Strong coupling yields abrupt synchronization transitions in coupled oscillators

TL;DR

This work addresses how strong coupling in delayed-coupled oscillators alters synchronization transitions beyond traditional phase reduction. By analyzing higher-harmonic phase interactions, conducting experiments with electrochemical oscillators, and performing numerical bifurcation analysis of delay-differential equations, it shows that in-phase and anti-phase bifurcations can exhibit distinct criticalities, leading to bistability with out-of-phase states. The results emphasize that amplitude dynamics can shape phase-locked patterns in ways a purely phase-based model cannot capture, underscoring the limitations of reduced-phase descriptions at strong coupling. The findings have broad implications for understanding and engineering synchronization in real-world networks where delays and non-sinusoidal oscillations are prevalent.

Abstract

Coupled oscillator networks often display transitions between qualitatively different phase-locked solutions -- such as synchrony and rotating wave solutions -- following perturbation or parameter variation. In the limit of weak coupling, these transitions can be understood in terms of commonly studied phase approximations. As the coupling strength increases, however, predicting the location and criticality of transition, whether continuous or discontinuous, from the phase dynamics may depend on the order of the phase approximation -- or a phase description of the network dynamics that neglects amplitudes may become impossible altogether. Here we analyze synchronization transitions and their criticality systematically for varying coupling strength in theory and experiments with coupled electrochemical oscillators. First, we analyze bifurcations analysis of synchrony and splay states in an abstract phase model and discuss conditions under which synchronization transitions with different criticalities are possible. Second, we illustrate that transitions with different criticality indeed occur in experimental systems. Third, we highlight that the amplitude dynamics observed in the experiments can be captured in a numerical bifurcation analysis of delay-coupled oscillators. Our results showcase that reduced order phase models may miss important features that one would expect in the dynamics of the full system.

Figures (6)

• Figure 1: Varying parameters demonstrates regions in which the bifurcations of equilibria $\psi=0,\pi$ have distinct criticality. Results are demonstrated for fixed $\gamma_2 = 0.2$, $\gamma_3 = 0.5$. a) Criticality coefficient \ref{['eq:CritZero']} for $\psi=0$ as the strength $r,s$ of the second and third harmonic is varied; blue indicates a continuous, red a discontinuous transition; a hollow circle indicates $r=s=0$ and a filled circle the parameter values in panel d). b) Criticality coefficient \ref{['eq:CritPi']} for $\psi=\pi$ as in panel a). c) Regions in which the bifurcations of the in-phase and anti-phase configurations have distinct criticalities: white indicates that the transition at $\psi=0$ is continuous while $\psi=\pi$ is discontinuous and black vice versa. Bifurcations of $\psi=0$ and $\psi=\pi$ have the same criticality in the grey regions. d) Pseudocontinuation plot for $r=-s=0.12$ as the parameter $\alpha$ is increased (green) or decreased (black); an approximate region of hysteresis/multistability due to the discontinuous transition of the $\psi=\pi$ solution is shaded in grey.
• Figure 2: Illustration of the time-delayed linear feedback experiment with a time series of the uncoupled system. a) Schematic of the experimental setup. CE: Counter electrode, RE: Reference electrode, and WE: Working electrodes. b) Diagram of the delay feedback schematic of the electrochemical experiment. The currents ($i_1$, $i_2$) of each nickel wires were measured and added to obtain a total current ($i_T$). The $i_T$ was fed back with a coupling strength ($K$), a delay, ($\tau$) and applied to the circuit potential ($V(t)$). c) Time series of the currents for the uncoupled ($K=0$) oscillators and without delay ($\tau = 0$). The blue and red lines correspond to oscillator 1 and 2, respectively.
• Figure 3: Scan under variation of the time delay $\tau$ at weak coupling strength ($K=-0.12V\per mA$) of the two electrode system. a) Time series of the current for the in-phase behavior for $\tau=0.081s$ (0.036 $\tau/T$). b) Time series of the current for the out-of-phase behavior for $\tau=0.60s$ (0.27 $\tau/T$). c) Time series of the current for the anti-phase behavior for $\tau=0.89s$ (0.39 $\tau/T$). In panels a)-c), the blue and red lines correspond to oscillators 1 and 2, respectively. d) Phase difference of the coupled oscillators as a function of the time delay. The green line is the phase difference corresponds to the forward scan ($\tau$ = 0 $\to$ 0.5 $\tau/T$), and the grey line to the backward scan ($\tau$ = 0.5 $\tau/T$$\to$ 0) with the direction indicated by the green and grey arrows, respectively.
• Figure 4: Phase difference under slow variation of the time delay close to the anti-phase solution for increasing coupling strengths. a) $K=-0.12V\per mA$, b) $K=-0.18V\per mA$, c) $K=-0.25V\per mA$ and d) $K=-0.50V\per mA$. The green line corresponds to the phase difference in the forward (green arrow) scan and the grey line to the backward (grey arrow) scan.
• Figure 5: Time series and phase difference of the currents in the bistability region for strong feedback, $K=-0.50V\per mA$, with time-delay $\tau=0.70s$ (0.31 $\tau/T$). The panels a) and c) corresponds to the out-of-phase configuration, while panels b) and d) correspond to the anti-phase configuration. The blue and red lines correspond to oscillator 1 and 2, respectively.