Bifurcation Study of Kuramoto Transition of Random Oscillator Network

We study the Kuramoto transition of oscillators in random network and Barabáshi-Albert network model. In both cases, the results of numerical simulation show good coincidence with the mean-field analysis. Synchronization is a widely spreading behaviour in nature. It is seen in the firing of neurons in our brain, flushing of fireflies, and some chemical reactions. One of the great progress of the study of synchronization was made by Kuramoto. 1) He showed that the dynamics of many kinds of oscillators can be reduced to the dynamics of phase oscillator, and analyzed the synchronization of it. One of the important results of his works is the discovery of the Kuramoto transition, the synchronization of globally coupled oscillators. However, global coupling is seldom found in nature. For example, a neuron in the brain is not connected to all other neurons. It is connected to a finite number of neurons, and the global coupling is far from real network of neurons. Recently, the study of complex network has been developed. 2) In the complex network, each node links to a finite number of other nodes. In this decade, many real networks such as the internet, metabolic networks, neurons in the brain, food web, and co-authorship of papers have been investigated. We have realized that many real networks have common structure, such as scale-free degree distribution P (k) ∝ k −γ , where P (k) is the distribution function of degree of nodes. Therefore, the study of synchronization in complex network is important problem to study the synchronization in real systems. The study in this regard had been mainly carried out by numerical simulations. After the simulation by Watts, 3) Hong et al. carried out detailed investigation of the Kuramoto transition in Watts-Strogatz model and obtained the phase diagram. 4) Moreno and Pacheco studied the synchronization in the Barabáshi-Albert(BA) network and concluded that the critical coupling constant K c is not zero in the BA network. 5) Recently, we developed the analytical theory of the Kuramoto transition in random networks. 6), 7) Using the mean-field approximation, we concluded that as the size of network approaches infinity, K c approaches 0 in the scale-free random network if γ ≤ 3. This result seems to contradict to that of Moreno and Pacheco. It is usually believed that the property of the BA network is similar to the γ = 3 random scale-free network. If K c …