Synchronisation under shocks: The Levy Kuramoto model

Abstract

We study the Kuramoto model of identical oscillators on Erclos-Renyi (ER) and Barabasi-Alberts (BA) scale free networks examining the dynamics when perturbed by a Levy noise. Levy noise exhibits heavier tails than Gaussian while allowing for their tempering in a controlled manner. This allows us to understand how 'shocks' influence individual oscillator and collective system behaviour of a paradigmatic complex system. Skewed alpha-stable Levy noise, equivalent to fractional diffusion perturbations, are considered, but overlaid by exponential tempering of rate lambda. In an earlier paper we found that synchrony takes a variety of forms for identical Kuramoto oscillators subject to stable Levy noise, not seen for the Gaussian case, and changing with alpha: a noise-induced drift, a smooth alpha dependence of the point of cross-over of synchronisation point of ER and BA networks, and a severe loss of synchronisation at low values of a. In the presence of tempering we observe both analytically and numerically a dramatic change to the alpha < 1 behaviour where synchronisation is sustained over a larger range of values of the 'noise strength' alpha, improved compared to the alpha > 1 tempered cases. Analytically we study the system close to the phase synchronised fixed point and solve the tempered fractional Fokker-Planck equation. There we observe that densities show stronger support in the basin of attraction at low a for fixed coupling, alpha and tempering lambda. We then perform numerical simulations for networks of size N = 1000 and average degree (d) over bar = 10. There, we compute the order parameter r as a function of sigma for fixed alpha and lambda. and observe values of r approximate to 1 over larger ranges of a for alpha < 1 and lambda not equal 0. In addition we observe drift of both positive and negative slopes for different alpha and lambda when native frequencies are equal, and confirm a sustainment of synchronisation down to low values of a. We propose a mechanism for this in terms of the basic shape of the tempered stable Levy densities for various alpha and how it feeds into Kuramoto oscillator dynamics and illustrate this with examples of specific paths. Show more