Efeitos de flutuações devidas a população finita na sincronização de osciladores globalmente acoplados

Abstract

In this work, we present a study on the influence of the finite number of oscillators on the synchronization of two models of globally coupled oscillators. The first is the Kuramoto's model with discrete phases and the second is the Yu's model, whose main characteristic is population growth. In both cases, we start from the microscopic dynamics of the number of oscillators in a given state and obtain the respective Langevin equations for the oscillator densities. For the discretized Kuramoto's model, we evaluated the temporal evolution of densities and compared our results with the continuous case, already studied in the literature in the mean field approach. Then, we used the finite size scale theory to obtain the critical exponents of the model, where we obtained values close to the universality class of an opinion dynamics model. In the study of Yu's model, which has three states, our analysis was restricted to the case in which the model has bistability. We studied, via non-linear dynamics, the bifurcation diagram, which indicated a subcritical Hopf bifurcation, that is, the model has two attractors: a fixed point and a limit cycle. We found that, unlike what occurs in models with a fixed population, bistability persists in the Yu's model and it has hysteresis. Thus, we realize that fluctuations do not break bistability in systems with a finite and growing population.