Estimate basin boundaries of resonant solitary states

Determine the basin boundaries of resonant solitary states in the Kuramoto model with inertia on complex networks, where a single solitary oscillator rotates at mean frequency ω_s distinct from the synchronized cluster’s mean frequency ω_sync and exchanges average power p_s with the cluster. In particular, ascertain practical estimation methods for basin boundaries in the frequency direction based on the turning points of the self-consistency function Z(ω) around a stable solution.

Background

The paper introduces a self-consistent framework to predict solitary states in the Kuramoto model with inertia on complex networks, deriving an explicit relation Z(ω_s)=P_z−α_z ω_s−p_s(ω_s) that links a solitary node’s mean frequency to the average power flow to the synchronized cluster via resonant excitation of network modes. While this framework captures existence and stability proxies of solitary states, the authors highlight the need to understand the basins of attraction of these states, which govern robustness to perturbations.

They note that basin boundaries depend on phase and frequency and suggest that (after averaging phase dependence) turning points of Z(ω) around a stable solution may provide an estimate of basin boundaries in the ω-direction. Establishing reliable methods to compute or approximate these boundaries would deepen insights into the resilience of synchronous operation under localized disturbances.