Figure 3

Theoretical stability plots for large N-dim oscillator system. A. Plots of error function \(\mathcal{R}(\varOmega , \delta ^{2})\) with varying fixed \(\delta > 0\) over \(\varOmega \in [\omega _{0}-g, \omega _{0}+g]\). The function is truncated between interval \([-0.5, 0.5]\) for visibility. There is a unique root \(\mathcal{R}(\varOmega , \delta ^{2}) = 0\) for each fixed \(\delta > 0\). B. Colour map of \(\operatorname {sgn}E(\varOmega , \delta ^{2})\) over states \((\varOmega , \delta ^{2}) \in [\omega _{0}-g, \omega _{0}+g] \times (0, 0.5^{2})\), along with the implicit solution curve (purple) \(\varOmega = \varOmega (\delta )\) parametrizing level set \(\mathcal{R}(\varOmega , \delta ^{2}) = 0\). Stable regions correspond to \(\operatorname {sgn}E(\varOmega , \delta ^{2}) = -1\) (blue) and unstable regions correspond to \(\operatorname {sgn}E(\varOmega , \delta ^{2}) = 1\) (red). The network synchronizes near a state \((\varOmega (\delta ), \delta ^{2})\) overlapping the stable region. C. Plot of stability term \(\text{s.} \log E(\varOmega , \delta ^{2})\) along the solution curve \(\varOmega = \varOmega (\delta )\) over \(\delta \in (0, 0.5)\). There is a small interval \(\delta \in (0.08, 0.1)\) for which \((\varOmega (\delta ), \delta ^{2})\) is in the stable region (blue). Other states are in the unstable region (red). D. Complex plot of non-zero eigenvalues of \(P(\lambda \mid \varOmega , \delta ^{2}) + Q(\lambda \mid \varOmega , \delta ^{2})\) on solution states \((\varOmega (\delta ^{2}), \delta ^{2})\) across varying \(\delta > 0\), scaled by s.log for visibility. The eigenvalues in plot D were computed at respective states \((\varOmega , \delta ^{2})\) in plot C indicated by the same colour. Power terms for polynomial \(Q(\lambda \mid \varOmega , \delta ^{2})\) were computed up to degree \(M = 3\). The parameters used for all plots are \(\alpha _{\tau }= 1.0\), \(g = 1.5\), \(\omega _{0} = 1.0\), \(\kappa = 80\), and \(\tau ^{0} = 0.1~\text{s}\)