Thesis advisor: Jan R. Engelbrecht / Thesis advisor: Renato E. Mirollo / Many phenomena in nature that involve ordering in time can be understood as collective behavior of coupled oscillators. One paradigm for studying a population of self-sustained oscillators is the Kuramoto model, where each oscillator is described by a phase variable, and interacts with other oscillators through trigonometric functions of phase differences. This dissertation studies $N$ identical Kuramoto oscillators in a general form \[ \dot{\theta}_{j}=A+B\cos\theta_{j}+C\sin\theta_{j}\qquad j=1,\dots,N, \] where coefficients $A$, $B$, and $C$ are symmetric functions of all oscillators $(\theta_{1},\dots,\theta_{N})$. Dynamics of this model live in group orbits of M\"obius transformations, which are low-dimensional manifolds in the full state space. When the system is a phase model (invariant under a global phase shift), trajectories in a group orbit can be identified as flows in the unit disk with an intrinsic hyperbolic metric. A simple criterion for such system to be a gradient flow is found, which leads to new classes of models that can be described by potential or Hamiltonian functions while exhibiting a large number of constants of motions. A generalization to extended phase models with non-identical couplings gives rise to richer structures of fixed points and bifurcations. When the coupling weights sum to zero, the system is simultaneously gradient and Hamiltonian. The flows mimic field lines of a two-dimensional electrostatic system consisting of equal amounts of positive and negative charges. Bifurcations on a partially synchronized subspace are discussed as well. / Thesis (PhD) — Boston College, 2017. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

Made available in DSpace on 2015-05-15T11:46:15Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1316113 bytes, checksum: fa392778ab5ea3463805913d86fe571f (MD5) Previous issue date: 2013-08-23 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Set a generalization of Möbius transformation and build a theory of inductive that may be an n-dimensional hyperbolic space. This theory allows for the inductive starting with n = 1, together with the extension notion of the Poincaré build a chain groups GM(n) transformation Möbius and spaces hyperbolic H2 members. We will see explicit formulas for the Poincaré bisectors in size 2. And may on models of hiperbolic space ball these bisectors coincide with the isometric spheres of isometries. We will be using explicit formulas of bissectors, to ge youself an algorithm, the DAFC, to obtain generators for Fuchsianos groups, which will be our study group. / Definir uma generalização do conceito de transformação de Möbius e construir uma teoria indutiva do que venha a ser um espaço hiperbólico de dimensão n. Essa teoria indutiva nos permite que se iniciando com n = 1, juntamente com a noção de extensão de Poincaré, construir uma cadeia de grupos GM(n) de transformação de Möbius e os espaços hiperbólicos H2 associados. Veremos fórmulas explícitas para os bissetores de Poincaré em dimensão 2. E que nos modelos de bola do espaço hiperbólico, esses bissetores coincidem com as esferas isométricas das isometrias. Iremos usar fórmulas explícitas dos bissetores, para obter-se um algoritmo, o DAFC, para obtenção de geradores para grupos Fuchsianos, que será nosso grupo em estudo.

The aim of this work is the study of arbitrary mobius transformations by means of simple complex transformations, namely: the Translation, the Rotation, the Homotetia (Contraction and Dilatation) and Inversion. The results obtained were applied in circles and straight line. At the end, we give the the alternative of studying mobius transformations via matrices. / O objetivo deste trabalho é estudar transformações de Möbius arbitrárias por meio de transformações complexas mais simples, a saber: a Translação, a Rotação, a Homotetia (Contração e Dilatação) e a Inversão. Os resultados obtidos foram aplicados em círculos e retas. No final, damos a alternativa de estudar transformações de Möbius via matrizes.

<p>I denna uppsats presenteras grundläggande delar av hyperbolisk geometri. Uppsatsen är indelad i två kapitel. I första kapitlet studeras Möbiusavbildningar på Riemannsfären. Andra kapitlet presenterar modellen av hyperbolisk geometri i övre halvplanet H, skapad av Poincaré på 1880-talet.</p><p>Huvudresultatet i uppsatsen är Gauss – Bonnét´s sats för hyperboliska trianglar.</p> / <p>In this thesis we present fundamental concepts in hyperbolic geometry. The thesis is divided into two chapters. In the first chapter we study Möbiustransformations on the Riemann sphere. The second part of the thesis deal with hyperbolic geometry in the upper half-plane. This model of hyperbolic geometry was created by Poincaré in 1880.</p><p>The main result of the thesis is Gauss – Bonnét´s theorem for hyperbolic triangles.</p>

I denna uppsats presenteras grundläggande delar av hyperbolisk geometri. Uppsatsen är indelad i två kapitel. I första kapitlet studeras Möbiusavbildningar på Riemannsfären. Andra kapitlet presenterar modellen av hyperbolisk geometri i övre halvplanet H, skapad av Poincaré på 1880-talet. Huvudresultatet i uppsatsen är Gauss – Bonnét´s sats för hyperboliska trianglar. / In this thesis we present fundamental concepts in hyperbolic geometry. The thesis is divided into two chapters. In the first chapter we study Möbiustransformations on the Riemann sphere. The second part of the thesis deal with hyperbolic geometry in the upper half-plane. This model of hyperbolic geometry was created by Poincaré in 1880. The main result of the thesis is Gauss – Bonnét´s theorem for hyperbolic triangles.

Cette thèse est rédigée par articles. Les articles sont rédigés en anglais et le reste de la thèse est rédigée en français. / Le travail établit une équivalence en termes de puissance entre les tests basés sur la alpha-distance de covariance et sur le critère d'indépendance de Hilbert-Schmidt (HSIC) avec fonction caractéristique de distribution de probabilité stable d'indice alpha avec paramètre d'échelle suffisamment petit. Des simulations en grandes dimensions montrent la supériorité des tests de distance de covariance et des tests HSIC par rapport à certains tests utilisant les copules. Des simulations montrent également que la distribution de Pearson de type III, très utile et moins connue, approche la distribution exacte de permutation des tests et donne des erreurs de type I précises. Une nouvelle méthode de sélection adaptative des paramètres d'échelle pour les tests HSIC est proposée. Trois simulations, dont deux sont empruntées de l'apprentissage automatique, montrent que la nouvelle méthode de sélection améliore la puissance des tests HSIC. Le problème de tests d'indépendance entre deux vecteurs est généralisé au problème de tests d'indépendance mutuelle entre plusieurs vecteurs. Le travail traite aussi d'un problème très proche à savoir, le test d'indépendance sérielle d'une suite multidimensionnelle stationnaire. La décomposition de Möbius des fonctions caractéristiques est utilisée pour caractériser l'indépendance. Des tests généralisés basés sur le critère d'indépendance de Hilbert-Schmidt et sur la distance de covariance en sont obtenus. Une équivalence est également établie entre le test basé sur la distance de covariance et le test HSIC de noyau caractéristique d'une distribution stable avec des paramètres d'échelle suffisamment petits. La convergence faible du test HSIC est obtenue. Un calcul rapide et précis des valeurs-p des tests développés utilise une distribution de Pearson de type III comme approximation de la distribution exacte des tests. Un résultat fascinant est l'obtention des trois premiers moments exacts de la distribution de permutation des statistiques de dépendance. Une méthodologie similaire a été développée pour le test d'indépendance sérielle d'une suite. Des applications à des données réelles environnementales et financières sont effectuées. / The main result establishes the equivalence in terms of power between the alpha-distance covariance test and the Hilbert-Schmidt independence criterion (HSIC) test with the characteristic kernel of a stable probability distribution of index alpha with sufficiently small scale parameters. Large-scale simulations reveal the superiority of these two tests over other tests based on the empirical independence copula process. They also establish the usefulness of the lesser known Pearson type III approximation to the exact permutation distribution. This approximation yields tests with more accurate type I error rates than the gamma approximation usually used for HSIC, especially when dimensions of the two vectors are large. A new method for scale parameter selection in HSIC tests is proposed which improves power performance in three simulations, two of which are from machine learning. The problem of testing mutual independence between many random vectors is addressed. The closely related problem of testing serial independence of a multivariate stationary sequence is also considered. The Möbius transformation of characteristic functions is used to characterize independence. A generalization to p vectors of the alpha -distance covariance test and the Hilbert-Schmidt independence criterion (HSIC) test with the characteristic kernel of a stable probability distributionof index alpha is obtained. It is shown that an HSIC test with sufficiently small scale parameters is equivalent to an alpha -distance covariance test. Weak convergence of the HSIC test is established. A very fast and accurate computation of p-values uses the Pearson type III approximation which successfully approaches the exact permutation distribution of the tests. This approximation relies on the exact first three moments of the permutation distribution of any test which can be expressed as the sum of all elements of a componentwise product of p doubly-centered matrices. The alpha -distance covariance test and the HSIC test are both of this form. A new selection method is proposed for the scale parameter of the characteristic kernel of the HSIC test. It is shown in a simulation that this adaptive HSIC test has higher power than the alpha-distance covariance test when data are generated from a Student copula. Applications are given to environmental and financial data.