Synchronization of phase oscillators with heterogeneous coupling: A solvable case (Q926802)

Subject of the paper is the following system of globally coupled phase oscillators \[ \phi_i'=\omega_i + \frac{1}{N} \sum_{j=1}^{N} k_i q_j \sin (\phi_j -\phi_i),\quad i=1,\dots,N, \] where \(k_i>0\) and \(q_j\geq 0\) determine the weighths of the interaction for each oscillator pair, \(\phi_i(t)\) and \(\omega_i\in \mathbb{R}\) are oscillator's phase and instantaneous frequency. The authors extend the Kuramoto theory for the transition to synchronization, which holds for the uniform coupling case, i.e. \(k_i=\mathrm{const}\) and \(q_j=\mathrm{const}\). Here, \(k_i\) can be interpreted as the coupling of the oscillator \(i\) with the mean field and \(q_i\) weights the contribution of the same oscillators to the mean field. The effect of correlations between \(k_i\) and \(q_i\) is investigated. In particular, it is shown that the synchronization can be completely inhibited when they are strongly anticorrelated.