Relaxation dynamics of Kuramoto model with heterogeneous coupling**Project supported by the National Natural Science Foundation of China (Grant Nos. 11905068, 11847013, 11175150, and 11605055), Postgraduate Research and Practice Innovation Project for Graduate Students of JiangSu Province, China (Grant No. KYCX18-2100), and the Scientific Research Funds of Huaqiao University, China (Grant No. 605-50Y17064).

Abstract

The Landau damping which reveals the characteristic of relaxation dynamics for an equilibrium state is a universal concept in the area of complex system. In this paper, we study the Landau damping in the phase oscillator system by considering two types of coupling heterogeneity in the Kuramoto model. We show that the critical coupling strength for phase transition, which can be obtained analytically through the balanced integral equation, has the same formula for both cases. The Landau damping effects are further explained in the framework of Laplace transform, where the order parameters decay to zero in the long time limit.