在此篇論文中，我們提供二維非線性動態系統之非振盪解的一個分類法，此分類法是依據解的漸近值作分類，同時我們也得到具有此漸近值之非振盪解的存在性的充分必要條件。 / In this thesis, we provide a classification scheme for nonoscillatory solutions of a class of two-dimensional dynamical systems in terms of their asymptotic values. In addition, we find the sufficient and necessary conditions for the existence of these solutions.

Aquesta tesi es situa en el marc de la teoriaqualitativadelssistemesd’equacionsdiferencials en el pla. Cada capítol conté un aspectediferent, però en totsells es tractenproblemes, la soluciódelsqualsestà basada en el rol que hi juguen les simetriesdiscretes i continues (reversibilitat o simetries de Lie) de campsvectorialsplans. A la introducció, es dóna un resumdelsresultatsmésconeguts i s’hiintrodueix la notació que es fa servir al llarg de la tesi. En el segon i tercer capítol, s’aborda el problema de trobarl’expressió explícita del canvilinealitzant o orbitalmentlinealitzantd#un camp vectorial suau a partir del coneixementd’uncommutador, en el cas de la linealització, o una simetria de Lie, en el cas de la linealització orbital. Cada capítol finalitzaambexemplesil.lustratius del procedimentconstructiudelscanvis. Al Capítol 5 s’apliquenelsresultatsdelscapítolsanteriors, combinatsamblinealitzacionsDarbouxianes. Concretament, es considera un sistema quadràtictipusLotka-Volterra i es caracteritzen les selles linealitzables i orbitalmentlinealitzablesmitjançant la troballadelscanvislinealitzants o orbitalmentlinealitzants. En el sisè capítol, s’utilitzal’existènciad’unàlgebra de simetriespuntuals de Lie per donar informació sobre l’existència i localitzaciód’òrbitesperiòdiques. En particular, quanl’àlgebra de simetriespuntuals de Lie d’unaequació diferencial escalar de segónordreautònoma i suau té dimensiómajor o igual a dos, definim les anomenadesfuncionsfonamentals que enspermeten estudiar les òrbitesperiòdiques al pla de fases. En el cas particular d’equacionspolinomials de Liénard, mostrem la no existència de cicles límit en tot el pla de fases. Finalment, al Capítol 7 es relacionen elssistemes reversibles amb el problema del centre aixícomamb el problema de la integrabilitat analítica. Consideremsistemesd’equacionsdiferencialsanalíticsamb centres degenerats i mostrem que poden transformar-se, després d’un reescalat del temps, en un sistema lineal i reversible. El coneixement de integralsprimeresens proporciona un procediment per caracteritzar, en alguns casos, la condició de reversibilitat del centre degenerat. D’altra banda, relacioneml’existència de integralsprimeresanalítiquesamb la reversibilitat orbital analítica en el cas de singularitatsdèbils no degenerades.

This dissertation aims at analyzing complex problems arising in the context of dynamical networks, proteomics, and disease prevention. First, a new graph-based method for proving global stability of synchronization in directed dynamical networks is developed. This method utilizes stability and graph theories to clarify the interplay between individual oscillator dynamics and network topology. Secondly, a graph-theoretical algorithm is proposed to predict Ca2+-binding site in proteins. The new algorithm enables us to identify previously-unknown Ca2+-binding sites, and deepens our understanding towards disease-related Ca2+-binding proteins at a molecular level. Finally, an optimization model and algorithm to solve a disease prevention problem are described at the population level. The new resource allocation model is designed to assist clinical managers to make decisions on identifying at-risk population groups, as well as selecting a screening and treatment strategy for chlamydia and gonorrhea patients under a fixed budget. The resource allocation model and algorithm can have a significant impact on real treatment strategy issues.

In this thesis, the dynamical system is used as a function on afinite group, to show how states change. We investigate the'numberof cycles' and 'length of cycle' under finite groups. Using grouptheory, fixed point, periodic points and some examples, formulas tofind 'number of cycles' and 'length of cycle' are derived. Theexamples used are on finite cyclic group Z_6 with respectto binary operation '+'. Generalization using finite groups ismade. At the end, I compared the dynamical system over finite cyclic groups with the finite non-cyclic groups and then prove the general formulas to find 'number of cycles' and 'length of cycle' for both cyclic and non-cyclic groups.

In this thesis, we study the normally elliptic singular perturbation problems including both finite and infinite dimensional cases, which could also be non-autonomous. In particular, we establish the existence and smoothness of O(1) local invariant manifolds and provide various estimates which are independent of small singular parameters. We also use our results on local invariant manifolds to study the persistence of homoclinic solutions under weakly dissipative and conservative per- turbations. We apply Semi-group Theory and Lyapunov-Perron Integral Equations with some careful estimates to handle the O(1) driving force in the system so that we can approximate the full system through some simpler limiting system. In the investigation of homoclinics, a diagonalization procedure and some normal form transformation should be first carried out. Such diagonalization procedure is not trivial at all. We discuss this issue in the appendix. We use Melnikov type analysis to study the weakly dissipative case, while the conservative case is based on some energy methods. As a concrete example, we have shown rigrously the persistence of homoclinic solutions of an elastic pendulum model which may be affected by damping, external forcing and other potential fields.

We introduce numerical methods for the analysis of random dynamical systems. The subdivision and the continuation algorithm are powerful tools which will be demonstrated for a system from ship dynamics. With our software package we are able to show that the well known safe basin is a moving fractal set. We will also give a numerical approximation of the attracting invariant set (which contains a local attractor) and its evolution.

We consider a general class of finite dimensional deterministic dynamical systems with finitely many local attractors each of which supports a unique ergodic probability measure, which includes in particular the class of Morse–Smale systems in any finite dimension. The dynamical system is perturbed by a multiplicative non-Gaussian heavytailed Lévy type noise of small intensity ε > 0. Specifically we consider perturbations leading to a Itô, Stratonovich and canonical (Marcus) stochastic differential equation. The respective asymptotic first exit time and location problem from each of the domains of attractions in case of inward pointing vector fields in the limit of ε-> 0 has been investigated by the authors. We extend these results to domains with characteristic boundaries and show that the perturbed system exhibits a metastable behavior in the sense that there exits a unique ε-dependent time scale on which the random system converges to a continuous time Markov chain switching between the invariant measures. As examples we consider α-stable perturbations of the Duffing equation and a chemical system exhibiting a birhythmic behavior.

We consider the problem of local linearization of power series defined over complete valued fields. The complex field case has been studied since the end of the nineteenth century, and renders a delicate number theoretical problem of small divisors related to diophantine approximation. Since a work of Herman and Yoccoz in 1981, there has been an increasing interest in generalizations to other valued fields like p-adic fields and various function fields. We present some new results in this domain of research. In particular, for fields of prime characteristic, the problem leads to a combinatorial problem of seemingly great complexity, albeit of another nature than in the complex field case. In cases for which linearization is possible, we estimate the size of linearization discs and prove existence of periodic points on the boundary. We also prove that transitivity and ergodicity is preserved under the linearization. In particular, transitivity and ergodicity on a sphere inside a non-Archimedean linearization disc is possible only for fields of p-adic numbers.

Les écoulements décollés sont omniprésents dans la nature comme dans les écoulements industriels (aérodynamique externe des véhicules, des bâtiments, écoulements autour d’aubes de turbines, aérodynamique interne dans des tuyaux...) où ils sont en général sources de nuisances (vibrations, bruit aéroacousitque, forces de traînée ou de portance). Les enjeux associés à la compréhension et à la maîtrise de tels écoulements, caractérisés par une bulle de recirculation, sont donc considérables.Un capteur "visuel" non invasif développé au laboratoire PMMH est d'abord amélioré afin d'accéder en temps réel aux champs de vitesses - et à leurs grandeurs dérivées - des écoulements rencontrés en soufflerie industrielle. Basé sur un algorithme de flot optique issu de la vision par ordinateur, cette approche expérimentale novatrice permet de faciliter les études paramétriques et peut être implémenté dans des boucles de contrôle réactif.Ensuite, les mesures obtenues pour un écoulement sur une plaque plane sont analysées dans le cadre de l'identification de système. Un modèle d’ordre réduit est alors construit par apprentissage, permettant de prédire la dynamique de la transition de la couche limite laminaire vers la turbulence.Enfin, le sillage pleinement turbulent derrière une géométrie modélisant une voiture simplifiée est caractérisé, de façon classique et en tant que système dynamique. Différentes modifications de l'écoulement à l'aide de micro-jets sont testées. Une loi de contrôle réactif consistant à suivre et forcer la recirculation est mise en œuvre avec succès. / Separated flows are ubiquitous in nature and industrial systems, such as diffusers, airfoils, air conditioning plants, moving vehicles... As the separation can strongly influence the performances of such devices, investigating their dynamics and their control is of great interest.A visual sensor developed at PMMH laboratory is first improved to measure in real time the velocity fields and its derived values for flows available in wind tunnels. Based on an optical flow algorithm from the computer vision domain, this new experimental approach makes easier parametric studies and may be used in closed-loop controls.The dynamics of the flow over a flat plate are then investigated. A system identification method - the dynamic observer - is successfully implemented to build a reduced-order model of the transient flow, which captures and predicts well the instabilities generated.Finally, the fully turbulent wake of the square-back Ahmed body is described. Dynamical system tools are applied to characterize it. Using continuous and pulsed micro-jets, different forcing strategies are analyzed. An opposition closed-loop control is implemented, tracking and driving the recirculation.

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.