Figure 5

Numerical plots of N-oscillator system. A. Plots of derivatives \(\theta _{i}'(t)\) over time. We see that all \(\theta _{i}'(t)\) converge to a common frequency, estimated to be \(\widehat {\varOmega} = 0.839\). B. Plots of sine phases \(\sin (\widehat {\phi}_{i}(t))\), where \(\widehat {\phi}_{i}(t) = \theta _{i}(t) - \widehat {\varOmega}t\). All oscillators appear to asymptotically phase-lock to one another. C. Plots of a sample of 50 adaptive delays \(\tau _{ij}(t)\) over time. Some delays \(\tau _{ij}(t)\) converge to some positive equilibrium \(\tau _{ij}^{E}\), while others decay to 0. D. Density of centralized phase-offsets \(\widehat {\Delta}_{i} = \widehat {\phi}_{i} - \overline{\phi }\), which was assumed to be Gaussian. The Gaussian curve \(N(0, \widehat {\delta}^{2})\) (black line) is fit over the density. E. Plot showing oscillators with randomized initial frequency and phase deviation \((\varOmega _{0}, \delta _{0})\) (yellow) synchronizing towards respective estimated asymptotic frequency and phase deviation \((\widehat {\varOmega}, \widehat {\delta})\) (magenta star) across 10 trials. The yellow diamond refers to the trial plotted in A, B, C. The numerical values are plotted directly over Fig. 3(B). As shown, the network synchronizes near the theoretical stable region. The parameters used were \(N = 50\), \(\varepsilon =0.01\), \(\alpha _{\tau }= 0.1\), \(g = 1.5\), \(\kappa =80\), \(\omega _{0} = 1.0\), \(\tau ^{0} = 0.1~\text{s}\)