Synchronization by Nonlinear Frequency Pulling

We analyze a model for the synchronization of nonlinear oscillators due to reactive coupling and nonlinear frequency pulling motivated by the physics of arrays of nanoscale oscillators. We study the model for the mean field case of all-to-all coupling, deriving results for the onset of synchronization as the coupling or nonlinearity increase, and the fully locked state when all the oscillators evolve with the same frequency.

Figure 1

Results for a triangular distribution of full width $w=2$. Panels (a)–(c) show the order parameter $R$ found in numerical simulations of 1000 oscillators for sweeps increasing and decreasing $\beta$ and for three representative values of $\alpha$. The symbols are time-averaged values and the error bars are the standard deviations in $R$ over the averaging time. Panel (d) shows the phase diagram deduced from sweeps at many values of $\alpha$ and numerical calculations of the linear instabilities: solid line, linear instability of the unsynchronized state; short-dashed line, saddle-node line deduced from jumps of $R$ in the numerical simulations [denoted by arrows in panels (a)–(c)]; long dashed line, linear instability of the fully locked solution (the large $R$ solution is fully locked to the right of this line).