Generalized coupling in the Kuramoto model

We propose a modification of the Kuramoto model to account for the effective change in the coupling constant among the oscillators, as suggested by some experiments on Josephson junction, laser arrays, and mechanical systems, where the active elements are turned on one by one. The resulting model is analytically tractable and predicts that both first and second order phase transitions are possible, depending upon the value of a new parameter that tunes the coupling among the oscillators. Numerical simulations of the model are in accordance with the analytical estimates, and in qualitative agreement with the behavior of Josephson junctions coupled via a cavity.

Figure 1

Behavior of the order parameter for a Josephson Junction array coupled to a cavity obtained by increasing (crosses, solid line) and decreasing (circles, dashed line) the number of active junctions. The arrows denote the direction of the transition. Parameters of the simulations are: $B=0.6$, $\beta =10$, ${\beta }_L=5000$, $Q=100$, $\Omega =⟨{\omega }_i⟩$. The disorder is set to $\gamma =0.05$.

Figure 2

Numerical and analytical results of Eq. (3) for various feedback strengths: $z=0.7$ (a), $z=2$ (b), $z=3$ (c). The solid lines represent the analytic prediction, the symbols represent numerical data obtained by increasing (filled circles) and decreasing (open circles) the coupling. The disorder is set to $\gamma =0.05$.

Figure 3

Minimum value for the coupling parameter $K_c$ (dashed line, left axis) for the onset of synchronization and the corresponding value of the order parameter $r_c$ (solid line, right axis). Circles refer to the numerically determined critical coupling obtained by increasing (filled symbols) and decreasing (open circles) the coupling. Crosses refer to an estimate of the order parameter at the onset of synchronization. The disorder is set to $\gamma =0.05$.