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

Muhammadhaji, Ahmadjan; Abdurahman, Abdujelil; Jiang, Haijun

doi:https://doi.org/10.1155/2017/5072308

Journal of Control Science and Engineering

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Abstract Introduction Preliminaries Conclusion Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2017 | Article ID 5072308 | https://doi.org/10.1155/2017/5072308

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

Ahmadjan Muhammadhaji,¹Abdujelil Abdurahman,¹and Haijun Jiang¹

Academic Editor: James Lam

Received27 Nov 2016

Revised25 Feb 2017

Accepted28 Feb 2017

Published21 Mar 2017

Abstract

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.

1. Introduction

In the last decades, complex networks have been deeply investigated in various types of subjects, such as biology, engineering, physics, and mathematics [1–3]. 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 [9–12]. 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 [22–28], 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 [25–30]. 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 [31–35]. 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 time-varying delay. In [39], FTS problem of a class of fractional-order 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 delay-independent 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.

Notations. In this paper, denotes the identity matrix and sgn represents the sign function. For a vector , is the vector of the form , and its Euclidean norm is defined as , where denotes transposition. denotes a matrix of -dimension, , and , where denotes the maximum eigenvalue of a symmetric matrix. The notation represents is symmetric and seminegative definite matrix.

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.

2. 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 is state variable of the th individual node, is a vector-valued continuous function, is a time-varying delay and for each , the positive constant is the coupling strength, the 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 ; otherwise, , and the diagonal elements of matrix are defined by , and is the control input.

The initial values of system (1) are given by where , , which denotes the Banach space of all continuous functions mapping 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.

() For every vector-valued function , there exist positive constants and such that

Definition 1. Assume that is any smooth dynamics. The complex network (1) is said to be synchronized onto the homogeneous state in a finite-time if, for a suitable designed feedback controller , there exists a constant such that and for , . The constant is called the settling time of system (1).

Lemma 2 (see [36]). Assume that is a positive definite continuous function. If satisfies the differential inequality where , are two constants, then, for any given , satisfies the following inequality: with given by

Lemma 3 (Jesen inequality [34]). If , are positive numbers and , then

3. Main Results

In this section, some effective control schemes are developed to finite-timely synchronize the complex network (1) to any smooth dynamics . First, we design controller of system (1) as the following form: where , , , is a positive constant determined later, is a tunable constant, and the real number satisfies .

Let synchronization error for . 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 , assume that the following inequality is satisfied: where , , , , , for and is minimum eigenvalue of Then, the controlled complex network (1) is synchronized to the given smooth dynamics in a finite-time where and .

Proof. Let and construct the following Lyapunov function: Calculating the time derivative of along the trajectory of system (12) leads toIn view of the definition of , we haveFrom the hypothesis (), we getwhere and . Since is the symmetric matrix, from inequality (14), we haveUsing Lemma 3, one has Hence,Thus, from (20) and the above inequality, we getThen, from Lemma 2, converges to zero in a finite-time , where is given by where . 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 holds and is a positive definite diagonal matrix with diagonal elements ; if the inequality is satisfied then the controlled complex network (1) is synchronized to the given smooth dynamics in a finite-time where and are the same as defined in Theorem 4.

Proof. Let and consider the following Lyapunov function: The time derivative of along the trajectory of (12) is From the hypothesis () and the definition of , we get where is the identity matrix and 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 , then system (1) is reduced to the following complex network: Simultaneously assumption () can be replaced by the following inequality.

() For all and , there exist constants such that

If we let where , then, the error system is the following form: Thus, for the FTS of controlled complex network (34), we have following corollaries.

Corollary 6. Assume that hypothesis holds. If satisfy inequality (14), 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 where and are the same as defined in Theorem 4.

Corollary 7. Under the hypothesis , assume that is a positive definite diagonal matrix. If 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 finite-time 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 letThen, for the FTS of complex system (1), we have following results.

Theorem 9. Assume that evolution functions in system (1) satisfywhere . If the error system (12) is controlled with controller (40), and its control strengths satisfywhere and are the same as in Theorem 4, then the controlled complex network (1) is finite-timely synchronized to the given smooth dynamics in settling time , where is the same as in (15).

Proof. Consider the following Lyapunov function: Taking the time derivative of along the solutions of (12), from (40) and (42), we haveThus, 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 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 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) and (46) lead toThus, 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 10 is now completed.

In the nonidentical complex network (1), if all the functions are the same, that is, for all , then system (1) is degenerated to the following identical complex network:Also, the synchronization state is chosen as a solution of a decoupled state satisfying 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 [34–37]. 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).

Corollary 11. Assume that hypothesis holds and satisfy inequality (14); then the controlled complex network (51) is globally finite-timely synchronized to the decoupled state (52) under controller (53). Moreover, the synchronization time is estimated by where and are the same as defined in Theorem 4.

In the controlled network (51), if the evolution function is not relevant to delay, that is, for , 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.

Corollary 12. Suppose that hypothesis holds. If satisfy inequality (14), then the controlled complex network (55) is finite-timely synchronized to the given decoupled state (56) under controller (57). Moreover, the synchronization time is estimated by where and are the same as defined in Theorem 4.

Remark 13. From the proofs of Theorems 4, 5, 9, and 10, we can see that the parameters , , and in controllers (11), (40), (53), and (57) play a central role in FTS of controlled complex networks. Inequality (20) indicates that the synchronization rate increases when , , or increases. On the other hand, whether network (1) can be synchronized or not relies on the value of , whereas the synchronization time depends on values of and and has nothing to do with the value of .

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 function can be used for discontinuous sign function to remove discontinuity. For example, controller (11) can be modified towhere for .

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 . 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 [41–44]. 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.

4. Numerical Simulations

In this section, two numerical examples and their simulations are given to demonstrate the feasibility of the proposed FTS schemes.

Example 1. Consider the nonidentical complex dynamical network described by following equation:where andHere, , , , , , andThe coupling configuration matric is chosen as

The time evolutions of delayed chaotic Lorenz system and chaotic Chua oscillator with initial conditions and for are shown in Figures 1 and 2.

Now we select Rössler system as the synchronized aim; it is given by

The chaotic behavior of Rössler system (65) with initial values is depicted in Figure 3.

By easy computation, we get , , and , , , and for Lorenz system and , , and for Chua oscillator. Letting and (see [40]), then we have and . Thus, take , , , and . Then, the hypothesis is satisfied for system (61). Using finite-time controller (11) and letting , and , it is not difficult to check that inequality (14) is satisfied with and . Therefore, according to Theorem 4, the controlled coupled network (61) is synchronized to the chosen orbit (65) in a finite-time. The synchronization errors are given in Figure 4.

Example 2. Consider a complex dynamic network consisting of two communities, in which each dynamical node is a chaotic neural network given bywhere , , , , , andThe 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

The inner connecting matrix and the matrix is selected as

The following periodic orbit is chosen as a synchronization aim:

It is not difficult to check that the hypothesis is satisfied for system (66) with , , , and . Choosing , , , and , then inequality (14) also holds. Use the following finite-time controller (see Remark 14):where , , . Then, from Theorem 4, the controlled coupled network (68) is synchronized to the chosen periodic orbit (70) in a finite-time. The synchronization errors are given in Figure 7, and the synchronization curves are depicted in Figure 8.

5. 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.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundations of China (Grant nos. 11601464 and 61473244) and the Natural Science Foundation of Xinjiang University (Starting Research Fund for the Xinjiang University Doctoral Graduates, Grant no. BS150202).

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Copyright © 2017 Ahmadjan Muhammadhaji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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