We study the synchronization problem of nonuniform second-order Kuramoto model with homogeneous dampings and frustration effects on an asymmetric network. More precisely, we focus on the second-order model defined on an asymmetric graph and present theories on the complete frequency synchronization. Due to the absence of the gradient flow structure, we develop novel energy functions to control the diameters of phase and frequency respectively, which allows us to construct first-order Gronwall-type inequalities. This eventually gives rise to the exponential convergence to the synchronized state in a regime in terms of large coupling strength, small inertia and frustration.
Zhu et al. (2026) studied this question.