Abstract
This paper is concerned with the impulsive synchronization problem of chaotic delayed neural networks. By employing Lyapunov stability theorem, impulsive control theory and linear matrix inequality (LMI) technique, several new sufficient conditions ensuring the asymptotically synchronization for coupled chaotic delayed neural networks are derived. Based on these new sufficient conditions, an impulsive controller is designed. Moreover, the stable impulsive interval of synchronized neural networks is objectively estimated by combining the MATLAB LMI toolbox and one of the two given equations. Two examples with numerical simulations are given to illustrate the effectiveness of the proposed method.
1. Introduction
During implementation of artificial neural networks, time delay is a very familiar phenomenon due to the finite switching speed of neurons and amplifiers. The existence of time delay possesses an important source in what cause instability and even chaotic behavior in some type of delayed neural networks (DNNs) if the parameters and time delays are appropriately chosen (see [1–6]). These kinds of chaotic neural networks have been successfully applied in chemical biology [7], combinatorial optimization [8], associative memory [9], biological simulation [10], and so on. Especially, the synchronization problem of chaotic delayed neural networks has been extensively studied over the past few decades due to its potential applications in many areas, such as secure communications [11–13], image encryption [14], image processing [15], and harmonic oscillation generation [16]. Hence, it is of great theoretical and practical significance to investigate synchronization problem of chaotic delayed neural networks.
A wide variety of schemes have been proposed for the synchronization of chaotic systems, for example, adaptive control [17], slide mode control [18], coupling control [19], feedback control [20], and impulsive control [21–25]. It is worth noting that impulsive control is characterized by the abrupt changes in the system dynamics at certain instants, which is an advantage in reducing the amount of information transmission and improving the security and robustness against disturbances especially in telecommunication network and power grid, orbital transfer of satellite. In addition, impulsive control allows the stabilization and synchronization of chaotic systems using only small control impulses. Thus, it has been widely used to synchronize chaotic neural networks (see [21, 23, 24]). In [21], Zhao and Zhang obtained some new criteria for the impulsive exponential antisynchronization of two chaotic delayed neural networks by establishing an integral delay inequality via the inequality method. Li et al. [23] investigated the synchronization scheme of coupled neural networks with time delays by utilizing the stability theory for impulsive functional differential equations. Zhang and Sun [24] studied the robust synchronization model of coupled delayed neural networks under general impulsive control. However, all above results do not efficiently utilize the so-called sector nonlinearity property of the activation functions of the neural networks, which leads to some conservatism of the results.
Motivated by the above discussions, the impulsive synchronization problem for chaotic delayed neural networks has not been fully investigated yet, which is still open and remains challenging. The aim of this paper is to study the synchronization of chaotic neural networks with time-varying delay. Some novel sufficient conditions which guarantee the coupled chaotic delayed neural networks can be asymptotically synchronized are derived based on Lyapunov stability theorem, impulsive control theory, and linear matrix inequality (LMI) technique. Moreover, the stable impulsive interval of synchronized neural networks is objectively estimated by combining the MATLAB LMI toolbox and one of the two given equations.
The organization of this paper is as follows. In Section 2, the impulsive synchronization problem is described and some necessary definitions and lemmas are given. Some new synchronization criteria are obtained in Section 3. In Section 4, two illustrative examples are given to show the effectiveness of the proposed method. Finally, conclusions are given in Section 5.
Notations. Let denote the set of real numbers, let denote the set of positive real numbers, and let and denote the dimensional Euclidean space and the set of all real matrices, respectively. denotes the set of positive integers. is the Euclidean norm of the vector . For any matrix , denotes that is a symmetric and positive definite matrix. If are symmetric matrices, then means that is a negative semidefinite matrix. denote the minimum and maximum eigenvalue of matrix , respectively. and mean the transpose of and the inverse of a square matrix . denotes the identity matrix with appropriate dimensions. is strictly increasing and , for , , for , , is continuous everywhere except at the finite number of points , where exist and . The notation always denotes the symmetric block in one symmetric matrix.
2. Problem Description and Preliminaries
The chaotic neural networks with variable delay can be described by where is the neuron state vector; is a positive diagonal matrix; and are the connection weight matrix and the delayed connection weight matrix, respectively; is the constant input vector; is the nonlinear neuron activation function which describes the behavior in which the neurons respond to each other; the time delay is bounded as , and the initial value condition is a piecewise right continuous function.
Throughout this paper we assume that satisfies assumption (H1).(see [26]) Each function is continuous, and there exist scalars and such that for any , where and can be positive, negative, or zero.
Based on assumption (H1), we set
Remark 1. In usual Lipschitz condition, it is assumed that . Clearly, the condition (H1) is quite general and includes the usual Lipschitz conditions as a special case.
To investigate the impulsive synchronization of chaotic neural networks and consider system (1) as the drive system, the corresponding response system is given by where is the neuron state vector of the response system, the initial value condition is a piecewise right continuous function.
At discrete time , the state variables of the drive system are transmitted to the response system as the control input such that the state vectors of the response system are suddenly changed at these instants. Therefore, the impulsive controlled response system can be written as where denotes the state jumping at impulsive time instant and are the right-hand and left-hand limits of the functions and at , respectively. Moreover, . Suppose that the discrete time sequence satisfies and . is the impulsive matrix.
Let be the synchronization error, and then we can obtain the error system between (1) and (5): where .
The following definitions and lemmas which are useful in deriving synchronization criteria are used in the paper.
In general, the impulsive functional differential equation can be described by where ensures that (7) has a zero solution. for each . For any is defined by . Assume that there exists a such that implies for all .
Definition 2. The function is said to belong to class if (i) is continuous in each of sets , and for each exists;(ii) is locally Lipschitzian in and for all .
Definition 3 (see [27]). For , the right and upper Dini’s derivatives of are defined as
Lemma 4 (see [28]). Assume that there exist , and such that (i) for ;(ii)for all , and , ;(iii)for any solution of (7), , implies that , where is locally integrable, is the inverse function of ;(iv) is nondecreasing and there exist constants and such that for all and and ;
then the zero solution of (7) is uniformly asymptotically stable.
Lemma 5 (see [29]). For any symmetric and positive definite matrix , the following inequality holds:
Lemma 6 (see [30]). If are real matrices with appropriate dimensions, then there exists a number such that
3. Main Results
In this section, we use the Lyapunov-like function to derive the asymptotically stability conditions of the zero solution of the error system (6), which implies that the drive system (1) and the response system (5) can be asymptotically synchronized.
Theorem 7. Assume that assumption holds. If there exist three symmetric and positive definite matrices , seven constants , , such that the following inequalities hold: where then the systems (1) and (5) are asymptotically synchronized.
Proof. From (11) and Lemma 5, we get
Let , and then .
For all ,
Let , and then .
For any solution of (6), if
that is,
specially, for , we have
For , the right and upper Dini’s derivatives of along the trajectory of system (6) are obtained as follows:
It follows from Lemma 6 that
By assumption , it is easy to see that
Combining (20) and (22) together yields
From (21) and (23), we get
Note that
From (26), we can, namely, obtain
Introducing (24), (25), (27), and (28) to (19), from condition of Theorem 7 and Schur complement [31], we obtain
Let and . Then
From Lemma 4, the zero solution of system (6) is asymptotically stable. Thus, system (1) and system (5) are synchronized. The proof of Theorem 7 is completed.
Remark 8. The stable impulsive interval is associated with the impulsive matrix and the choice of parameter . In order to reduce the man-made misleading during the prediction of stable impulsive interval, here only fix parameter and the impulsive matrix , and then let . Firstly, can be obtained by Theorem 7 via MATLAB LMI toolbox. Finally, can be calculated by following algebraic equations: or linear matrix inequalities: In [22, 25], the parameters and which are corresponding to parameter had been selected subjectively, which may cause result to lack fidelity. In addition, parameters are selectable variables which can increase the flexibility of the possible outcomes. Therefore, our results is more objective in some situations.
From the control point of view, in order to obtain the synchronization between the drive system (1) and the controlled response system (5), we design as the impulsive controller [32]. Let in Theorem 7, the following corollary holds.
Corollary 9. Assume that assumption holds. If there exist three symmetric and positive definite matrices , seven constants , , such that the following inequalities hold: where then the systems (1) and (5) are asymptotically synchronized.
In particular, when in (H1), the following corollary holds.
Corollary 10. Consider system (6) satisfying assumption with . If there exist three symmetric and positive definite matrices , seven constants , , such that the following inequalities hold: where then the systems (1) and (5) are asymptotically synchronized.
Remark 11. The sufficient conditions in Theorem 7 and Corollaries 9 and 10 are all independent of the delay parameter but rely on the maximum impulsive interval and the impulsive matrix , which plays a fundamental role when the size of the delay is unknown.
4. Numerical Results
In order to illustrate the feasibility of our above-established criteria in the preceding sections, we provide two concrete examples. Throughout the simulations, we use the IMEX implicit Euler method.
Example 12. Consider a two-dimensional chaotic delayed neural networks as the drive system (1), where the initial condition ; then , and the parameter matrices are given as follows:
The corresponding response system is designed as (5), where the initial condition . From [33], we know that the system (1) of Example 12 has a chaotic attractor which can be seen from Figure 1. Figures 1(a)–1(d) show the time response of state variables and the phase plots of the drive system and response system in Example 12 without impulsive control.
By choosing and then using MATLAB LMI toolbox and (31), we can obtain the following feasible solutions to LMIs and in Corollary 9:
Thus, from Corollary 9 we can calculate that the maximum impulsive interval satisfies .
With the same initial conditions as given above and , Figures 2(a)–2(d) depict the time response of state variables, the synchronization errors, and the phase plots of the drive system and response system in Example 12 under the impulsive controller .
(a)
(b)
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(d)
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Remark 13. In Table 1, we list corresponding to different and algorithms when the impulsive matrix is given. For each , the is obtained by solving the LMIs in Corollary 9 and (31) or (32). From the results in Table 1, we can see that the obtained maximum stable impulsive interval by using (31) is better than (32) when the impulsive matrix is given. And when are chosen in interval , we will have the same results which are chosen as .
Example 14. Consider the Ikeda-type neural network [34] as the drive system (1); when , the system (1) exhibits chaotic behavior (see Figure 3(a)). And the initial condition . It is easy to obtain that .
The corresponding response system is designed as (5), where the initial condition .
By choosing and then using MATLAB LMI toolbox and (31), we can obtain the following feasible solutions to LMIs and in Corollary 10:
Hence, the designed impulsive controller is
With the same initial conditions as given above, the simulations of the synchronization error in Example 14 under the impulsive controller are shown in Figure 3(b).
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(b)
5. Conclusion
In this paper, the impulsive synchronization problem of chaotic delayed neural networks has been investigated. Some new criterions which ensure that the coupled chaotic delayed neural networks can be asymptotically synchronized have been derived in terms of linear matrix inequalities (LMIs) by using Lyapunov stability theorem, impulsive control theory, and linear matrix inequality (LMI) technique. The desired impulsive controller which is with respect to the stable impulsive interval and the impulsive matrix is established, its existence can be verified effectively by the simulations. It is worthwhile to mention that the positive constants set can increase the flexibility for the design of the impulsive controller. Moreover, the stable impulsive interval can be calculated combining MATLAB LMI toolbox and one of the two given equations objectively.
Finally, two illustrative examples are given to show the applicability and usefulness of the proposed results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to thank the editor and the anonymous reviewers for their careful comments and suggestions to improve the quality of the paper. This work is jointly supported by Opening Fund of Artificial Intelligence Key Laboratory of Sichuan Province under Grant no. 2011RK01, Opening Fund of Geomathmathematics Key Laboratory of Sichuan Province under Grant no. scsxdz2011010, the Scientific Research Fund of Sichuan University of Science and Engineering under Grant no. 2011PY08, and the Graduate Innovation Fund of Sichuan University of Science and Engineering under Grant no. y2013021.