Investigating Bifurcation Points of Complex Network Synchronization

Abstract

This paper investigates the relationship between synchronization, system dynamics, and bifurcation points. To investigate the synchronization of dynamical systems, first, two oscillators are considered. Then networks with different structures are generated. The Master Stability Function (MSF) and synchronization criterion are computed for various oscillator dynamics in each network. The results of this paper can be categorized into two groups. In the first group, three points around the bifurcation points of the systems (before, near, and after the bifurcation point) are investigated that show no specific variation in the MSF diagram or its zero-crossing. Then it is revealed that the zero-crossing of the MSF diagram is related to the bifurcation points of the synchronization measure. In the second group, the types of synchronization transitions and MSFs for two periodic and chaotic dynamics of the studied systems are investigated for different networks. The results show that the type of synchronization transition can be different by changing the dynamics or network structure. However, in all networks and dynamics, the zero-crossing of MSF is completely matched with the bifurcation points of synchronization measure.