Figure 2

Theoretical stability plots for two-oscillator system. A. Plot of error functions \(R_{\kappa }(\varOmega )\) with varying fixed gain \(\kappa = 0\) (magenta), \(\kappa = 20\) (yellow), \(\kappa = 30\) (blue). All roots \(\varOmega \in [\omega _{0} - g, \omega _{0})\) of \(R_{\kappa }(\varOmega )\) are potential synchronization frequencies for the two-oscillator system. The number of roots Ω for \(\mathcal{R}_{\kappa }(\varOmega ) = 0\) increase with larger κ. B. The plasticity gain is set to \(\kappa = 30\). Plot of the real part of the non-zero branches \(\lambda _{1}(\varOmega )\), \(\lambda _{2}(\varOmega )\) (orange, cyan) of the polynomial root equation \(P_{\varOmega }(\lambda ) + Q_{\varOmega }(\lambda ) = 0\) over \(\varOmega \in [\omega _{0} - g, \omega _{0})\). Ticks on the Ω-axis (blue) indicate the frequencies \(\varOmega _{i}\) solving \(\mathcal{R}_{\kappa }(\varOmega _{i}) = 0\) where the system can synchronize. The plotted branches imply that the oscillators will synchronize at \(\varOmega = \varOmega _{1}, \varOmega _{3}\), and avoid the unstable frequency \(\varOmega = \varOmega _{2}\) with \(\operatorname {Re}\lambda _{1}(\varOmega _{2}) > 0\). C, D. Error heatmaps with \(\varOmega = \varOmega _{1}, \varOmega _{2}\), respectively, approximate the distribution of eigenvalues \(\lambda \in \mathbb {C}\) solving \(P_{\varOmega }(\lambda ) + Q_{\varOmega }(\lambda )e^{-\lambda \tau } = 0\) near \(\lambda = 0\), scaled and normalized for visibility. Spots near zero error (white) suggest potential eigenvalue locations. Markers plot the eigenvalues \(\lambda _{0} = 0, \lambda _{1}(\varOmega ), \lambda _{2}(\varOmega )\) (blue, orange, cyan) for \(\tau = 0\). The heatmap in D indicates an eigenvalue λ near \(\lambda _{1}(\varOmega _{2}) > 0\), which implies instability at \(\varOmega = \varOmega _{2}\). All other eigenvalues λ appear to be distributed either at \(\lambda = 0\) or on the left-side of the imaginary axis. Here, \(\varOmega _{1} = 0.626\) and \(\varOmega _{2} = 0.783\). For all plots, \(\alpha _{\tau }= 0.5\), \(g = 1.5/2\), \(\omega _{0} = 1.0\), \(\kappa = 30\), and \(\tau ^{0} = 0.1~\text{s}\)