Synchronisation under shocks: The Lévy Kuramoto model

We study the Kuramoto model of identical oscillators on Erdős–Rényi (ER) and Barabasi–Alberts (BA) scale free networks examining the dynamics when perturbed by a Lévy noise. Lévy 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 α-stable Lévy noise, equivalent to fractional diffusion perturbations, are considered, but overlaid by exponential tempering of rate λ. In an earlier paper we found that synchrony takes a variety of forms for identical Kuramoto oscillators subject to stable Lévy noise, not seen for the Gaussian case, and changing with α: a noise-induced drift, a smooth α 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 α. In the presence of tempering we observe both analytically and numerically a dramatic change to the α<1 behaviour where synchronisation is sustained over a larger range of values of the ‘noise strength’ σ, improved compared to the α>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 α for fixed coupling, σ and tempering λ. We then perform numerical simulations for networks of size N=1000 and average degree d̄=10. There, we compute the order parameter r as a function of σ for fixed α and λ and observe values of r≈1 over larger ranges of σ for α<1 and λ≠0. In addition we observe drift of both positive and negative slopes for different α and λ when native frequencies are equal, and confirm a sustainment of synchronisation down to low values of α. We propose a mechanism for this in terms of the basic shape of the tempered stable Lévy densities for various α and how it feeds into Kuramoto oscillator dynamics and illustrate this with examples of specific paths.