Phase transition to synchronization of coupled oscillators in the Kuramoto model

Synchronization is a widespread phenomenon in nature and in our daily life. In this thesis we study the Kuramoto model which deals with the phenomenon of synchronization in the case of a system of coupled limit-cycle oscillators. We first treat the model in the absence of noise, then we take into account the relevance of noise. In order to make the treatment less difficult we make some assumptions and we apply the mean-field theory. We consider the model with a finite number of oscillators and a continuum model found in the limit of an infinite population. We find out that synchronization between oscillators depends on the constant K, which appears in the governing equations and which sizes the coupling strength between oscillators. In fact, in both cases, with and without noise, a phase transition occurs: for small coupling, the system is incoherent, with the oscillators which run independently, but when the coupling exceeds a certain threshold, the system spontaneously synchronizes. We show that transition, in the continuum model, takes the form of a bifurcation of a discrete eiegenvalue from a continuous spectrum. Studying the stability property of the incoherent state in the case of no noise, we discover that it is unstable above threshold but neutrally stable below threshold. Instead, in the presence of noise, the incoherent state can be also linearly stable below threshold. In this work we explore all of these properties of the Kuramoto model through theoretical analysis and simulations.