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

Abstract

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.

1. 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 𝑡 is often the voice from the transmitter side at time 𝑡−𝜏 [10]. So, in many cases, it is more reasonable to require the slave system to synchronize the master system with a time delay 𝜏. 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.

2. Problem Formulation and Preliminaries

In this paper, we consider the chaotic time-delay systems described by 𝑥̇(𝑡)(𝑡)(𝑥(𝑡(𝑥(𝑥(𝑡)(𝑡)=𝐶𝑥+𝐴𝑓))+𝐵𝑔𝑡−𝜏)),𝑡>0,=𝜑,−𝜏≤𝑡≤0, (1) where 𝑥∈𝑅^𝑚 denotes the state vector, 𝐶, 𝐴, and 𝐵∈𝑅^𝑚×𝑚 are constant matrices, 𝜏 is the time delay, 𝑓,𝑔∶𝑅^𝑚→𝑅^𝑚 are nonlinear functions satisfying the Lipschitz condition; namely, there exist positive constants 𝐿_𝑓, 𝐿_𝑔 such that 𝑓(𝛼)−𝑓𝛽≤𝐿_𝑓,𝑔(𝛼)𝛽𝛼−𝛽−𝑔≤𝐿_𝑔𝛼−𝛽,∀𝛼,𝛽∈𝑅^𝑛. (2)

In order to lag-synchronize system (1) by means of periodically intermittent feedback control, we assume that the corresponding slave system as 𝑦̇(𝑡)(𝑡)𝑦(𝑡)𝑦()(𝑡)𝑦(𝑡)(𝑡)=𝐶𝑦+𝐴𝑓+𝐵𝑔𝑡−𝜏+𝑢,𝑡>0,=𝜓,−𝜏≤𝑡≤0, (3) where 𝑦∈𝑅^𝑚 denotes the state vector, and 𝑢(𝑡) is the intermittent linear state feedback control gain defined as follows: 𝑢(𝑡)=𝑘𝑥()(𝑡)()𝑡−𝜃−𝑦,𝑛𝑇≤𝑡<𝑛𝑇+𝜎𝑇,0,𝑛𝑇+𝜎𝑇≤𝑡<𝑛+1𝑇, (4) where 𝑘 denotes control strength, 0<𝜎<1denotes switching rate, 𝑇 denotes control period, and 𝜃 is the transmittal delay.

Defining the lag synchronization error between the systems (1) and (3) as 𝑒(𝑡)=𝑦(𝑡)−𝑥(𝑡−𝜃), we have the following error dynamical system: 𝑒̇(𝑡)=𝑦̇(𝑡)−𝑥̇()(𝑡)𝑓𝑦(𝑡)(𝑥(𝑔𝑦()(𝑥((𝑡).𝑡−𝜃=𝐶𝑒+𝐴−𝑓𝑡−𝜃))+𝐵𝑡−𝜏−𝑔𝑡−𝜏−𝜃))+𝑢 (5)

Under the control of the form (4), the system (5) can be rewritten as 𝑒̇(𝑡)=()𝑒(𝑡)𝑓𝑦(𝑡)(𝑥(𝑔𝑦()(𝑥(,𝑒̇(𝑡)(𝑡)𝑓𝑦(𝑡)(𝑥(𝑔𝑦()(𝑥(,()𝐶−𝑘𝐼+𝐴−𝑓𝑡−𝜃))+𝐵𝑡−𝜏−𝑔𝑡−𝜏−𝜃))𝑛𝑇≤𝑡<𝑛𝑇+𝜎𝑇,=𝐶𝑒+𝐴−𝑓𝑡−𝜃))+𝐵𝑡−𝜏−𝑔𝑡−𝜏−𝜃))𝑛𝑇+𝜎𝑇≤𝑡<𝑛+1𝑇. (6)

In order to define the initial condition of system (6), we supplement the state 𝑥(𝑡) on [−𝜃−𝜏,−𝜏] as 𝑥(𝑡)=𝜑(−𝜏),−𝜏−𝜃≤𝑡<−𝜏. (7)

Introduce a new notation 𝜑(𝑡) as 𝜑(𝑡)=𝜑(𝑡)𝜑(),−𝜏≤𝑡≤0,−𝜏,−𝜏−𝜃≤𝑡≤−𝜃. (8) Then initial condition of system (6) is defined by 𝑒(𝑡)=𝜓(𝑡)−𝜑(𝑡−𝜃),−𝜏≤𝑡≤0. (9)

To establish the main result of this paper, the following preliminaries are necessary.

Lemma 1 (see [14]). Given any real matrices Σ₁, Σ₂, Σ₃ of appropriate dimensions and a scalar 𝑠>0, such that 0<Σ₃=Σ^𝑇₃. Then the following inequality holds: Σ^𝑇₁Σ₂+Σ^𝑇₂Σ₁≤𝑠Σ^𝑇₁Σ₃Σ₁+𝑠⁻¹Σ^𝑇₂Σ₃⁻¹Σ₂. (10)

Lemma 2 (Halanay inequality [15]). Let 𝜔∶[𝜇−𝜏,∞)→[0,∞) be a continuous function such that 𝜔̇(𝑡)≤−𝑎𝜔(𝑡)+𝑏max𝜔_𝑡 (11) is satisfied for 𝑡≥𝑢. If 𝑎>𝑏>0, then 𝜔(𝑡)≤max𝜔_𝜇exp−𝛾𝑡−𝜇,𝑡≥𝜇, (12) where max𝜔_𝑡=sup_{𝑡−𝜏≤𝜃≤𝑡}𝜔(𝜃), and 𝛾>0 is the smallest real root of the equation 𝑎−𝛾−𝑏exp𝛾𝜏=0. (13)

Lemma 3 (see [16]). Let 𝜔 be a nonnegative function defined on the interval [𝑡₀−𝜏,∞) and continuous on the subinterval [𝑡₀,∞). Assume that 𝜔̇(𝑡)≤𝑎𝜔(𝑡)+𝑏𝜔(𝑡−𝜏) (14) is satisfied for 𝑡≥𝑡₀. If 𝑎>0, 𝑏>0, then 𝜔(𝑡)≤𝜔_𝑡0exp𝜂𝑡−𝑡₀+𝜏,𝑡≥𝑡₀, (15) where 𝜔_𝑡0=sup_{𝑡0−𝜏≤𝜃≤𝑡0}𝜔(𝜃) and 𝜂>0 is the unique root of the equation 𝑎+𝑏𝑒^−𝜂𝜏=𝜂.

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 ℎ₁>0, ℎ₂>0, and 𝑠_𝑖>0 (𝑖=1,2,3,4,5,6) such that(i)𝐶+𝐶^𝑇−2𝑘𝐼+𝑠₁𝐴𝐴^𝑇+𝑠₁⁻¹𝐿²_𝑓𝐼+𝑠₂𝐵𝐵^𝑇+ℎ₁𝐼≤0,(ii)𝐶+𝐶^𝑇+𝑠₃𝐴𝐴^𝑇+𝑠₃⁻¹𝐿²_𝑓𝐼+𝑠₄𝐵₂𝐵^𝑇₂−ℎ₂𝐼≤0,(iii)ℎ₁>𝑠₂⁻¹𝐿²_𝑔, and 𝛾(𝜎𝑇−𝜏)−𝜂(𝑇−𝜎𝑇+𝜏)>0, where 𝛾 is the unique positive root of the equation ℎ₁−𝛾−𝑠₂⁻¹𝐿²_𝑔exp{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)^𝑇𝑒(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: 𝑉̇(𝑡)=2𝑒(𝑡)^𝑇𝑒̇(𝑡)=2𝑒(𝑡)^𝑇)𝑒(𝑡)𝑓𝑦(𝑡)(𝑥(𝑔𝑦()(𝑥((𝑡)[(𝐶−𝑘𝐼+𝐴−𝑓𝑡−𝜃))+𝐵𝑡−𝜏−𝑔𝑡−𝜏−𝜃))=𝑒^𝑇𝐶^𝑇𝑒(𝑡)(𝑡)+𝐶−2𝑘𝐼+2𝑒^𝑇𝐴𝑓𝑦(𝑡)(𝑥((𝑡)−𝑓𝑡−𝜃))+2𝑒^𝑇𝐵𝑔𝑦(𝑡−𝜏)−𝑔(𝑥(𝑡−𝜏−𝜃)). (17)

Using Lemma 1, we have the following estimation: 2𝑒(𝑡)^𝑇𝐴𝑓𝑦(𝑡)−𝑓(𝑥(𝑡−𝜃))≤𝑠₁𝑒(𝑡)^𝑇𝐴𝐴^𝑇𝑒(𝑡)+𝑠₁⁻¹𝑓𝑦(𝑡)−𝑓(𝑥(𝑡−𝜃))^𝑇×𝑓𝑦(𝑡)−𝑓(𝑥(𝑡−𝜃))≤𝑠₁𝑒(𝑡)^𝑇𝐴₂𝐴^𝑇₂𝑒(𝑡)+𝑠₁⁻¹𝐿²_𝑓𝑒(𝑡)^𝑇𝑒(𝑡)=𝑒(𝑡)^𝑇𝑠₁𝐴₂𝐴^𝑇₂+𝑠₁⁻¹𝐿²_𝑓𝐼𝑒(𝑡). (18)

Similarly, we have the following estimation for another term: 2𝑒(𝑡)^𝑇𝐵𝑔𝑦(𝑡−𝜏)−𝑔(𝑥(𝑡−𝜏−𝜃))≤𝑠₂𝑒(𝑡)^𝑇𝐵𝐵^𝑇𝑒(𝑡)+𝑠₂⁻¹𝐿²_𝑔𝑒(𝑡−𝜏)^𝑇𝑒(𝑡−𝜏). (19)

Substituting these into (17), we have the following: 𝑉̇(𝑡)≤𝑒(𝑡)^𝑇𝐶+𝐶^𝑇𝑒(𝑡)(𝑡)−2𝑘𝐼+𝑒^𝑇𝑠₁𝐴𝐴^𝑇₂+𝑠₁⁻¹𝐿²_𝑓𝐼𝑒(𝑡)+𝑠₂𝑒(𝑡)^𝑇𝐵𝐵^𝑇𝑒(𝑡)+𝑠₂⁻¹𝐿²_𝑔𝑒(𝑡−𝜏)^𝑇𝑒()(𝑡)𝑡−𝜏=𝑒^𝑇𝐶+𝐶^𝑇−2𝑘𝐼+𝑠₁𝐴𝐴^𝑇+𝑠₁⁻¹𝐿²_𝑓𝐼+𝑠₂𝐵𝐵^𝑇+ℎ₁𝐼𝑒(𝑡)−ℎ₁𝑒(𝑡)^𝑇𝑒(𝑡)+𝑠₂⁻¹𝐿²_𝑔𝑒(𝑡−𝜏)^𝑇𝑒(𝑡−𝜏)≤−ℎ₁𝑒(𝑡)^𝑇𝑒(𝑡)+𝑠₂⁻¹𝐿²_𝑔𝑒(𝑡−𝜏)^𝑇𝑒(𝑡−𝜏)≤−ℎ₁𝑉(𝑡)+𝑠₂⁻¹𝐿²_𝑔𝑉(𝑡−𝜏). (20)

Similarly, when 𝑛𝑇+𝜎𝑇≤𝑡<(𝑛+1)𝑇, one obtains 𝑉̇(𝑡)=2𝑒(𝑡)^𝑇𝑒̇(𝑡)=2𝑒(𝑡)^𝑇(𝑡)𝑓𝑦(𝑡(𝑥(𝑔𝑦()(𝑥((𝑡)𝐶𝑒+𝐴)−𝑓𝑡−𝜃))+𝐵𝑡−𝜏−𝑔𝑡−𝜏−𝜃))≤𝑒^𝑇𝐶+𝐶^𝑇𝑒(𝑡)+𝑒(𝑡)^𝑇𝑠₃𝐴𝐴^𝑇+𝑠₃⁻¹𝐿²_𝑓𝐼𝑒(𝑡)+𝑠₄𝑒(𝑡)^𝑇𝐵𝐵^𝑇𝑒(𝑡)+𝑠₄⁻¹𝐿²_𝑔𝑒(𝑡−𝜏)^𝑇𝑒()(𝑡)𝑡−𝜏=𝑒^𝑇𝐶+𝐶^𝑇+𝑠₃𝐴𝐴^𝑇+𝑠₃⁻¹𝐿²_𝑓𝐼+𝑠₄𝐵𝐵^𝑇−ℎ₂𝐼𝑒(𝑡)+ℎ₂𝑒(𝑡)^𝑇𝑒(𝑡)+𝑠₄⁻¹𝐿²_𝑔𝑒(𝑡−𝜏)^𝑇𝑒(𝑡−𝜏)≤ℎ₂𝑒(𝑡)^𝑇𝑒(𝑡)+𝑠₄⁻¹𝐿²_𝑔𝑒(𝑡−𝜏)^𝑇𝑒(𝑡−𝜏)≤ℎ₂𝑉(𝑡)+𝑠₄⁻¹𝐿²_𝑔𝑉(𝑡−𝜏). (21)

Therefore, 𝑉̇(𝑡)≤−ℎ₁𝑉(𝑡)+𝑠₂⁻¹𝐿²_𝑔𝑉(),𝑉̇(𝑡)𝑡−𝜏𝑛𝑇≤𝑡<𝑛𝑇+𝜎𝑇;≤ℎ₂𝑉(𝑡)+𝑠₄⁻¹𝐿²_𝑔𝑉(),()𝑡−𝜏𝑛𝑇+𝜎𝑇≤𝑡<𝑛+1𝑇. (22)

By Lemmas 2 and 3, following the same line of argument of the proof of Theorem 1 of [16], we can obtain ‖𝑥(𝑡)‖²(𝑥(𝑡≤=𝑉))𝜔(0)−𝛾()(𝑡exp𝜎𝑇−𝜏−𝜂𝑇−𝜎𝑇+𝜏)𝑇,forany𝑡≥0. (23)

This implies that the error system (6) is globally exponentially stable, and the following estimate holds: ‖𝑥(𝑡≤𝜙−1)‖exp2𝛾()(𝑡𝜎𝑇−𝜏−𝜂𝑇−𝜎𝑇+𝜏)𝑇,𝑡≥0. (24)

The proof is thus completed.

For computational purpose, we now present a numerically tractable lag synchronization conditions. Let 𝑠₁=𝐿_𝑓𝜆_max𝐴𝐴^𝑇,𝑠₂=𝑒^0.5𝛾𝜏𝐿_𝑔𝜆_max𝐵𝐵^𝑇,𝑠₃=𝑠₁,𝑠₄=𝑒^{−0.5𝜂𝜏}𝐿_𝑔𝜆_max𝐵𝐵^𝑇,ℎ^∗₁=2𝑘−𝜆_max𝐶+𝐶^𝑇−2𝜆_max𝐴𝐴^𝑇𝐿_𝑓−𝑒^0.5𝛾𝜏𝜆_max𝐵𝐵^𝑇𝐿_𝑔,ℎ^∗₂=𝜆_max𝐶+𝐶^𝑇+2𝜆_max𝐴𝐴^𝑇𝐿_𝑓+𝜆_max𝐵𝐵^𝑇𝐿_𝑔+1, (25) where ℎ^∗₁≥ℎ₁, ℎ^∗₂≤ℎ₂.

One obtains from Theorem 4 the following corollary.

Corollary 5. If there exist constants 𝛾>0, k and 𝜎 satisfying 0<𝜎<1 such that 𝑘+𝜆_max(𝐶)−𝜆_max𝐴𝐴^𝑇𝐿²_𝑓−𝑒^0.5𝛾𝜏𝜆_max𝐵𝐵^𝑇𝐿²_𝑔−12𝛾()()𝛾=0,𝜎𝑇−𝜏−𝜂𝑇−𝜎𝑇+𝜏>0, (26) where 𝜂>0 is the root of the equation −𝜆_max(𝐶)+𝜆_max𝐴𝐴^𝑇𝐿²_𝑓+𝑒^{−0.5𝜂𝜏}𝜆_max𝐵𝐵^𝑇𝐿²_𝑔−12𝜂=0. (27) 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 𝑇, Corollary 5 allows us to estimate the feasible region 𝐷 of control parameters (𝑘,𝜎). One observes that the control strength 𝑘 can be estimated as follows: 𝑘>𝑘^∗=−𝜆_max(𝐶)+𝜆_max𝐴𝐴^𝑇𝐿²_𝑓+𝑒^0.5𝛾𝜏𝜆_max𝐵𝐵^𝑇𝐿²_𝑔+12𝛾. (28) From (32), we can estimate the feasible region 𝐷 of control parameters (𝑘,𝜎) 𝐷=(𝑘,𝜎)∣𝑘>𝑘^∗=−𝜆_max(𝐶)+𝜆_max𝐴𝐴^𝑇𝐿²_𝑓+𝑒^0.5𝛾𝜏𝜆_max𝐵𝐵^𝑇𝐿²_𝑔+12𝛾,0<𝜎<1. (29)

4. 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.

Example 7. Consider the Lu neural oscillator described by following delayed differential equations [17] 𝑥̇(𝑡)=−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑔(𝑥(𝑡−1)), (30) where ,,,𝐶=1001𝐴=3.05.00.12.0𝐵=−2.50.20.1−1.5 (31) and 𝑓(𝑥(𝑡))=𝑔(𝑥(𝑡))=tanh(𝑥(𝑡)).

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 by 𝑦̇(𝑡)=−𝐶𝑦(𝑡)+𝐴𝑓𝑦(𝑡)+𝐵𝑔𝑦(𝑡−1), (32) where 𝑓(𝑦(𝑡))=𝑔(𝑦(𝑡))=tanh(𝑦(𝑡)).

Figure 1 

The chaotic attractor of the Lu oscillator described by (30) with initial value 𝑥1(𝑠)=0.2, 𝑥2(𝑠)=−0.5, for 𝑠∈[−1,0].

In this example, one observes that 𝐿_𝑓=𝐿_𝑔=1, and 𝜏=1. For numerical simulation, we select 𝑇=8, 𝜎=0.8, 𝜃=0.01, and 𝑘=20 and plot the lag synchronization error curve as shown in Figure 2.

Figure 2 

Synchronization error curve with 𝑇=8, 𝜎=0.8, and 𝑘=20.

5. 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.