Hopf Bifurcation In Two Groups Of Kuramoto Oscillators With Two Delays

The Kuramoto model is a classic phase oscillator model which is applied to study synchronization.Considering that it takes time lag to transfer and process information between oscillators,this paper mainly focus on the Hopf bifurcation problem of the two-family Kuramoto oscillators with two delays.First,in this paper,we reduce the mathematical model of the two-family Kuramoto oscillators by introducing two complex-valued order parameters,and the model is reduced onto the Ott-Antonsen manifold,then we obtain the equation that the order parameters must satisfy.For the convenience of calculation,the Lorentzian distribution is chosen as the density function to simplify the model,and finally a system of delay differential equations is obtained.By studying the characteristic equation of the delay differential equation,we find the stable switching curve,determine the direction of the stable switching curve,analyze the stability of the equilibrium point according to the stable switching curve,and obtain the stability of the incoherent state of the original Kuramoto model.Secondly,we prove the existence of Hopf bifurcation,and then use the center manifold theorem and normal form method to reduce the system onto the center manifold.By analyzing the Hopf bifurcation property of the model,we obtain the property of Hopf bifurcation of the Kuramoto model.Finally,we select the parameters that meet the conditions and conduct numerical simulation.We select three groups of parameters: the coupling strength in the first two groups of parameters is relatively small,and the coupling strength of the third group of parameters is relatively large.The simulations indicate that the stability switching curves of the first two parameters are families of closed curves.The original system is synchronized inside the closed curves,and the asynchronized outside theclosed curves.When fixing the value of transmission lag,and changing reaction time lag,the system will exhibite "synchronization-incoherent…" switch.The stability switching curve of the third set of parameters is a continuous curve.The reduced system is unstable in the entire plane,so the original Kuramoto model is synchronized in the entire plane.It can be seen that when the coupling strength is large enough,the time lag will not change the synchronization state of the system.