We consider a class of complex networks with both delayed and nondelayed coupling. In particular, we consider the situation for both time delay-independent and time delay-dependent complex dynamical networks and obtain sufficient conditions for their asymptotic synchronization by using the Lyapunov-Krasovskii stability theorem and the linear matrix inequality (LMI). We also present some simulation results to support the validity of the theories.

A complex dynamical network is a large set of interconnected nodes that represent the individual elements of the system and their mutual relationships. Owing to their immense potential for applications to various fields, complex networks have been intensively investigated in the past decade in areas as diverse as mathematics, physics, biology, engineering, and even the social sciences [

Time delays are an important consideration for complex networks although these were usually ignored in early investigations of synchronization and control problems [

The main contributions of this paper are two-fold. Firstly, we present a more general model for networks with both delayed and nondelayed couplings and derive criteria for their asymptotical synchronization. Secondly, we apply the Lyapunov-Krasovskii theorem and the LMIs to ensure the inevitable attainment of the required synchronization.

The rest of the paper is organized as follows. In Section

In general, a linearly coupled ordinary differential equation system (LCODES) can be described as follows:

If all the eigenvalues of a matrix

A complex network with delayed and nondelayed coupling (

A matrix

If

If

If

The linear matrix inequality (LMI)

Consider the delayed differential equation

The functional

Let

For all positive-definite matrices

Consider the delayed dynamical network (^{,} respectively. If the

In this section, we derive the conditions for the asymptotic synchronization of time-delayed coupled dynamical networks when they are either time-dependent or time-independent.

Consider the general time delayed and non-time delayed complex dynamical network (

For each fixed

The following corollaries follow immediately from the above theorem.

Consider the general non-time delayed complex dynamical network

From Lemma

Consider the general time delayed complex dynamical network

The results of [

The above analysis is applicable to a general system with arbitrary time delays. A simpler synchronization scheme, however, could be applied to systems with time delays that are already known and are small in value.

Consider the general time delayed and non-time delayed complex dynamical network (

For each fixed

Finally, we have

Consider the general time delayed complex dynamical network (

The proof can be found in [

The above criteracould be applied to networks with different topologies and different size. We put two examples to illustrate the validity of the theories.

We use a three-dimensional stable nonlinear system as an example to illustrate the main results, Theorem

Synchronization evolution

We use a 4-nodes networks model as another example to illustrate the Theorem

By using Theorem

Synchronization evolution

This paper considered a class of complex networks with both time delayed and non-time delayed coupling. We derived, respectively, a sufficient criterion for time delay-dependent and time delay-independent asymptotic synchronization which are more general than those obtained in previous works. These asymptotic synchronization results were obtained by using the Lyapunov-Krasovskii stability theorem and the linear matrix inequality. Two simple examples were also used to validate the theoretical analysis.

The authors thank the referees and the editor for their valuable comments on this paper. This work is supported by the National Natural Science Foundation of China (Grant no. 61273220), Guandong Education University Industry Cooperation Projects (Grant no. 2009B090300355), and the Shenzhen Basic Research Project (JC201006010743A, JC200903120040A).