Synchronization in kuramoto oscillators under single external oscillator

Abstract

In this chapter we study the influence of a single strongly attractively coupled external oscillator on a system of coupled Kuramoto oscillators. First we go through the original method used by Kuramoto to solve this system of coupled oscillators. Then we use a later approach developed by Ott and Antonsen. We will use this approach first to solve the original system and show that the results match. Next we will solve a variations of the this system using Ott-Antonsen method, after which we will use it to solve our particular system. We consider a variation of the Kuramoto system which shows multiple regions of synchronization. First we observe the effects of attractive and repulsive couplings. Next we qualitatively study the effect of the initial frequency distribution of the internal oscillators, both the mean and the standard deviation of different distributions like the Gaussian and Lorentzian distributions, on these synchronization regions.