We can add noise to the Kuramoto model by modifying it as follows;
Interestingly, one can still determine the value of Kc in a system exhibiting noise. Stability analysis techniques make it possible to locate the transition from stability to instability, and hence, calculate the value of Kc [4].
Assuming we use noise chosen from a normal Gaussian distribution of mean zero and width β / Δt Where β is the strength of the noise and Δt is the size of the time-steps used in the simulation, then we can calculate the critical value as: Kc = β2 + 2γ (where γ is defined from the Lorentzian distribution governing the natural frequencies) [4].
Where ηi(t) is an independent and randomly generated value, chosen from a Gaussian distribution of noise, to be applied as a perturbation to oscillator i at time t. The other variables are defined as for the standard Kuramoto model.Adding noise to the Kuramoto model makes it more difficult for the oscillators to synchronise near the critical coupling value, effectively increasing the 'guaranteed' critical value Kc of the system in a noiseless environment [3]. In a synchronised system, the net effect of noise is to alter the synchronised frequency of the oscillators.
Interestingly, one can still determine the value of Kc in a system exhibiting noise. Stability analysis techniques make it possible to locate the transition from stability to instability, and hence, calculate the value of Kc [4].
Assuming we use noise chosen from a normal Gaussian distribution of mean zero and width β / Δt Where β is the strength of the noise and Δt is the size of the time-steps used in the simulation, then we can calculate the critical value as: Kc = β2 + 2γ (where γ is defined from the Lorentzian distribution governing the natural frequencies) [4].
