An investigation of phenomena observed in scale-free coupled circle map

Abstract

In this study, phenomena observed in scale-free coupled circle map are investigated. The circle map is a one-dimensional discrete-time dynamical system which exhibits various kinds of behavior as the parameters change. As the high-dimensional coupled circle map, the coupled map lattice and the globally coupled map have been studied. However, the scale-free coupled circle map has not been well investigated so far. We study the model in which one of the circle maps corresponding to a hub node of the scale-free network has the parameter which leads the single circle map to generate high-order periodic points, and the parameter values of the other maps are set to converge to a fixed point. Changing the coupling strength between each map, we investigated the synchronization in the scale-free coupled circle map by calculating the value of the order parameter. The result of this study elucidated that when the coupling strength between each map was negative, all circle maps in the network behaved chaotic whatever the parameter value of the hub node was set to. That means the phenomena was generated because of the scale-free network structure itself but not the state of the hub node. On the other hand, the coupling strength was positive, the behavior observed in the network was based on the state of the hub node. In addition, the result showed the periodic point observed in the hub node could change to the fixed point after the scale-free network generated. The result suggests that the phenomena such as chaos and periodic oscillation could change to be converged into stable fixed point by forming the scale-free network and appropriately controlling the parameters.