Noise and Synchronization in Kuramoto-type Networks

Bachelor thesis (2023)

Authors

J.G.A. Dessing Electrical Engineering, Mathematics and Computer Science

Contributors

J.L.A. Dubbeldam Mathematical Physics - EEMCS (supervisor 1)
A.J.L. Adam ImPhys/Adam group - AS (supervisor 2)
Faculty
Electrical Engineering, Mathematics and Computer Science
Copyright: © 2023 Jostijn Dessing
More Info
expand_more

Published Date

2023

Language

English

Reuse Rights

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons.

Faculty

Electrical Engineering, Mathematics and Computer Science

Abstract

The Kuramoto model (KM) is a well known mathematical model of coupled oscillators that is frequently used to study synchronization phenomena. In this bachelor thesis we investigate the effects of noise on synchronization in Kuramoto-type networks.

In the first part we follow the methods of Maggi and Paoluzzi [1], but include detailed in between steps, to obtain an analytical expression for the critical coupling strength, $k_c$, of the KM in the thermodynamic limit under the influence of time-correlated noise (i.e non-white noise). The coupling strength, $k$, is a parameter in the KM that essentially determines to what extent oscillators influence each other. When $k > k_c$ we start to see synchronization. Our numerical simulations agree with the results found in [1], in that the analytical expression for $k_c$ holds up for low values of correlation time, but quickly breaks down as correlation time increases.

In the second part of this thesis we consider a Kuramoto-type adaptive dynamical network that is also investigated in Fialkowski et al. [2]. The dynamical phenomena that are observed in [2] are also present in our simulations. We explain these dynamical phenomena with the help of a variety of plots. Subsequently, the Kuramoto-type adaptive dynamical network is expanded to include white noise terms in the coupling dynamics. We find, through the use of simulation, that under the same conditions as in [2], synchronization is observed for significantly lower values of coupling strength. This result is explained qualitatively. Simulations also show, that for specific values of noise strength, the
hysteric behaviour observed in [2] is not present.

Other conclusions, like the degree to which the noise can reduce the coupling strength required for full synchronization, or beyond what value of noise strength full synchronization can no longer occur, are unable to be drawn. Additional simulation work and a further analytical work is recommended for an ensuing study.