Lag Synchronization of Coupled Delayed Chaotic Neural Networks by Periodically Intermittent Control

and Applied Analysis 3 3. Main Results In this section, we shall stabilize lag synchronization error system based on Lyapunovmethod and the intermittent control technique. Theorem 4. Suppose that there exist positive scalars h 1 > 0, h 2 > 0, and s i > 0 (i = 1, 2, 3, 4, 5, 6) such that (i) C + CT − 2kI + s 1 AA T + s −1 1 L 2 f I + s 2 BB T + h 1 I ≤ 0, (ii) C + CT + s 3 AA T + s −1 3 L 2 f I + s 4 B 2 B T 2 − h 2 I ≤ 0, (iii) h 1 > s −1 2 L 2 g , and γ(σT − τ) − η(T − σT + τ) > 0, where γ is the unique positive root of the equation h 1 − γ − s −1 2 L 2 g exp{γτ} = 0. 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: V (t) = e(t) T e (t) . (16) When nT ≤ t < nT + σT, the derivative of (16) with respect to time t along the trajectories of the first subsystem of the system (6) is calculated and estimated as follows: ?̇? (t) = 2e(t) T ̇ e (t) = 2e(t) T [(C − kI) e (t) + A (f (y (t)) − f (x (t − θ))) +B (g (y (t − τ)) − g (x (t − τ − θ)))] = e(t) T (C T + C − 2kI) e (t) + 2e(t) T A (f (y (t)) − f (x (t − θ))) + 2e(t) T B (g (y (t − τ)) − g (x (t − τ − θ))) . (17) Using Lemma 1, we have the following estimation: 2e(t) T A (f (y (t)) − f (x (t − θ))) ≤ s 1 e(t) T AA T e (t) + s −1 1 (f (y (t)) − f (x (t − θ))) T × (f (y (t)) − f (x (t − θ))) ≤ s 1 e(t) T A 2 A T 2 e (t) + s −1 1 L 2 f e(t) T e (t) = e(t) T [s 1 A 2 A T 2 + s −1 1 L 2 f I] e (t) . (18) Similarly, we have the following estimation for another term: 2e(t) T B (g (y (t − τ)) − g (x (t − τ − θ))) ≤ s 2 e(t) T BB T e (t) + s −1 2 L 2 g e(t − τ) T e (t − τ) . (19) Substituting these into (17), we have the following: ?̇? (t) ≤ e(t) T (C + C T − 2kI) e (t) + e(t) T [s 1 AA T 2 + s −1 1 L 2 f I] e (t) + s 2 e(t) T BB T e (t) + s −1 2 L 2 g e(t − τ) T e (t − τ) = e(t) T [C + C T − 2kI + s 1 AA T +s −1 1 L 2 f I + s 2 BB T + h 1 I] e (t) − h 1 e(t) T e (t) + s −1 2 L 2 g e(t − τ) T e (t − τ) ≤ −h 1 e(t) T e (t) + s −1 2 L 2 g e(t − τ) T e (t − τ) ≤ −h 1 V (t) + s −1 2 L 2 g V (t − τ) . (20) Similarly, when nT + σT ≤ t < (n + 1)T, one obtains ?̇? (t) = 2e(t) T ̇ e (t) = 2e(t) T [Ce (t) + A (f (y (t)) − f (x (t − θ))) +B (g (y (t − τ)) − g (x (t − τ − θ)))] ≤ e(t) T (C + C T ) e (t) + e(t) T [s 3 AA T + s −1 3 L 2 f I] e (t) + s 4 e(t) T BB T e (t) + s −1 4 L 2 g e(t − τ) T e (t − τ) = e(t) T [C + C T + s 3 AA T +s −1 3 L 2 f I + s 4 BB T − h 2 I] e (t) + h 2 e(t) T e (t) + s −1 4 L 2 g e(t − τ) T e (t − τ) ≤ h 2 e(t) T e (t) + s −1 4 L 2 g e(t − τ) T e (t − τ) ≤ h 2 V (t) + s −1 4 L 2 g V (t − τ) . (21)


Introduction
After the seminal works of Pecora and Carroll [1], 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 [3], projective synchronization [4], phase synchronization [5], lag synchronization [6], and anticipating synchronization [7]. 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 − [10]. 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. [11] investigated lag synchronization between unidirectional coupled delayed Ikeda systems by using feedback control techniques. Li et al. [12] considered lag synchronization problem with an application in secure communication. Zhou et al. [13] 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.

Problem Formulation and Preliminaries
In this paper, we consider the chaotic time-delay systems described bẏ 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 aṡ where ∈ denotes the state vector, and ( ) is the intermittent linear state feedback control gain defined as follows: where denotes control strength, 0 < < 1denotes switching rate, denotes control period, and is the transmittal delay. Defining the lag synchronization error between the systems (1) Under the control of the form (4), the system (5) can be rewritten aṡ In order to define the initial condition of system (6), we supplement the state ( ) on [− − , − ] as Introduce a new notation ( ) aŝ Then initial condition of system (6) is defined by To establish the main result of this paper, the following preliminaries are necessary.
Abstract and Applied Analysis 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 ℎ 1 > 0, ℎ 2 > 0, and > 0 ( = 1, 2, 3, 4, 5, 6) such that Proof. Consider the following Lyapunov function: When ≤ < + , the derivative of (16) with respect to time along the trajectories of the first subsystem of the system (6) is calculated and estimated as follows: 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 + ≤ < ( + 1) , one obtainṡ By Lemmas 2 and 3, following the same line of argument of the proof of Theorem 1 of [16], we can obtain for any ≥ 0.
This implies that the error system (6) is globally exponentially stable, and the following estimate holds: The proof is thus completed.

Corollary 5.
If there exist constants > 0, k and satisfying 0 < < 1 such that where > 0 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 ( , )

Numerical Example
In this section, we take Lu neural oscillator [17] 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.
This model was investigated by Lu in [17] where it is shown to be chaotic as shown in Figure 1. The corresponding slave system is given bẏ where ( ( )) = ( ( )) = tanh( ( )).
In this example, one observes that = = 1, and = 1. For numerical simulation, we select = 8, = 0.8, = 0.01, and = 20 and plot the lag synchronization error curve as shown in Figure 2.

Conclusions
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.