Exponential Synchronization of Two Nonlinearly Coupled Complex Networks with Time-Varying Delayed Dynamical Nodes

and Applied Analysis 3 3. Exponential Synchronization Criteria In this section, we will make drive-response complex dynamical networks with nondelayed and delayed coupling achieving global exponential synchronization by using adaptive controlling method. Define the synchronization errors e i (t) = y i (t) − x i (t) (i = 1, 2, . . . , N); the following errors dynamics system is obtained: ̇ e i = f (t, y i (t) , yi (t − τ (t))) − f (t, xi (t) , xi (t − τ (t)))


Introduction
Generally speaking, a complex network consists of a large number of interconnected nodes by edges, where a node is a fundamental unit having specific contents and exhibiting dynamical behavior, typically.As we all have known, the complex network models widely exist in real world, such as Internet, World Wide Web, biological neural networks, social connection networks, global economic markets, and ecosystems.Since the discovery of some typical complex networks such as the random networks, small-world networks and scale-free networks, many scientists and engineers from various fields, for instance, mathematics, physics, engineering, and biology, have paid increasing attention to the studies of complex networks.
In past few decades, the control and synchronization problem of networks coupled with complex dynamical systems, especially chaotic systems, has been extensively investigated in various fields due to its many potential applications [1][2][3][4][5][6][7][8][9][10].Many kinds of synchronization have been proposed, such as complete synchronization, lag synchronization, cluster synchronization, projective synchronization, and generalized synchronization.Since chaotic systems defy synchronization, how to design effective controllers for synchronizing coupled chaotic systems becomes an important and challenging problem.Many effective methods including pinning control [11][12][13][14], adaptive control [15][16][17][18][19][20], impulsive control [21][22][23][24][25][26], and intermittent control [27][28][29] have been adopted to design proper controllers.Inner synchronization, that is, the synchronization of all the nodes within a network, has been investigated recently.As a matter of fact, there exist other kinds of network synchronization in real world, for example, the synchronization phenomenon between two or more complex networks, which was called outer synchronization [30].Therefore, how to synchronize between different networks is an interesting and challenging work.Li et al. [30] pioneered in studying the outer synchronization between two unidirectionally coupled complex networks and derived analytically a criterion for them having the identical topological structures.Tang et al. [31] discussed the synchronization between two complex dynamical networks with nonidentical topological structures via using adaptive control method.Li et al. [32] studied the synchronization between two networks with different topology structures and different dynamical behaviors with open-plus-closed-loop method.Wu et al. [33] studied the problem of generalized outer synchronization 2 Abstract and Applied Analysis between two complex dynamical networks with different topologies and diverse node dynamics.Li et al. [34] discussed the outer synchronization of coupled discrete-time network.The adaptive-impulsive synchronization between two complex networks with nondelayed and delayed coupling was discussed in [35], but the delay is constant; moreover, the inner connecting function is linear and the delay in the dynamical nodes is ignored.In the real world, time delays are ubiquitous in natural and artificial systems.In much of the literature, time delays in the couplings are considered; however, the time delays in the dynamical nodes, which are more complex, are still relatively unexplored.To the best of our knowledge, the problem of synchronization between two nonlinearly coupled complex networks with time-delayed dynamical nodes is seldom discussed.
Motivated by the above discussions, the aim of this paper is to discuss the problem of exponential synchronization between two nonlinearly coupled dynamical networks with identical time-delayed dynamical nodes via adaptive control.Particularly, the coupling matrices are not assumed to be symmetric or irreducible.Based on the Lyapunov function method and mathematical analysis, synchronization criteria are derived analytically.Numerical examples are used finally to illustrate the usefulness of synchronization conditions.The above proposed scheme herein will be very useful for practical engineering applications.
The rest of this paper is organized as follows.In Section 2, model description and preliminaries are given.In Section 3, some sufficient conditions for the exponential synchronization are derived with the adaptive method.In Section 4, one illustrative example is given for supporting the theoretical results.Finally, conclusions are given in Section 5.
We refer to model (1) as the drive complex dynamical network, and the response complex network with control can be rewritten in the following form: where   () = ( 1 (),  2 (), . . .,   ())  ∈   is the response state vector of the th node and   ( = 1, 2, . . ., ) are the controllers to be designed later.
Before presenting the derivation of the main results, the definition, the assumptions, and lemmas are introduced as follows.
Remark 4. Assumption 3 gives some requirements for the dynamics of isolated node in network.It is easy to verify that many chaotic systems with delays or without delays satisfy Assumption 3, for example, Chua's oscillator, Rössler's system, the Lorenz system, Chen's system, and Lü's system as well as the delayed Lorenz system, delayed Hopfield neural networks, and delayed cellular neural networks, and so on.
Proof.We construct the Lyapunov function as follows: where  *  is sufficiently large positive constants to be determined.
On the other hand, we have Therefore, one has Thus, the errors vector () → 0, that is, the network (1) and network (2), are globally exponentially asymptotically synchronous.
In a special case, when the networks are all linearly coupled, that is, ℎ(  ()) = Γ 1   (), (  ( − ())) = Γ 2   ( − ()), the network (1) degenerates into the following: and the response complex network with control is given by We have Corollary 7 for the networks ( 16) and ( 17 where   are the feedback strength and   > 0 are arbitrary constants.Then, the drive-response networks can globally exponentially asymptotically synchronize. If the time-varying delay in the network ( 1) is the constant delay, we can obtain the following results.

Theorem 8. Suppose Assumptions 2 and 3 hold. Use the following adaptive controllers and updated laws:
where   are the feedback strength and   > 0 are arbitrary constants.Then, the drive-response networks can globally exponentially asymptotically synchronize.
Proof.We construct the Lyapunov function as follows: where  *  is a positive constants to be determined.The rest of the proof is similar to that of Theorem 6 and is omitted here.
Similar to Corollary 7, we have Corollary 9.

Corollary 9. Suppose Assumptions 2 and 3 hold. Use the following adaptive controllers and updated laws:
where   are the feedback strength and   > 0 are arbitrary constants.Then, the drive-response network can globally exponentially asymptotically synchronize.
Remark 10.It is noted that the configuration matrices  and  do not need to be symmetric, diffusive, or irreducible.This means that the networks can be undirected or directed networks and may also contain isolated nodes or clusters.Therefore, the network structures here are very general and the results can be applied to a great many complex dynamical networks.
Remark 11.The feedback strengths   are automatically adapted to the suitable constant, which depends on the initial values.The constants   can be chosen properly to adjust the synchronization speed.The larger the constants   the faster the achievement of synchronization of the driveresponse nonlinearly coupled networks with time-varying delayed dynamical nodes.
Remark 12.Some stability criteria for the exponential synchronization between drive and response nonlinearly coupled networks with time-varying delays are derived, which can also be applied to the complex network with single time delay.Thus, the results presented in this paper improve and generalize the corresponding results of recent works.Moreover, our designed synchronization controller is not only robust but also easy to implement.

Numerical Simulation
In this section, numerical simulation is given to verify and demonstrate the effectiveness of the proposed method for exponentially synchronizing two nonlinearly coupled complex networks with time-delayed dynamical nodes.Consider the time-delayed Lorenz chaotic system as node dynamics.It is described by which has a chaotic attractor when  = 10,  = 8/3,  = 28, and  = 5.See Figure 1. where ) . ( For any state vectors   of the time-delayed Lorenz chaotic system, there exists a constant  satisfying ‖  ‖ ≤  since chaotic attractor is bounded. To satisfy Assumption 3, consider that one can always find  > 0 such that || ≤ ( 2 /2) + where  1 ,  2 can be determined by choosing appropriate parameters   > 0,  = 1, 2, 3. Now, we consider two nonlinearly coupled complex dynamical networks (1) and ( 2) with coupling delay For simplicity, in the numerical simulations, we assume the time-varying delay () =   /(1 +   ); then τ  4, the larger the constants   the faster the convergence to synchronization.The numerical results show that adaptive scheme for the exponential synchronization of the drive-response nonlinearly coupled complex networks is effective in all the theorems and corollaries.

Conclusion
In this paper, the adaptive controllers have been proposed to study the global exponential synchronization between two nonlinearly coupled complex networks with time-varying delay dynamical nodes.By constructing the appropriate Lyapunov functions, some criteria are derived.In particular, the coupling matrices are not symmetric and irreducible.Numerical results demonstrate that the proposed approach is effective and feasible.In the analysis and simulation study of this paper, we fully considered the impact of the time delay element to the exponential synchronization of the drive-response coupled networks.In order to obtain the synchronization criteria, we did not take into account the environment factors, for example, noise, on the networks, which often affects the synchronization process of the driveresponse coupled dynamical networks.Therefore, in the near future work, we will further investigate the exponential synchronization problem of drive-response nonlinearly coupled dynamical network with noise.

Figure 1 :
Figure 1: Chaos phase portrait of the time-delayed Lorenz system.

Figure 3 :
Figure 3: State subvariables for node  = 3 between the drive and response coupled networks.