An exploration of the Kuramoto model

Abstract

This work is a bachelor thesis featuring a collection of explorations of the Kuramoto model in the presence of noise in one and two dimensions. The model was explored analytically and numerically. It contains a study of literature and new analytical derivations. An attempt to link the fraction of oscillators to the order parameter resulted in experience through the process of deriving the result, but a plot of the first few terms of this expansion showed it not being close. The assumption on which the expansion was based however was shown to be valid in simulations. More successfully an upper bound on the probability of synchronization was found. This probability was found on a ring consisting of N oscillators, and verified of a ring of 3 oscillators. The results was a bound better than one that was already calculated, though the method of verification used was not ideal. An analysis of the stability of solutions on this 3-ring resulted a visual representation of entrainment and an amount of stable solutions possible, which was 1. This result was in accordance with the more general result following from the investigation of a system of coupled rings with arbitrary sizes of either ring. We then expanded this system to conclude that the stability of the entire system could be split into the stability of either ring and the possibility of having a solution for the linking phase difference. We elaborated on general practices when numerically analyzing SDEs and integrations methods. The Kuramoto model proved to be kind in the sense that the Euler-forward and Milstein scheme coincided. In a more physics-centered approach we studied the Ising and XY model. This prompted literature study on the Kosterlitz Thouless transition, through analysing the spin-wave Hamiltonian and the energy of free vortex of a pair of vortices. We also then explored conservation of the vortex-pair behaviour in a noise including Kuramoto model, which verified a duality in partition functions as described in ref. [1].