Fig. 1: From discontinuous to extreme synchronization transitions.
Panels show classical Kuramoto order parameter (1) as a function of coupling strength. a Continuous synchronization phase transition in the Kuramoto model with unimodal natural frequency distribution. b Discontinuous synchronization phase transition in the Kuramoto model with bimodal natural frequency distribution. Phase transitions with defined transition point Kc in (a) and (b) emerge only in the thermodynamic limit N → ∞. c Recently experimentally observed discontinuous synchronization in a finite (N = 200) system of photo-chemical Belousov-Zabotinsky reactions (inset), modeled via FitzHugh-Nagumo fast-slow oscillators (main panel), data reproduced from18. d Extreme synchronization transitions in finite-N systems of complexified Kuramoto units, visible already for N = 8. The inset displays r vs. system size N just above the critical coupling at K = 1.05Kc, in log-log scales with red dots for panel (a), blue dots for panel (b), and purple dots for panel (d). See Supplementary Information for details of the parameter settings.
