Characterizing generalized synchronization in complex networks

Shuguang Guan,Zonghua Liu,Xiaofeng Gong,Kun Li,C-H Lai

doi:10.1088/1367-2630/12/7/073045

Abstract

Recently, it was shown that generalized synchronization (GS) can generally occur in systems of networked oscillators. In this paper, we further characterize the states of GS by both theoretical analysis and numerical experiments. We show that the entrainment of local dynamics in a network can be characterized by the conditional Lyapunov exponent of the local oscillator. Meanwhile, different types of states of GS can be identified by analyzing the Lyapunov exponent spectra of the coupled system. Most importantly, we further provide direct evidence demonstrating that node dynamics in a network in a chaotic GS state can indeed achieve functional relations, although they may not directly connect to each other in typical complex networks.

Highlights

It was shown that generalized synchronization (GS) can generally occur in systems of networked oscillators
Combining the results in figure 10, we can conclude that, for networked non-identical oscillators in the chaotic GS (CGS) state, i.e. when all the oscillators have been entrained, the node dynamics can achieve GS in the strict sense, they may not be directly connected to each other in the network. We carry out both theoretical analysis and numerical experiments to characterize the GS occurring on a scale-free network
It is shown that the entrainment of oscillators in networks can be characterized by the conditional Lyapunov exponent of the local dynamics

Summary

Dynamical model

The dynamical models in this study are the same as in [37]. For self-consistency, we here briefly describe them in the following. For the local node dynamics, we typically choose the chaotic Lorenz oscillator, Fi (xi ) = [10(yi − xi ), ri xi − yi − xi zi , xi yi − (8/3)zi ]T,. We choose the Barabási–Albert (BA) model that has been extensively used to represent the scale-free networks [38]. In such a model, the network grows from the initial m0 nodes. A node’s probability of being selected is directly proportional to its degree This is the rule that is called preferential attachment. In this way, a scale-free network with degree distribution satisfying power law can be generated

The conditional Lyapunov exponents

The chaotic GS states

Identifying functional relations among node dynamics

Summary