Modeling, Analysis, and Applications of Complex SystemsView this Special Issue
Lag Synchronization of Coupled Delayed Chaotic Neural Networks by Periodically Intermittent Control
We investigate the lag synchronization of coupled neural networks with time delay. Some sufficient conditions for lag synchronization will be derived by using Lyapunov stability theory and intermittent control. Compared to existing results, some less conservative conditions are derived to guarantee the stabilization of error system. The analytical results are confirmed by numerical simulations.
After the seminal works of Pecora and Carroll , the idea of synchronization of chaotic systems has received a great deal of interest among researchers from various fields. Over the past decades, several different regimes of chaos synchronization have been investigated, for example, complete synchronization [1, 2], generalized synchronization , projective synchronization , phase synchronization , lag synchronization , and anticipating synchronization . In recent years, Wen et al. have studied the stability analysis and synchronization of the memristor-based recurrent neural networks [8, 9]. On the other hand, it has been shown that the complete synchronization of chaos is practically impossible for the finite speed of signals. Chaotic lag synchronization appears as a coincidence of shift-in-time states of interactive systems. It is just synchronization lag that makes lag synchronization practically available. For instance, in the telephone communication system, the voice one hears on the receive side at time is often the voice from the transmitter side at time . So, in many cases, it is more reasonable to require the slave system to synchronize the master system with a time delay . Thus, it is of great importance to study lag synchronization and some results have been reported in this research area. Shahverdiev et al.  investigated lag synchronization between unidirectional coupled delayed Ikeda systems by using feedback control techniques. Li et al.  considered lag synchronization problem with an application in secure communication. Zhou et al.  studied lag synchronization of coupled chaotic delayed neural networks by using adaptive feedback control techniques.
In this paper, we have formulated the lag synchronization problem for chaotic neural networks by means of periodically intermittent control. Our interest focuses on the class of intermittent control with time duration; namely, the control is activated in certain nonzero time intervals and inactivated in other time intervals. Also, we will remove this limitation and design a general periodically intermittent controller for chaotic neural networks.
The rest of the paper is organized as follows. In Section 2, we formulate the problem of lag synchronization of coupled systems. In Section 3, a general convergence criterion for the lag synchronization error of coupled systems is established. Also, some corollaries and remarks are listed to show the advantage of this paper. In Section 4, numerical examples are given to show the theoretical results, which is followed by the conclusions in Section 5.
2. Problem Formulation and Preliminaries
In this paper, we consider the chaotic time-delay systems described by where denotes the state vector, , , and are constant matrices, is the time delay, are nonlinear functions satisfying the Lipschitz condition; namely, there exist positive constants , such that
In order to lag-synchronize system (1) by means of periodically intermittent feedback control, we assume that the corresponding slave system as where denotes the state vector, and is the intermittent linear state feedback control gain defined as follows: where denotes control strength, denotes switching rate, denotes control period, and is the transmittal delay.
In order to define the initial condition of system (6), we supplement the state on as
Introduce a new notation as Then initial condition of system (6) is defined by
To establish the main result of this paper, the following preliminaries are necessary.
Lemma 1 (see ). Given any real matrices , , of appropriate dimensions and a scalar , such that . Then the following inequality holds:
Lemma 2 (Halanay inequality ). Let be a continuous function such that is satisfied for . If , then where , and is the smallest real root of the equation
Lemma 3 (see ). Let be a nonnegative function defined on the interval and continuous on the subinterval . Assume that is satisfied for . If , , then where and is the unique root of the equation .
3. Main Results
In this section, we shall stabilize lag synchronization error system based on Lyapunov method and the intermittent control technique.
Theorem 4. Suppose that there exist positive scalars , , and () such that(i),(ii),(iii), and , where is the unique positive root of the equation . Then, error system (6) is globally exponentially stable; that is, the master system (1) and the slave system (3) achieve lag synchronization.
Proof. Consider the following Lyapunov function:
Using Lemma 1, we have the following estimation:
Similarly, we have the following estimation for another term:
Substituting these into (17), we have the following:
Similarly, when , one obtains
This implies that the error system (6) is globally exponentially stable, and the following estimate holds:
The proof is thus completed.
For computational purpose, we now present a numerically tractable lag synchronization conditions. Let where , .
One obtains from Theorem 4 the following corollary.
Corollary 5. If there exist constants , k and satisfying such that where is the root of the equation Then, the origin of system (6) is globally exponentially stabilized; that is, the master system (1) and the slave system (3) achieve lag synchronization.
Remark 6. Given the period , Corollary 5 allows us to estimate the feasible region of control parameters (). One observes that the control strength can be estimated as follows: From (32), we can estimate the feasible region of control parameters ()
4. Numerical Example
In this section, we take Lu neural oscillator  as an example to show the validity of the proposed results. The programs DDE23 in MATLAB are used to solve numerically the delay differential equations.
Example 7. Consider the Lu neural oscillator described by following delayed differential equations  where and .
In this example, one observes that , and . For numerical simulation, we select , , , and and plot the lag synchronization error curve as shown in Figure 2.
In this paper, we have formulated the lag synchronization problem for chaotic neural networks by means of periodically intermittent control. Stochastic synchronization criteria were established based on Lyapunov stability theory and linear matrix inequality techniques. Numerical simulations have showed the validity of theoretical result.
R. Mainieri and J. Rehacek, “Projective synchronization in three-dimensional chaotic systems,” Physical Review Letters, vol. 82, no. 15, pp. 3042–3045, 1999.View at: Google Scholar
M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Physical Review Letters, vol. 76, no. 11, pp. 1804–1807, 1996.View at: Google Scholar
M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “From phase to lag synchronization in coupled chaotic oscillators,” Physical Review Letters, vol. 78, no. 22, pp. 4193–4196, 1997.View at: Google Scholar
S. P. Wen and Z. G. Zeng, “Exponential stability analysis of memristor-based recurrent neural networks with time-varying delays,” Neurocomputing, vol. 97, pp. 233–240, 2012.View at: Google Scholar
A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, San Diego, Calif, USA, 1966.View at: MathSciNet
H. T. Lu, “Chaotic attractors in delayed neural networks,” Physics Letters A, vol. 298, pp. 109–116, 2002.View at: Google Scholar