Operational tracking loss in nonautonomous second-order oscillator networks

Abstract

We study when a network of coupled oscillators with inertia ceases to follow a time-dependent driving protocol coherently, using a simplified graph-based model motivated by inverter-dominated energy systems. We show that this loss of tracking is diagnosed most clearly in the frequency dynamics, rather than in phase-based observables. Concretely, a tracking ratio built from the frequency-disagreement observable $E_ω(t)$ and normalized by the instantaneous second-order modal decay rate yields a robust protocol-dependent freeze-out time whose relative dispersion decreases with system size. Graph topology matters substantially: the resulting freeze-out time is only partly captured by the algebraic connectivity $λ_2$, while additional structural descriptors, particularly Fiedler-mode localization and low-spectrum structure, improve the explanation of graph-to-graph variation. By contrast, phase-sector observables develop strong non-monotonic and underdamped structure, so simple diagonal low-mode relaxation closures are not quantitatively reliable in the same regime. These results identify the frequency sector as the natural operational sector for nonautonomous tracking loss in second-order oscillator networks and clarify both the usefulness and the limits of reduced spectral descriptions in this setting.

Figures (8)

• Figure 1: Measured freeze-out time extracted from the frequency-disagreement observable $E_\omega$ as a function of the protocol timescale $\tau$, using the second-order normalized tracking ratio \ref{['eq:romega']}. Error bars show the standard error over realizations. The monotonic increase with $\tau$ demonstrates that the frequency-based diagnostic is sensitive to the rate of nonautonomous driving.
• Figure 2: Frequency-based tracking ratio $r_\omega(t)$ for representative fast and slow protocols. The dashed horizontal lines indicate the thresholds used in the operational extraction of $t_*$, and the vertical line marks the ensemble-mean freeze-out time. The slow protocol delays the loss of coherent tracking substantially relative to the fast one.
• Figure 3: Comparison of operational freeze-out times obtained from different observables. The frequency-disagreement observable $E_\omega$ produces the clearest and most monotonic protocol dependence. The combined observable $E_\theta+E_\omega$ shows a similar but weaker trend, while the phase-sector observable by itself is less informative in the second-order regime.
• Figure 4: Size scaling of the frequency-based freeze-out time for WS graphs. The operational crossover remains well defined across system sizes and retains a clear dependence on the protocol timescale.
• Figure 5: Coefficient of variation of the frequency-based freeze-out time as a function of system size for WS graphs. The decrease with $N$ indicates that the diagnostic becomes sharper and less noisy in larger networks. In the fast-ramp case, the initial spike reflects a launch-transient effect of the logarithmic derivative and does not by itself define the operational freeze-out event.