Controlling and enhancing synchronization through adaptive phase lags

We compare two methods for controlling synchronization in the Kuramoto model on an undirected network. The first is by driving selected oscillators at a desired frequency by coupling to an external driver, and the second is by including adaptive lags-or dynamical frustrations-within the Kuramoto interactions, with the lags evolving according to a dynamics as a function of the reference frequency with an associated time constant. Performing numerical simulations with random regular graphs, we find that above a certain connectivity driving via adaptive lags allows for stronger alignment to the external frequency at a lower value of the time constant compared to the corresponding coupling strength for the externally driven model. Numerical results are supported by equilibrium analysis based on a fixed-point ansatz for frequency synchronized clusters where we solve the spectrum of the associated Jacobian matrix. We find that at low connectivity the external driving mechanism is successful down to lower densities of controlled oscillators where the adaptive lag approach is Lyapunov unstable at all densities. As connectivity increases, however, the adaptive lag mechanism shows stability over similar ranges of density to the external driving and proves superior in terms of tighter splays of oscillators. In particular, the threshold for instability for the adaptive lag model shows robustness against variations in the associated time constant down to lower densities of controlled oscillators. A simple intuitive model emerges based on the interaction between splayed clusters close to a critical point.