On the exponential synchronization for the asymmetric second-order Kuramoto model
Tingting Zhu, Xiongtao Zhang
TL;DR
The paper addresses exponential synchronization for a nonuniform second-order Kuramoto ensemble with inertia and frustration on an asymmetric depth-two network. It introduces two energy functionals that bound phase and frequency diameters, then derives Gronwall-type inequalities to prove phase cohesiveness and exponential decay of the frequency diameter under large coupling and small inertia/frustration. The key contributions are the phase-cohesiveness result via the first energy and the exponential synchronization result via the second energy, providing rigorous conditions under which complete frequency synchronization is achieved. The approach advances understanding of inertial, non-symmetric oscillator networks and offers a framework potentially extensible to other second-order networked systems.
Abstract
In this paper, 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 with depth no greater than two 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.