Effects of assortative mixing in the second-order Kuramoto model

Assortativity-induced explosive synchronization in a complex neuronal network

In this study, we consider a scale-free network of nonidentical Chialvo neurons, coupled through electrical synapses. For sufficiently strong coupling, the system undergoes a transition from completely out of phase synchronized to phase synchronized state. The principal focus of this study is to investigate the effect of the degree of assortativity over the synchronization transition process. It is observed that, depending on assortativity, bistability between two asymptotically stable states allows one to develop a hysteresis loop which transforms the phase transition from second order to first order. An expansion in the area of hysteresis loop is noticeable with increasing degree-degree correlation in the network. Our study also reveals that effective frequencies of nodes simultaneously go through a continuous or sudden transition to the synchronized state with the corresponding phases. Further, we examine the robustness of the results under the effect of network size and average degree, as well as diverse frequency setup. Finally, we investigate the dynamical mechanism in the process of generating explosive synchronization. We observe a significant impact of lower degree nodes behind such phenomena: in a positive assortative network the low degree nodes delay the synchronization transition.

Low-frequency oscillations in coupled phase oscillators with inertia

This work considers a second-order Kuramoto oscillator network periodically driven at one node to model low-frequency forced oscillations in power grids. The phase fluctuation magnitude at each node and the disturbance propagation in the network are numerically analyzed. The coupling strengths in this work are sufficiently large to ensure the stability of equilibria in the unforced system. It is found that the phase fluctuation is primarily determined by the network structural properties and forcing parameters, not the parameters specific to individual nodes such as power and damping. A new "resonance" phenomenon is observed in which the phase fluctuation magnitudes peak at certain critical coupling strength in the forced system. In the cases of long chain and ring-shaped networks, the Kuramoto model yields an important but somehow counter-intuitive result that the fluctuation magnitude distribution does not necessarily follow a simple attenuating trend along the propagation path and the fluctuation at nodes far from the disturbance source could be stronger than that at the source. These findings are relevant to low-frequency forced oscillations in power grids and will help advance the understanding of their dynamics and mechanisms and improve the detection and mitigation techniques.

Effects of network robustness on explosive synchronization

Current studies have shown that there is a positive correlation between the network assortativity and robustness and that the assortativity also plays an important role in explosive synchronization. In this paper, taking the network robustness as a global property, we investigate its significance as well as the influence of its interaction with the assortativity on explosive synchronization. Our numerical results demonstrate that explosive synchronization is suppressed in extreme situations of both the robustness and assortativity. In addition, through appropriate adjustments of them, a maximum hysteresis area between the forward and backward transitions can be reached. Furthermore, our results might also provide reference for those who are interested in effects of network structure on synchronization, though this problem is still challenging as we show in the discussion.

Influence of stochastic perturbations on the cluster explosive synchronization of second-order Kuramoto oscillators on networks

Explosive synchronization in networked second-order Kuramoto oscillators has been well studied recently and it is revealed that the synchronization process is featured by cluster explosive synchronization. However, little attention has been paid to the influence of noise or perturbation. We here study this problem and discuss the influences of noise and perturbation. For the former, we interestingly find that noise has significant influence on the cluster explosive synchronization of those nodes with smaller degrees, i.e., their synchronization will change from the first-order to second-order transition and the critical points for both the forward and backward synchronization depend on the strength of noise. Especially, when the strength of noise is in an optimal range, a synchronization of the nodes with smaller degrees will be induced in the region of coupling strength where they do not display synchronization in the absence of noise. For the latter, we find that the effect of perturbation is similar to that of noise when its duration W is small. However, the perturbation will induce a change from cluster explosive synchronization to explosive synchronization when W is large. Furthermore, a brief theory is provided to explain the influence of perturbations on the critical points.

Multistable states in a system of coupled phase oscillators with inertia

We investigate the generalized Kuramoto model of globally coupled oscillators with inertia, in which oscillators with positive coupling strength are conformists and oscillators with negative coupling strength are contrarians. We consider the correlation between the coupling strengths of oscillators and the distributions of natural frequencies. Two different types of correlations are studied. It is shown that the model supports multistable synchronized states such as different types of travelling wave states, π state and another type of nonstationary state: an oscillating π state. The phase distribution oscillates in a confined region and the phase difference between conformists and contrarians oscillates around π periodically in the oscillating π state. The different types of travelling wave state may be characterized by the speed of travelling wave and the effective frequencies of oscillators. Finally, the bifurcation diagrams of the model in the parameter space are presented.

Interplay of degree correlations and cluster synchronization

We study the evolution of coupled chaotic dynamics on networks and investigate the role of degree-degree correlation in the networks' cluster synchronizability. We find that an increase in the disassortativity can lead to an increase or a decrease in the cluster synchronizability depending on the degree distribution and average connectivity of the network. Networks with heterogeneous degree distribution exhibit significant changes in cluster synchronizability as well as in the phenomena behind cluster synchronization as compared to those of homogeneous networks. Interestingly, cluster synchronizability of a network may be very different from global synchronizability due to the presence of the driven phenomenon behind the cluster formation. Furthermore, we show how degeneracy at the zero eigenvalues provides an understanding of the occurrence of the driven phenomenon behind the synchronization in disassortative networks. The results demonstrate the importance of degree-degree correlations in determining cluster synchronization behavior of complex networks and hence have potential applications in understanding and predicting dynamical behavior of complex systems ranging from brain to social systems.