Analytic relation between coupling phases and locking bandwidth in large oscillator networks

Derive an analytic relation that characterizes how the coupling phase parameters β_{k,j} determine the size of the frequency-locking (synchronization) bandwidth in networks with an arbitrary number K of nonlinear oscillators governed by the Slavin–Tiberkevich universal auto‑oscillator model, where pairwise coupling strengths Ω_{k,j} are random. The relation should generalize beyond special two‑oscillator cases and provide explicit dependence of the locking bandwidth on β_{k,j} for large, randomly coupled arrays.

Background

In the universal nonlinear oscillator model, the effective phase dynamics includes coupling phases β_{k,j}, which are known to influence synchronization bandwidth in two-oscillator settings. For two nearly identical oscillators, appropriate β choices can compensate nonlinear frequency shifts and maximize synchronization, but these insights do not straightforwardly extend to large networks.

The paper notes the absence of a known analytic relation linking β{k,j} to the locking bandwidth in networks with an arbitrary number of oscillators and random couplings Ω{k,j}. Establishing such a relation would guide the design of coupling phases to optimize synchronization in practical large‑scale oscillator arrays, impacting hardware implementations and training stability under Equilibrium Propagation.