The subject of the paper is the system of coupled phase oscillators \[ \theta_i' = g_i(\theta_1,\dots,\theta_n), \quad 1\leq i \leq n, \] where \(\theta_i\) is the (scalar) phase of the \(i\)th oscillator. The functions \(g_i\) are \(2\pi\)-periodic with respect to all variables and have the following form \[ g_i(\theta_1,\dots,\theta_n) = S \biggl(\theta_i,\sum_{j=1}^n c_{ij}f(\theta_i,\theta_j)\biggr). \] The author obtains conditions for existence and stability of synchronized oscillations, i.e., such solutions that \(\theta_i(t)=\overline \theta(t)\) for all \(i\) and \[ \overline {\theta}(t+T) = \overline {\theta}(t) + 2\pi \tag{1} \] for some \(T\) and all \(t\).
The methods of analysis are similar to the “master stability function” approach by L. M. Pecora and T. L. Carroll [Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80, 2109–2112 (1998)], which was applied to the general case of coupled oscillators. Nevertheless, the specific context of this paper, i.e., that \(\theta_i\) are \(2\pi\)-periodic and (1) is fulfilled, allows one to obtain rigorous results in a closed form for the stability and existence of synchronized solutions.
A study of specific network configurations is given as an example: two- and three-oscillator network, cyclically coupled networks, and two populations of coupled oscillators.