Synchronous Analysis Of A Class Of Second-order Kuramoto Oscillator Systems

Synchronization of complex networks has been one of the important issues worldwide,which is also an important means to study the dynamical behavior of linear and non-linear systems.The main concern in the study of the synchronization phenomenon in the real world is to establish mathematical model and design effective control protocol.The nonlinear coupled Kuramoto model in complex networks can well explain the synchronization phenomenon in real life,and the model has important both theoretical value and practical application in mathematics,physics,biology,engineering,society,and other fields.The dynamics of Kuramoto model is usually closely related to the coupling strength between the oscillators,the communication time delay,the phase shift,and the network topology features and so on.This paper studies the effects of the communication time delay and the phase shifts on the synchronization performance,some conditions on the synchronization are obtained respectively.Numerical simulations further verify the theoretical results.The main contributions are as follows:(1)For a class of inhomogeneous Kuramoto oscillator systems with inertia coefficient and damping coefficient,a simple first-order Kuramoto model is extended to the second-order case,and the synchronization condition of the second-order model is obtained by the singular perturbation analysis theory,and the two synchronization conditions are studied by comparison.Numerical simulations further test the correctness of the synchronization conditions.(2)The synchronization of a class of second-order Kuramoto oscillator systems with communication time delay and phase delay is investigated.Some sufficient conditions are obtained via constructing some assumptions on the natural frequency,the inertia coefficient,time delay,and phase shift.The correctness of synchronization sufficient condition is verified via some numerical simulations.(3)The phase and frequency synchronization of second-order Kuramoto models with asymmetric delays of phase delay is investigated.By restricting the time difference of the same phase delay,some constraints on the natural frequency,inertia coefficient,time delay and phase shift are presented,and some sufficient conditions for phase and frequency synchronization are obtained via mathematical induction and differential equation theory.