The theory of thermodynamic phase transitions has played a central role both in theoretical physics and in dynamical systems for several decades. One of its fundamental results is the classification of various physical models into equivalence classes with respect to the scaling behavior of solutions near the critical manifold. From that point of view, systems characterized by the same set of critical exponents are equivalent, regardless of how different the original physical models might be. For non-equilibrium phase transitions, the current theoretical framework is much less developed. In particular, an equivalent classification criterion is not available, thus requiring a specific analysis of each model individually. In this thesis, we propose a potential classification method for time-dependent dynamical systems, namely comparing the possible deformations of the original problem, and identifying dynamical systems which share the same deformation space. The specific model on which this procedure is developed is the Kuramoto model for interacting, disordered oscillators. Studied in the mean-field limit by a variety of methods, its associated synchronization phase transition appears as an appropriate model for cooperative phenomena ranging from coupled Josephson junctions to self-ordering patterns in biological and social systems. We investigate the geometric deformation of the dynamical system into the space of univalent maps of the unit disk, related to the Douady-Earle extension and the Denjoy-Wolff theory, and separately the algebraic deformation into the space of nonlinear sigma models for unitary operators. The results indicate that the Kuramoto model is representative for a large class of non-equilibrium synchronization models, with a rich phase-space diagram.

Ces travaux de thèse s’intéressent aux outils de simulation rapide de la fabrication d’un pneumatique. L'objectif est de prédire le mouvement et la position des éléments constitutifs du pneumatique, lors de leur mise en conformation, en prenant en compte les déformations induites par les actions mécaniques associées. Ces travaux s’appuient sur la mise en place d’un algorithme visant à modéliser le gonflement d’un outillage et la mise en place d’un modèle de déformation d’un système de formes géométriques aux caractéristiques mécaniques différentes. L'algorithme de gonflement de l’outillage se base sur la théorie des membranes inextensibles et son industrialisation est validée pour des dimensions standards de pneumatiques. La méthode Masse-Ressort est retenue pour réaliser la déformation de formes géométriques afin d’obtenir des résultats de déformation en temps réel. Dans ces travaux, cette méthode est adaptée aux matériaux d’un pneumatique à partir de travaux de caractérisations géométrique et mécanique validés par comparaison aux Éléments Finis. En conclusion, la modélisation développée permet une description à chaque étape du procédé de fabrication. / This thesis deals with fast simulation tools used to manufacture of a tire. The goal is to predict the displacement and position of the components of a tire taking into account the deformations induced by the associated mechanical actions. This work is based on the implementation of an algorithm for modeling the inflation of a tool and the development of a deformation model in order to compute the deformation of geometric shapes taking into account different mechanical properties. The algorithm to model the inflation of the tool is based on the theory of inextensible membranes and its industrialization is validated for standard sizes of tires. Mass-Spring method is used to achieve the deformation of geometric shapes in order to compute deformation in real time. In this work, this method is suitable for materials of a tire from geometric and mechanical characterizations validated by comparison with FEM. To conclude, the developed modelization allows a description for each step of the manufacturing process.

This master thesis is engaged in problematics of creation and compensation of geometric distortions in 1D agarose electrophoresis. This master thesis analyze the problematics of cause of these distortions and summarize the theory needed for compensation of these distortions. Based on acquired theory and created electrophoretic phantoms, the master thesis contains several suggestions for compensation of incurred distortions. These suggestions are recreated into functions, which are connected into a functional user interface for gel image analysis and geometric distortions compensation.