Osciladores de fase são sistemas dinâmicos dissipativos com aporte externo de energia, e matematicamente caracterizados pela presença de ciclos-limite, ao longo dos quais a dinâmica pode ser descrita por um ângulo de fase. No presente artigo, consideramos fenômenos de sincronização num sistema de muitos osciladores de fase acoplados. Adotamos uma descrição qualitativa, baseada num modelo paradigmático proposto por Y. Kuramoto em 1975. Discutimos, à luz dessa descrição, algumas aplicações em sistemas físicos (metrônomos e pedestres numa ponte pênsil) e biológicos (neurônios e vagalumes piscando).

Phase oscillators are dissipative dynamic systems with external energy input, and mathematically characterized by the presence of limit cycles, along which the dynamics can be described by a phase angle. In the present article, we consider synchronization phenomena in a system of many phase-coupled oscillators. We adopt a qualitative description, based on a paradigmatic model proposed by Y. Kuramoto in 1975. We discuss, in the light of this description, some applications in physical systems (metronomes and pedestrians on a suspension bridge) and biological (neurons and fireflies blinking).

Resumo Os relógios de pêndulo e metrônomos têm em comum o fato de serem osciladores auto-excitados, que exibem oscilações estacionárias, onde um aporte externo de energia compensa as perdas devidas ao amortecimento. No plano de fase, estas oscilações são representadas por trajetórias fechadas chamadas ciclos-limite, e o movimento sobre elas pode ser parametrizado por um ângulo de fase. Neste trabalho investigamos a dinâmica resultante do acoplamento de dois osciladores de fase, usando equações médias para os casos de ressonância. Os resultados dessa análise são utilizados para descrever qualitativamente fenômenos de travamento de fase e sincronização em osciladores de fase, e que explicam resultados observados no acoplamento de relógios de pêndulo e metrônomos.

Abstract Pendulum clocks and metronomes have in common the fact that they are self-excited oscillators, exhibiting stationary oscillations where an external energy input compensates losses due to damping. In the phase plane these oscillations are represented by closed trajectories called limit cycles, and the motion along them can be parameterized by a phase angle. In this work we investigate the dynamics resulting from the coupling between two phase oscillators, using average equations for the resonant cases. The results from this analysis are used to qualitatively describe phase locking and synchronization phenomena in phase oscillators, explaining results observed in the coupling of pendulum clocks and metronomes.

<p>O presente trabalho analisa o modelo Hamiltoniano de ondas de deriva que descreve o transporte caótico de partículas no confinamento do plasma. Com apenas uma onda de deriva o sistema é integrável e apresenta órbitas regulares. Ao adicionarmos uma segunda onda, o sistema pode ou não ser integrável dependendo da velocidade de fase de cada onda. Com a mesma velocidade de fase o sistema ainda é integrável. Quando as velocidades de fase entre as duas ondas são diferentes, o sistema apresenta comportamento caótico. A este modelo acrescentamos uma pequena dissipação. Na presença de uma fraca dissipação, para certas condições iniciais, observamos órbitas transientes que convergem para atratores periódicos.</p>

<p>This paper analyses the Hamiltonian model of drift waves which describes the chaotic transport of particles in the plasma confinement. With one drift wave the system is integrable and it presents stable orbits. When one wave is added the system may or may not be integrable depending on the phase of each wave velocity. If the two waves have the same phase velocity, the system is integrable. When the phase velocities between the two waves are different, the system shows chaotic behaviour. In this model we add a small dissipation. In the presence of a weak dissipation, for different initial conditions, we observe transient orbits which converge to periodic attractors.</p>

Computer simulations of partial differential equations of mathematical physics typically lead to some kind of high-dimensional dynamical system. When there is chaotic behavior we are faced with fundamental dynamical difficulties. We choose as a paradigm of such high-dimensional system a kicked double rotor. This system is investigated for parameter values at which it is strongly non-hyperbolic through a mechanism called unstable dimension variability, through which there are periodic orbits embedded in a chaotic attractor with different numbers of unstable directions. Our numerical investigation is primarily based on the analysis of the finite-time Lyapunov exponents, which gives us useful hints about the onset and evolution of unstable dimension variability for the double rotor map, as a system parameter (the forcing amplitude) is varied.

In this work it is considered the effect of an ergodic magnetic limiter on the plasma confined in the TCABR tokamak. The poloidal distribution of the limiter currents is determined taking into account the toroidal geometry of this tokamak. The plasma equilibrium field is analytically obtained by solving the Grad-Schlüter-Shafranov equation in polar toroidal coordinates. The perturbing limiter field is obtained as a vacuum field by solving the Laplace equation. Supposing the limiter action as a sequence of delta-function pulses, a symplectic map is derived by using a Hamiltonian formulation for the perturbed field line. Examples are given to illustrate the influence of this design on the topology of the perturbed field line structure.

An equation for MHD stationary equilibrium of rotating plasmas in the azimuthal direction is derived in the case of an orthogonal curvilinear coordinate system. The basic assumptions we made are: (i) there is an ignorable coordinate so that surface quantities are independent of it; (ii) the entropy is a surface quantity.

We derive a system of equations to describe stationary axisymmetric MHD equilibria characterized by toroidal and poloidal flows as well as plasma anisotropy due to strong magnetic field and eventual auxiliary heating methods. The system consists of a nonlinear partial differential equation for the poloidal magnetic flux function and an algebraic Bernoulli-type equation, which relates plasma density with several surface functions. We analyse the ellipticity of the equation and the plasma density bifurcation as the Alfvén-Mach number is changed.