Fig. 4: Extreme synchronization emerges via Hopf bifurcation.

Panels (a–c) display the eigenvalues (open circles) of the Jacobian matrix evaluated at the locked state z^*. The pair of eigenvalues relevant to the bifurcation is highlighted by filled red disks. It crosses the imaginary axis with increasing ∣K∣, indicating a Hopf bifurcation. For panel (b), we choose ∣K∣ = 0.47, close to but slightly above the critical coupling strength. d–f show the order parameter as a function of time after a transient period, t₀ = 3000, with the system state initiated by a random perturbation of order 10⁻¹ away from each locked state evaluated in (a–c), respectively. Additional oscillations visible in (e) and (f) are transient phenomena due to small negative real parts of eigenvalues. All panels for \(\alpha=\frac{\pi }{2}-0.01\), i.e., β = 0.01 and N = 128.