Synchronization in the two networks-frustrated coupled oscillators with a noisy attractive-repulsive frequencies

We investigate the correlation effects of the two networks combined with an attractive-repulsive frequency altered by noise on the mutual synchronization of the two coupled Kuramoto models with parametric random phase-shift properties. The necessity and significance of collective behavior between ensembles of interacting oscillators and their rich phenomenology offer an idealization of several disciplines in which mutual synchronization competes with force. In this paper, we derive the phase-locked states and identify the significant synchronization transition points analytically with exact boundary conditions for the correlated and uncorrelated joint distributions, their stability, and bifurcation diagrams. We find that a perfect and imperfect supercritical to subcritical Hopf bifurcation transition occurs depending on the synchronic transition points for the correlated cases, characterized by the power scales and the largest eigenvalues of the networks. Moreover, we show the powerful interplay of force, noise, frustration, and network on the synchronization transitions of the two populations and their compromise between the correlated and uncorrelated joint probability distributions. The intensity and transmissibility of noise, in particular, vary within and between populations.