Operational Emergence of a Global Phase under Time-Dependent Coupling in Oscillator Networks

Abstract

Collective synchronization is often summarized by a complex order parameter $R e^{iΨ}$, implicitly treating the global phase $Ψ$ as a meaningful macroscopic coordinate. Here we ask when $Ψ$ becomes \emph{operationally well-defined} in oscillator networks whose coupling varies in time. We study damped (and optionally inertial) phase-oscillator models on graphs with time-dependent coupling $K(t)$, covering standard Kuramoto dynamics as a limit and including network and spatial topologies relevant to engineered settings. We propose an operational emergence criterion: a macroscopic phase is emergent only when it is robustly estimable, which we quantify via gauge-fixed phase-lag fluctuations under weak noise and finite sampling. This yields a quantitative threshold controlled by $NR^2$ and makes explicit why $Ψ$ is ill-posed in incoherent states even when formally definable. Nonautonomous coupling introduces a ramp timescale that competes with relaxation. Using a Laplacian-mode reduction near coherence, we derive a graph-spectral rate criterion: ordering tracks the protocol when $K(t)λ_2$ dominates the ramp rate, while faster ramps induce freeze-out. Numerically, we extract an operational freeze-out time from an energy-based tracking diagnostic and show that, for non-spatial networks, the residual incoherence at freeze-out collapses when plotted against the spectral protocol parameter $λ_2τ$ across Erdős--Rényi and small-world graph families. Finally, on periodic lattices we show that topological sectors and defect-mediated ordering obstruct complete alignment, leading to protocol-dependent, long-lived partially synchronized states and systematic deviations from spectral collapse.

Figures (9)

• Figure 1: Schematic of the nonautonomous phase-oscillator network \ref{['eq:model']}. Nodes carry phases $\theta_i(t)$ and couple along edges with weights $A_{ij}$ under a time-dependent protocol $K(t)$. A dashed reference node represents optional entrainment to $\phi(t)$ with strength $\eta(t)$.
• Figure 2: Rate-controlled ordering on the main graph. Time evolution of the Kuramoto order-parameter magnitude $R(t)$ for several ramp times $\tau$ under the nonautonomous protocol $K(t)=K_0(1+t/\tau)^{-\alpha}$ (parameters as in Sec. \ref{['sec:numerics']}). Fast ramps suppress coherence, while slower ramps rapidly produce near-complete synchronization.
• Figure 3: Operational stability of the global phase. Lagged, gauge-fixed phase variance $V_\Psi(\Delta t)=\mathrm{Var}[\mathrm{wrap}(\Psi(t_0+\Delta t)-\Psi(t_0))]$ for a representative slow ramp, anchored at $t_0$ (shown in the plot). The smooth growth with $\Delta t$ indicates slow diffusion of the emergent reference phase once coherence is established.
• Figure 4: Phase robustness versus coherence. Lagged phase variance $V_\Psi$ plotted against the coherence proxy $NR^2$ along trajectories (with mild smoothing in time for visualization, as in the code). Phase uncertainty is strongly suppressed as $NR^2$ increases, supporting the operational-emergence criterion.
• Figure 5: Disagreement-energy tracking ratio. The tracking diagnostic $r_E(t)=\left[-\tfrac{1}{2}\partial_t\ln\|\theta-\bar{\theta}\|^2\right]/\left[K(t)\lambda_2\right]$ for a representative ramp time (here $\tau=2$), with horizontal reference thresholds $r_{\rm high}$ and $r_{\rm low}$. In the linear tracking regime one expects $r_E(t)\approx 1$; deviations quantify loss of adiabatic tracking and motivate the operational freeze-out definition used to extract $t_\ast$.