Operational tracking loss in nonautonomous second-order oscillator networks

TL;DR

This paper investigates how coupled second-order oscillators lose their ability to follow time-dependent inputs, highlighting the importance of frequency dynamics and graph structure in understanding this transition.

Contribution

It introduces a frequency-based tracking ratio to diagnose loss of coherence and analyzes how graph topology influences this phenomenon, extending understanding beyond phase-based observables.

Findings

Frequency dynamics better diagnose tracking loss than phase observables.

Graph topology, especially Fiedler-mode localization, significantly affects freeze-out time.

Frequency sector analysis clarifies limits of spectral reductions in 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 $E_\omega(t)$ 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 $\lambda_2$, 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.