Figure 4

Numerical plots for two-oscillator system. For plots A, B, C, two trials with different initial conditions are graphed. Trial 1, 2 (purple, orange) starts with initial frequency and phase difference \((\varOmega _{0}, \Delta _{0}) = (0.473, 0.402), (0.727, 0.860)\), respectively. A. Plots of derivatives \(\theta _{1}'(t)\), \(\theta _{2}'(t)\) over time. For each trial, \(\theta _{1}'(t)\) (dashed) and \(\theta _{2}'(t)\) (dotted) converge to a common value Ω̂ asymptotically. Each trial of oscillators entrain to a different frequency, implying the existence of multiple synchronization frequencies. B. Plots of sine phases \(\sin (\widehat {\phi}_{i}(t))\), where \(\widehat {\phi}_{i}(t) = \theta _{i}(t) - \widehat {\varOmega}t\). For both trials, the oscillators asymptotically phase-lock with \(\widehat {\phi}_{i}(t) \rightarrow \widehat {\phi}_{i}\), \(i=1\) (dotted), \(i=2\) (dashed). The phase-lock difference \(\widehat {\Delta}_{12} = \widehat {\phi}_{2} - \widehat {\phi}_{1}\) is also different for the two trials. C. Plots of adaptive delays \(\tau _{12}(t)\) (dashed), \(\tau _{21}(t)\) (dotted) over time. For each trial, delay \(\tau _{12}(t)\) converges to some positive equilibrium \(\tau ^{E}\), and delay \(\tau _{21}(t)\) decays to 0. D. Plots showing where two-oscillators with randomized initial conditions \((\varOmega _{0}, \Delta _{0})\) (orange) synchronize towards in terms of asymptotic frequency and phase-offset \((\widehat {\varOmega}, \widehat {\Delta}_{12})\) (magenta) across 80 trials. Each of the two trials in A, B, C have their initial condition plotted as a diamond marker of matching colour. Theoretical synchronization states \((\varOmega , \Delta _{12})\) given by Eqs. (20) and (22) are also plotted (blue). Trials converge to two states \((\widehat {\varOmega}, \widehat {\Delta}_{12}) = (0.625, 0.522), (0.916, 0.111)\), which align with the two theoretically stable states shown in Fig. 2. The parameters used for all plots are \(\alpha _{\tau }= 0.5\), \(\varepsilon = 0.01\), \(g = 1.5/2\), \(\kappa = 30\), \(\omega _{0} = 1.0\), \(\tau ^{0} = 0.1~\text{s}\)