Extreme synchronization transitions

Abstract

Across natural and human-made systems, transition points mark sudden changes of order and are thus key to understanding overarching system features. Motivated by recent experimental observations, we here uncover an intriguing class of transitions in coupled oscillators, extreme synchronization transitions, from asynchronous disordered states to synchronous states with almost completely ordered phases. Whereas such a transition appears like discontinuous or explosive phase transitions, it exhibits markedly distinct features. First, the transition occurs already in finite systems of N units and so constitutes an intriguing bifurcation of multi-dimensional systems rather than a genuine phase transition that emerges in the thermodynamic limit N → ∞ only. Second, the synchronization order parameter jumps from moderate values of the order of N^-1/2 to values extremely close to 1, its theoretical maximum, immediately upon crossing a critical coupling strength. We analytically explain the mechanisms underlying such extreme transitions in coupled complexified Kuramoto oscillators. Extreme transitions may similarly occur across other systems of coupled oscillators as well as in certain percolation processes. In applications, their occurrence impacts our ability of ensuring or preventing strong forms of ordering, for instance in biological and engineered systems.

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 K_c 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 from. 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 = 1.05K_c, 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.

Fig. 2. Extreme features of the synchronization transition.

a The gap 1−r of the order parameter r to its maxmimal value 1 for fixed ∣K∣ = 1.5. Direct numerical observations (open circles) agree well with our approximate, second-order prediction (8) asymptotically as β = π/2 − α → 0⁺ (red line) and even for the full range of α ∈ (0, π/2) (inset). b With increasing α, the critical coupling strength decreases and the jumps in order parameter r (color-coded r ∈ [0.8, 1]) become increasingly extreme. The white area indicates an incoherent state with r of the order of N^−1/2. The black solid curve indicates the critical coupling ∣K_c∣. In (a) and (b), N = 80 across all observations.

Fig. 3. Disorder moves to additional variables.

a The order parameter r is depicted as a function of ∣K∣ for both real parts, i.e. the phase-variables (solid disk) and the other variables (the imaginary parts, open circles) for N = 128 and β = 0.01. As the y_ν are unbounded, we define phase-like variables θ_μ by a stereographic projection via $\cos {\theta }_{\mu }:=\frac{1-y_{\mu }^2}{1+y_{\mu }^2}$ and $\sin {\theta }_{\mu }:=\frac{2y_{\mu }}{1+y_{\mu }^2}$ for each μ and evaluate $r=∣\frac{1}{N}{\sum }_{\mu =1}^N{e}^{\mathrm{i}{\theta }_{\mu }}∣$, in analogy to (1). b Complex locked states in the complex plane for N = 80 and ∣K∣ = 3.0 move with increasing α values from curves ① for α = 0 and ② for α = 1.5 to curve ③ for $\alpha =\frac{\pi }{2}$. c Local angles φ of the curves around the origin are depicted as a function of α with gray solid guiding line indicating ∣φ(α)∣ = α as emerges for N = 2 up to corrections $\mathcal{O}\left(\mid K{\mid }^{-1}\right)$.

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.

Fig. 5. Extreme transitions in other systems.

a Systems of relaxation (Van der Pol) oscillators and (b) of phase-amplitude (Stuart-Landau) oscillators with Gaussian natural frequency distribution, both exhibit discontinuous transitions in finite systems with an order parameter close to its maximum immediately past the transition point. Exact governing equations, parameters, and a definition of phases entering the order parameter r in (1) are detailed in the Supplementary Information.