market. The results validate the dynamic system identification method when tested in simulated data, and corroborate the Wheaton (1999) model as a predictive model in real data. In addition, the results indicate that a model that is able to understand the dynamic evolution of the static parameters of the Wheaton (1999) model, could improve its results as a predictive tool.

Les systèmes de contrôle en réseau sont un champ actif de recherche, où les différentes composantes du réseau sont spatialement distribué et tentent d'atteindre un objectif global. Ils apparaissent naturellement lors l'interaction d'un système piloté par ordinateur avec le monde physique.Avec les systèmes de contrôle en réseau une classe connexe des systèmes est décrit par les systèmes Cyber-physique, où les capacités de calcul embarqué peuvent interagir avec le monde physique.Dans ce travail, nous allons considérer la tâche classique d'observation et d'estimation et étudier les cas où les contraintes induite par le réseau nécessite une adaptation des mécanismes classique d'observation et d'estimation.Dans les système de contrôle en raison de limitation des capteurs (pour des raisons pratiques telles que la réduction des coûts) certains états ou paramètre du système ne sont pas connus. Dans ce contexte, la notion classique d'observabilitéexprime la capacité de déduire de la mesure les valeurs d'intérêt.Premièrement nous considérons le problème de la réduction de l'échantillonnage par l'utilisation de échantillonnage événementiel et ce pour plusieurs classes de systèmes. Ensuite, une procédure d'estimation et de contrôle sera proposé pour résoudre le problème du consensus dans un système multi-agent.Considérant enfin une dynamique de véhicule plus complexe, nous nous concentrons sur l'estimation du coefficient de frottement de la route pour résoudre un problème de suivi. / Network control systems is an active field of study where interacting component spatially distributed try to achieve a global goal. They naturally emerge from the interaction of computer driven mechanism and the physical world.Along with network control system a related class of systems is described by the so called: Cyber-physical systems, where integrated physical computational capabilities can interact.In this work we will consider the classical task of observation and estimation and investigate cases where network induced constraint calls for adapted observation and estimation scheme.In control system due to limitation in sensors ( for practical reason such as cost reduction) all the value of interest (whether the some unmeasured state or unknown parameter)are unknown. The classical notion of observabilityaccount for the ability to deduce from measurement those value of interest.First sampling reduction by use of event trigger will be studied for several class of systems. Then an estimation and control scheme will be establish to solve the problem of consensus in a multi agents system.Finally considering a more complex vehicle dynamic we focus on the estimation of tire road friction coefficient to solve a tracking problem.

abstract: The dynamics of a fluid flow inside 2D square and 3D cubic cavities under various configurations were simulated and analyzed using a spectral code I developed. This code was validated against known studies in the 3D lid-driven cavity. It was then used to explore the various dynamical behaviors close to the onset of instability of the steady-state flow, and explain in the process the mechanism underlying an intermittent bursting previously observed. A fairly complete bifurcation picture emerged, using a combination of computational tools such as selective frequency damping, edge-state tracking and subspace restriction. The code was then used to investigate the flow in a 2D square cavity under stable temperature stratification, an idealized version of a lake with warmer water at the surface compared to the bottom. The governing equations are the Navier-Stokes equations under the Boussinesq approximation. Simulations were done over a wide range of parameters of the problem quantifying the driving velocity at the top (e.g. wind) and the strength of the stratification. Particular attention was paid to the mechanisms associated with the onset of instability of the base steady state, and the complex nontrivial dynamics occurring beyond onset, where the presence of multiple states leads to a rich spectrum of states, including homoclinic and heteroclinic chaos. A third configuration investigates the flow dynamics of a fluid in a rapidly rotating cube subjected to small amplitude modulations. The responses were quantified by the global helicity and energy measures, and various peak responses associated to resonances with intrinsic eigenmodes of the cavity and/or internal retracing beams were clearly identified for the first time. A novel approach to compute the eigenmodes is also described, making accessible a whole catalog of these with various properties and dynamics. When the small amplitude modulation does not align with the rotation axis (precession) we show that a new set of eigenmodes are primarily excited as the angular velocity increases, while triadic resonances may occur once the nonlinear regime kicks in. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

We formalize and analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuoustime but not in discrete-time.