Finite-Time Synchronization of Complex Dynamical Networks with Time-Varying Delays and Nonidentical Nodes

In this paper, we investigated the finite-time synchronization (FTS) problem for a class of time-delayed complex networks with nonidentical nodes onto any uniformly smooth state. By employing the finite-time stability theorem and designing two types of novel controllers, we obtained some simple sufficient conditions for the FTS of addressed complex networks. Furthermore, we also analyzed the effects of control variables on synchronization performance. Finally, we showed the effectiveness and feasibility of our methods by giving two numerical examples.


Introduction
In the last decades, complex networks have been deeply investigated in various types of subjects, such as biology, engineering, physics, and mathematics [1][2][3].In general, a complex network can be depicted as a large set of nodes connected by edges, where every node is a basic unit with specific dynamics.In fact, the structure of many types of natural and artificial systems, such as genetic networks, metabolic pathways, social networks, electrical power distribution networks, and World Wide Web (WWW), can be modeled via complex networks.
Synchronization is unique in nature and can play an extremely vital role in many fields of science including biology, climatology, sociology, and ecology [4,5].Over the past two decades, synchronization of complex networks with identical dynamical topology has been broadly studied in different fields of engineering and sciences owing to the fact that they not only can well depict a great number of natural phenomena, but also have many useful applications in biological systems [6], secure communication [7], image processing [8], and so on.For this reason, many useful methods have been introduced for the synchronization of complex networks without control [9][10][11][12].However, sometimes we cannot achieve the synchronization of network without adding any controller to the dynamics of individual node.Thus, how to synchronize the complex networks by designing a suitable controller is seen to be a most significant topic in both theory and application.As a result, many useful approaches have been developed to achieve chaos stabilization and chaos synchronization, such as adaptive control [13,14], pinning control [15], impulsive control [16,17], sliding mode control [18], and intermittent control [19,20].
Time delays inevitably exist in natural and man-made systems and cannot be neglected; for instance, delay effects cannot be ignored in the communication systems due to the limited switching speed of the hardware [21].So far, in most of the existing works the networks with coupling time delays were considered.However, the time delays in the dynamical nodes [22][23][24][25][26][27][28], which can be more complicated, are still neglected in most of the existing works.In addition, it maybe unpractical to always use the hypothesis that all of the network nodes are the same since some real-world complex networks can be modeled by using different dynamical nodes [25].Taking a software community network or metabolic network as an example, the dynamics of any two nodes in different communities are totally different, while the dynamics of each individual node in every community can be described by the same functional units.When the dynamics of nodes in a complex network are allowed to be nonidentical, the synchronization approaches for networks consisting of identical nodes will not work anymore.Thus, it is of great importance to develop new synchronization approaches for time-delayed complex networks with nonidentical nodes [25][26][27][28][29][30].In [26], some exponential synchronization criteria for a class of complex networks with nonidentical nodes were established via combining the local intermittent controller with the open-loop controller.In [27], the outer synchronization of two complex networks with nonidentical dynamical nodes and coupling time delays was investigated by employing the adaptive control technique.In [28], by using pinning control scheme, the cluster synchronization of complex dynamical networks with time-delayed coupling and dynamic nodes was studied.
Many of the results mentioned above have been employed to ensure the asymptotical or exponential stability of error dynamics, which means that the synchronization between the controlled system and the desired system can only be achieved in the infinite horizon.In practical application, however, it is more desirable that the synchronization aim is achieved in a finite-time.As is known to all, the finite-time control technique has been used as an efficient approach to realize the synchronization in a given time due to the fact that the finite-time control schemes may lead to better system performance such as better robustness and disturbance rejection properties [31][32][33][34][35].In [36], the authors investigated the FTS problem for a class of complex networks with stochastic noise perturbations.In [37], the FTS between two complex networks with general coupling was studied by employing two different types of controllers, that is, periodically intermittent control and impulsive control.In [38], based on the finite-time stability theory and nonlinear control theory, the authors were concerned with FTS synchronization of class of nonidentical drive-response chaotic systems with timevarying delay.In [39], FTS problem of a class of fractionalorder memristor-based neural networks with time delays was studied by using Laplace transform method and generalized Gronwall's inequality technique.However, it is worth pointing out that most of the existing results on the FTS of complex networks are concerned with those identical nodes and delayindependent or constant delays, and very little attention has been paid to solving the problem which emerged from complex networks with nonidentical dynamical nodes and time-varying delays.
Inspired by the above analysis, in the paper, we deal with the problem of FTS for a class of complex networks with nonidentical nodes and time-varying delays.By employing the well-known finite-time stability theorem, designing two types of novel controllers, and using some inequality techniques, we develop some simple but useful sufficient criteria for the FTS of the addressed network.
The rest of the paper is organized as follows.In Section 2, the FTS problem of a complex network with nonidentical nodes and time-varying delays is introduced, and some related definitions and preliminary lemmas are given.Next section is devoted to investigating the FTS of the addressed networks.In Section 4, two numerical examples with their simulations are given to demonstrate the effectiveness of the obtained theoretical results.Finally, we conclude the paper with some general conclusions in Section 5.

Preliminaries
In the paper, we consider the following controlled complex networks model consisting of  nonidentical nodes, in which every node is an -dimensional dynamical system described by where   () = ( 1 (),  2 (), . . .,   ())  is state variable of the th individual node,   :  + ×   ×   →   is a vectorvalued continuous function,   () is a time-varying delay and 0 ≤   () ≤   for each  ∈ I, the positive constant  is the coupling strength, the Γ ∈  × is the inner connecting matrix between nodes, and the constant matrix  = (  ) × denotes the linear coupling configuration of the network, where   is given as follows: if there is a connection between node  and node  ( ̸ = ), then   > 0; otherwise,   = 0, and the diagonal elements of matrix  are defined by   = − ∑  =1, ̸ =   , and   () is the control input.The initial values of system (1) are given by where  = max ∈I {  },   () = ( 1 (),  2 (), . . .,   ())  ∈ ([−, 0],   ), which denotes the Banach space of all continuous functions mapping [−, 0] into   with norm defined by The main aim of this paper is to finite-timely synchronize the states of networks (1) onto any smooth dynamics (), where () ∈   can be any desired state: equilibrium point, a nontrivial periodic orbit, or even a chaotic orbit.That is, by designing a suitable feedback controller in system (1), there exists time  * such that Throughout this paper, for system (1), we assume that the following hypothesis is satisfied.
Let synchronization error   () =   () − () for  ∈ I.Then, according to system (1) and the control law (11), the error dynamical system can be derived as where It is not difficult to see that the global FTS of the controlled complex network (1) is realized when the zero solution of the error system (12) is globally finite-time stable and this can be ensured by the following theorems.Theorem 4.Under hypothesis ( 1 ), assume that the following inequality is satisfied: where =  and â = ( min /‖Γ‖ 2 )  ,  min is minimum eigenvalue of Γ  .Then, the controlled complex network ( 1) is synchronized to the given smooth dynamics () in a finite-time where Proof.Let () = (  1 (),   2 (), . . .,    ())  and construct the following Lyapunov function: Calculating the time derivative of () along the trajectory of system (12) leads to In view of the definition of   (), we have where Since Â is the symmetric matrix, from inequality ( 14), we have Using Lemma 3, one has Hence, Thus, from (20) and the above inequality, we get Then, from Lemma 2, () converges to zero in a finite-time  * , where  * is given by where  = (1 + )/2.Therefore, according to Definition 1, the controlled complex network (1) is finite-timely synchronized in the settling time  * .The proof of Theorem 4 is now completed.
Theorem 5. Suppose that the hypothesis ( 1 ) holds and Γ is a positive definite diagonal matrix with diagonal elements  1 ,  2 , . . .,   ; if the inequality is satisfied then the controlled complex network ( 1) is synchronized to the given smooth dynamics () in a finite-time where (0) and  are the same as defined in Theorem 4.
Proof.Let () = (  1 (),   2 (), . . .,    ())  and consider the following Lyapunov function: The time derivative of () along the trajectory of ( 12 From the hypothesis ( 1 ) and the definition of   (), we get where   is the identity matrix and ẽ () = [ẽ 1 (), ẽ2 (), . . ., ẽ ()]  is the th column vector of the ().From inequality (25), we have From the analysis in Theorem 4, we have Thus, together with (20), Lemma 3, and the above inequality, we get By Lemma 2, () converges to zero in a finite-time  * , where  * is given by Therefore, according to Definition 1, the controlled complex network (1) is finite-timely synchronized in the settling time  * .The proof of Theorem 5 is now completed.
In the controlled network (1), if the evolution function   =   (,   ()) for  ∈ I, then system (1) is reduced to the following complex network: Simultaneously assumption ( 1 ) can be replaced by the following inequality.
(  1 ) For all   ,   ∈   and  ∈ I, there exist constants   such that If we let where   () =     () −  sgn(  ())|  ()|  , then, the error system is the following form: Thus, for the FTS of controlled complex network (34), we have following corollaries.

Corollary 7.
Under the hypothesis (  1 ), assume that Γ is a positive definite diagonal matrix.If  1 ,  2 , . . .,   satisfy inequality (25), then the controlled complex network ( 34) is finite-timely synchronized to the given smooth dynamics () under controller (36).Moreover, the synchronization time is estimated by Remark 8.In Theorems 4 and 5, by using special finitetime controller, we achieved the FTS of nonidentical complex network (1) onto any smooth goal dynamics ().However, the used control law   () is somehow expensive and not easily applicable, especially if the governing functions   of system (1) satisfy some special conditions.Below, we will modify the adaptive laws   () to improve the applicability of our results.Suppose that the evolution functions   are bounded and let Then, for the FTS of complex system (1), we have following results.
Proof.Consider the following Lyapunov function: Taking the time derivative of () along the solutions of ( 12), from ( 40) and ( 42 Thus, from Lemma 2, () converges to zero in a finite-time  * , where  * is given by Therefore, according to Definition 1, the controlled complex network ( 1) is finite-timely synchronized in the settling time  * .The proof of Theorem 9 is now completed.
Theorem 10.Suppose that Γ is a positive definite diagonal matrix and the evolution functions   in system (1) satisfy inequality (41).If the error system ( 12) is controlled with controller (40), and its control strengths   satisfy then the controlled complex network ( 1) is finite-timely synchronized to the given smooth dynamics ().Moreover, the synchronization time is estimated by where (0) and  are the same as defined in Theorem 4.
Proof.Consider the following Lyapunov function: Calculating the time derivative of () along the trajectory of system (12) and using (40) Therefore, according to Definition 1, the controlled complex network ( 1) is finite-timely synchronized in the settling time  * .The proof of Theorem 10 is now completed.
In the nonidentical complex network (1), if all the functions   are the same, that is,   =  for all  ∈ I, then system (1) is degenerated to the following identical complex network: (52) then the FTS problem of the complex network (1) becomes the FTS problem of a complex network with identical nodes studied in much of the literatures, such as [34][35][36][37].Additionally, the finite-time control scheme ( 11) is degenerated to Evidently, the following criteria can be directly derived to ensure the FTS of the complex network with identical nodes (51).
In the controlled network (51), if the evolution function  is not relevant to delay, that is,  = (,   ()) for  ∈ I, then system (51) is reduced to following complex network: In addition, the synchronization state () is chosen as a solution of a decoupled state satisfying Let Then, the error system is the following form: Thus, for the FTS of controlled complex network (55), we have following result.
Remark 14.Since control inputs ( 11), ( 40), (53), and (57) contain the discontinuous sign function, an undesirable chattering may appear as a hard switcher.To avoid the chattering, the continuous tanh function can be used for discontinuous sign function to remove discontinuity.For example, controller (11) can be modified to where   > 0 for  ∈ I.
Remark 15.In [37], the FTS between two identical complex networks with delayed and nondelayed coupling was investigated by using two different types of controller, that is, periodically intermittent controller and impulsive controller.In [40], the FTS of complex dynamical networks with or without coupling delay was studied by employing the Lyapunov function method and well-known finite-time stability theorem.However, the proposed controllers   () in [37,40] are very complex and   () → ∞ as ‖  ()‖ → 0. Thus, the designed control laws in [37,40] are too expensive and not easily applicable.Evidently, the improper technique somewhat affects the validity of results obtained in [37,40].To avoid this, a different technique is used in this paper to achieve the FTS of considered nonidentical complex system.
Remark 16.In the past few years, FTS has drawn an increasing attention due to the fact that it requires considered chaotic networks to be synchronized in finite-time rather than merely asymptotically.In the literature, so far, there are many excellent results on the TNS of complex networks, neural networks, and multiagent systems based on the continuous or discontinuous control approaches [41][42][43][44].However, there are very few results to address the problem of the FTS for complex networks with delays and nonidentical nodes.Generally, different nodes in a complex network have different features including dynamic behaviors and initial conditions, which may result in their different control protocols to optimize synchronization time.In this paper, we employ the finite-time control technique to achieve the synchronization of a class of complex networks with nonidentical dynamical nodes and time-varying delays.Obviously, the results obtained in this paper can fasten the synchronization speed in great extent than the methods of asymptotical or exponential synchronization [22,28].Therefore, our results are more conducive and the extension and improvement of the existing results can be seen.

Numerical Simulations
In this section, two numerical examples and their simulations are given to demonstrate the feasibility of the proposed FTS schemes.
Example 2. Consider a complex dynamic network consisting of two communities, in which each dynamical node is a chaotic neural network given by where The time evolutions of each dynamical node in (66) are depicted in Figures 5 and 6.Now, we consider the following controlled complex networks given by   (68) The inner connecting matrix Γ = diag(1.5,2)  and the matrix  is selected as  The following periodic orbit is chosen as a synchronization aim:  () = (4 sin (0.5) , 3 cos ())  . (70) It is not difficult to check that the hypothesis ( 1 ) is satisfied for system (66) with

Conclusion
In this paper, we investigate the FTS of nonidentical complex networks with time-varying delays.By using finite-time stability theorem, inequality techniques, and designing suitable controllers, we develop some simple but useful sufficient criteria on the FTS of the addressed model.Our results of the FTS technique have optimal convergence time than the asymptotical and exponential synchronization methods.Finally, we gave two examples to demonstrate the effectiveness and feasibility of the developed FTS schemes.

Figure 4 :Figure 5 :
Figure 4: Time evolution of synchronization errors for (61) with different initial values.

32 Figure 6 :
Figure 6: The chaotic behavior of each node in the second cluster in (66).

Figure 7 :
Figure 7: Time evolution of synchronization errors for (26) with different initial values.

Figure 8 :
Figure 8: Finite-time synchronization curves of   for (68) with different initial values.