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 2015-05-15T11:46:06Z (GMT). No. of bitstreams: 1 ArquivoTotal.pdf: 937197 bytes, checksum: 5c6f34c8e250983d1071fcc18f9f9fb7 (MD5) Previous issue date: 2013-03-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study systems of elliptic equations of gradient and hamiltonean types by variational methods whose domains is the whole RN. More specifically, we use critical point theorems of the mountain pass and linking types to prove results of existence of non-trivial solutions to these problems. / Neste trabalho estudamos sistemas de equações elípticas dos tipos gradiente e hamiltoniano via técnicas variacionais em domínios não limitados. Mais especificamente, utilizamos teoremas de ponto crítico do tipo passo da montanha e linking para provar existência de solução não trivial para estes problemas.

Neste trabalho, estudamos uma generalização dos semigrupos gradientes, os semigrupos gradiente-like, algumas de suas propriedades e a sua invariância por pequenas perturbações; isto é, pequenas perturbações de sistemas gradiente-like continuam sendo gradiente-like. Como consequência da caracterização dos atratores para este tipo de sistema, estudamos a atração exponencial de atratores. Por fim, estudamos o concetio de dimensão de Hausdorff e dimensão fractal de atratores e apresentamos alguns resultados sobre este assunto, e estudamos a construção de uma nova classe de atratores, os atratores exponenciais fractais / In this work, we study a generalization of gradient discrete semigroups, the gradientlike semigroups, some of its properties and its invariance under small perturbations; that is, small perturbations of gradient-like semigroups are still gradient-like semigroups. As a consequence of the characterization of the attractors for this sort of semigroups, we study the exponential attraction of attractors. Finally, we study some concepts of Hausdorff dimension and fractal dimension and present some results about this subject, and we studied the construction of a new class of attractors, the exponential fractal attractors

Neste trabalho, estudamos uma generalização dos semigrupos gradientes, os semigrupos gradiente-like, algumas de suas propriedades e a sua invariância por pequenas perturbações; isto é, pequenas perturbações de sistemas gradiente-like continuam sendo gradiente-like. Como consequência da caracterização dos atratores para este tipo de sistema, estudamos a atração exponencial de atratores. Por fim, estudamos o concetio de dimensão de Hausdorff e dimensão fractal de atratores e apresentamos alguns resultados sobre este assunto, e estudamos a construção de uma nova classe de atratores, os atratores exponenciais fractais / In this work, we study a generalization of gradient discrete semigroups, the gradientlike semigroups, some of its properties and its invariance under small perturbations; that is, small perturbations of gradient-like semigroups are still gradient-like semigroups. As a consequence of the characterization of the attractors for this sort of semigroups, we study the exponential attraction of attractors. Finally, we study some concepts of Hausdorff dimension and fractal dimension and present some results about this subject, and we studied the construction of a new class of attractors, the exponential fractal attractors

The mathematical theory of homogenization deals with the rigorous derivation of effective models from partial differential equations with rapidly-oscillating coefficients. In this thesis we deal with modeling and homogenization of random heterogeneous media. Namely, we obtain stochastic homogenization results for certain evolutionary gradient systems. In particular, we derive continuum effective models from discrete networks consisting of elasto-plastic springs with random coefficients in the setting of evolutionary rate-independent systems. Also, we treat a discrete counterpart of gradient plasticity. The second type of problems that we consider are gradient flows. Specifically, we study continuum gradient flows driven by λ-convex energy functionals. In stochastic homogenization the derived deterministic effective equations are typically hardly-accessible for standard numerical methods. For this reason, we study approximation schemes for the effective equations that we obtain, which are well-suited for numerical analysis. For the sake of a simple treatment of these problems, we introduce a general procedure for stochastic homogenization – the stochastic unfolding method. This method presents a stochastic counterpart of the well-established periodic unfolding procedure which is well-suited for homogenization of media with periodic microstructure. The stochastic unfolding method is convenient for the treatment of equations driven by integral functionals with random integrands. The advantage of this strategy in regard to other methods in homogenization is its simplicity and the elementary analysis that mostly relies on basic functional analysis concepts, which makes it an easily accessible method for a wide audience. In particular, we develop this strategy in the setting that is suited for problems involving discrete-to-continuum transition as well as for equations defined on a continuum physical space. We believe that the stochastic unfolding method may also be useful for problems outside of the scope of this work.