Ordinary differential equations of various types appear in the mathematical modelling in mechanics. Differential equations obtained are usually rather complicated nonlinear equations. However, using suitable approximations of nonlinearities, one can derive simple equations that are either well known or can be studied analytically. An example of such "approximative" equation is the so-called Duffing equation. Hence, the question on the existence of a periodic solution to the Duffing equation is closely related to the existence of periodic vibrations of the corresponding nonlinear oscillator.

In this thesis, central configurations, regularization of Simultaneous binary collision, linear stability of Kepler orbits, and index theory for symplectic path are studied. The history of their study is summarized in section 1. Section 2 deals with the following problem: given a collinear configuration of 4 bodies, under what conditions is it possible to choose positive masses which make it central. It is always possible to choose three positive masses such that the given three positions with the masses form a central configuration. However, for an arbitrary configuration of 4 bodies, it is not always possible to find positive masses forming a central configuration. An expression of four masses is established depending on the position x and the center of mass u, which gives a central configuration in the collinear four body problem. Specifically it is proved that there is a compact region in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of the compact region, it is always possible to choose positive masses to make the configuration central. The singularities of simultaneous binary collisions in collinear four-body problem is regularized by explicitly constructing new coordinates and time transformation in section 3. The motion in the new coordinates and time scale across simultaneous binary collision is at least C^2. Furthermore, the behavior of the motion closing, across and after the simultaneous binary collision, is also studied. Many different types of periodic solutions involving single binary collisions and simultaneous binary collisions are constructed. In section 4, the linear stability is studied for the Kepler orbits of the rhombus four-body problem. We show that, for given four proper masses, there exists a family of periodic solutions for which each body with the proper mass is at the vertex of a rhombus and travels along an elliptic Kepler orbit. Instead of studying the 8 degrees of freedom Hamilton system for planar four-body problem, we reduce this number by means of some symmetry to derive a two degrees of freedom system which then can be used to determine the linear instability of the periodic solutions. After making a clever change of coordinates, a two dimensional ordinary differential equation system is obtained, which governs the linear instability of the periodic solutions. The system is surprisingly simple and depends only on the length of the sides of the rhombus and the eccentricity e of the Kepler orbit. In section 5, index theory for symplectic paths introduced by Y.Long is applied to study the stability of a periodic solution x for a Hamiltonian system. We establish a necessary and sufficient condition for stability of the periodic solution x in two and four dimension.

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In this thesis, we introduce two new classes of differential equations, which essentially extend, in several directions, impulsive differential equations and equations on time scales. Basics of the theory for quasilinear systems are discussed, and particular results are obtained so that further investigations of the theory are guaranteed. Applications of the newly-introduced systems are shown through a center manifold theorem, and further, Hopf bifurcation Theorem is proved for a three-dimensional discontinuous dynamical system.

Cette thèse porte sur l’étude des solutions périodiques du problème des N-tourbillons à vorticité positive. Ce problème, formulé par Helmholtz il y a plus de 160 ans, possède une histoire très riche et reste un domaine de recherche très actif. Pour un nombre quelconque de tourbillons et sans contrainte sur les vorticités, ce système n’est pas intégrable au sens de Liouville : on ne peut trouver de solution périodique non triviale par des méthodes explicites. Dans cette thèse, à l’aide de méthodes variationnelles, nous prouvons l’existence d’une infinité de solutions périodiques non triviales pour un système de N tourbillons à vorticités positives. De plus, lorsque les vorticités sont des nombres rationnels positifs, nous montrons qu’il n’existe qu’un nombre fini de niveaux d’énergie sur lesquels un équilibre relatif pourrait exister. Enfin, pour un système de N-tourbillons identiques, nous montrons qu’il existe une infinité de chorégraphies simples. / This thesis focuses on the study of the periodic solutions of the N-vortex problem of positive vorticity. This problem was formulated by Helmholtz more than 160 years ago and remains an active research field. For an undetermined number of vortices and general vorticities the system is not Liouville integrable and periodic solutions cannot be determined explicitly, except for relative equilibria. By using variational methods, we prove the existence of infinitely many non-trivial periodic solutions for arbitrary N and arbitrary positive vorticities. Moreover, when the vorticities are positive rational numbers, we show that there exists only finitely many energy levels on which there might exist a relative equilibrium. Finally, for the identical N-vortex problem, we show that there exists infinitely many simple choreographies.

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In this thesis, we study the existence of random periodic solutions for both nonlinear dissipative stochastic functional differential equations (SFDEs) and semilinear nondissipative SFDEs in C([-r,0],R^d). Under some sufficient conditions for the existence of global semiflows for SFDEs, by using pullback-convergence technique to SFDE, we obtain a general theorem about the existence of random periodic solutions. By applying coupled forward-backward infinite horizon integral equations method, we perform the argument of the relative compactness of Wiener-Sobolev spaces in C([0,τ],C([-r,0]L²(Ω))) and the generalized Schauder's fixed point theorem to show the existence of random periodic solutions.

This thesis extends the basic ordinary differential equations (ODE) course, specifically considering perturbations of ODEs. We introduce uniformly asympto- tic approximation and uniformly ordered approximation. We provide a perturba- tion-based method of computing derivatives of ODE solutions with respect to: an initial value, a parameter, and initial time. We present the method of averaging, error estimate, and a theorem about the existence and stability of a periodic so- lution to ODEs in periodic standard form. Furthermore, we apply the method of averaging to determine the period of a periodic solution of Duffing equation without forcing or damping. All the terms and methods of perturbation theory used in the thesis are accompanied with examples. 1

La compréhension du cycle de la matière organique du sol (MOS) est un outil majeur dans la lutte contre le réchauffement climatique, la préservation de la biodiversité ainsi que dans la consolidation de la sécurité alimentaire. Dans ce contexte, cette thèse porte sur la modélisation et l'analyse mathématique de modèles de la dynamique du carbone organique dans le sol.Dans le chapitre 2, nous avons étudié la robustesse et les propriétés mathématiques d'un modèle non linéaire (MOMOS). Nous avons montré que si les données sont périodiques nous obtenons l'existence d'une solution périodique attractive. Le chapitre3 est consacré à la validation mathématique d'un modèle spatialisé basé sur les équations de MOMOS, auxquels nous avons ajouté des opérateurs de diffusion et de transport. L'effet de l'hétérogénéité spatiale sur ce modèle est étudié dans le chapitre4 en utilisant des techniques d'homogénéisation. Suivant la méthodologie de Bosattaet Agren, nous dérivons un autre modèle à qualité continue, qui prend en compte l'effet de l'âge sur la décomposition de la MOS. Le chapitre 5 contient la validation mathématique et expérimentale du modèle. Enfin, nous considérons dans les chapitres6 et 7, un modèle incluant l'effet de la chemotaxie. Nous montrons l'existence, la positivité et l'unicité des solutions dans des domaines suffisamment réguliers de dimension inférieure ou égale à 3. / Understanding the soil organic matter (SOM) cycle is a major tool in the effort toreduce global warming, to preserve biodiversity and to improve food safety strategies.In this context, this thesis is about modelling and mathematical analysis of thedynamics of the organic carbon in soil.In chapter 2, we validate mathematically a nonlinear soil organic carbon model(MOMOS) and we prove that, if data is periodic, then there is a unique attractiveperiodic solution. In chapter 3, we focus on the mathematical validation of a spatialmodel derived from MOMOS and where we used diffusion and transport operators.We prove also the existence of a periodic solution. In addition, the effect of soilheterogeneities on the model is studied in chapter 4 using homogenization techniques.Moreover, following the Bosatta and Agren methodology, we derive a continuousquality model taking in consideration the effect of age on the quality of SOM. Wevalidate the model mathematically and experimentally in chapter 5. Finally, weconsider in chapters 6 and 7 another model that takes into account the chemotaxismovement of soil microorganisms. We prove mainly the existence and uniqueness of apositive solution in a regular spatial domain of dimension less or equal to 3.

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