Adaptive synchronization of chaotic neural networks with time delays via delayed feedback control

Transcription

1 2017 º 12 È 31 4 ½ Dec Communication on Applied Mathematics and Computation Vol.31 No.4 DOI /j.issn Adaptive synchronization of chaotic neural networks with time delays via delayed feedback control LUO Yujing, LI Xiaolin (College of Sciences, Shanghai University, Shanghai , China) Abstract This paper is concerned with the synchronization of two delayed chaotic neural networks with a new adaptive feedback controller which includes state coupling control and delayed state coupling control. By constructing a new Lyapunov functional, we obtain a new adaptive synchronization criteria based on the Lyapunov s stability theory, the linear matrix inequality (LMI) technique and the adaptive feedback control technique. An example and its numerical simulation are given to illustrate the effectiveness of the results. Key words adaptive synchronization; delayed chaotic neural network; delayed feedback control 2010 Mathematics Subject Classification O193 Chinese Library Classification 34D06 ÜÑÐ ÑÐÐÖĐÒÛÔ ( ²Ç ) ÝÚ Á Î É Ã ÏÄ; Å Êà ³À¼» ³À¼»ÆƲ LMI ÏÄ ÍƲ «Åµ Î ÏÄ Ä Ì ¹ ±ÂË«Å ÏÄ Î Î Í 2010 ÓÙ O193 Õ 34D06 Øß A Þ (2017) Received ; Revised Project supported by the National Natural Science Foundation of China ( ) Corresponding author LI Xiaolin, research interests are neural network theory and its applications. xlli@i.shu.edu.cn

2 444 Communication on Applied Mathematics and Computation Vol Introduction Synchronization which means agreement or correlation of different processes in time is an important property for dynamical systems. In application, there are different types of synchronization such as chaotic synchronization [1-8], phase synchronization [9], and cluster synchronization [10]. Especially, the chaotic synchronization in nonlinearly dynamical systems has widely been investigated for decades due to its potential applications in many areas such as secure communication, image processing, and harmonic oscillation generation. Therefore, different kinds of schemes have been studied for chaotic synchronization such as adaptive control [2,3,6-8], feedback control [1-3,11-12], and pinning control [2,13]. In practice, there usually exists delay in spreading of the information due to the finite speed of transmission as well as traffic congestions. The introducing of delay into neural networks is of necessity for practical design and their application in real world. Since Marcus and Westervelt firstly introduced the simple delay into neural networks, the delayed neural networks which are indeed special complex networks have been reported with much complex and unpredicted behaviors including stable equilibrium, periodic oscillations, bifurcation and chaotic attractors [14]. Qiao, et al. [1] studied the synchronization of coupled delayed systems via the unidirectional linear error feedback approach. For a special case where the information on the size of delay is available, Yu and Cao [15] studied the synchronization of stochastic delayed neural networks by using a delayed feedback controller. In many real networks, only output signals can be measured, so the synchronization of chaotic delayed neural networks with nonlinear state feedback has been investigated by different approaches, see [1,13]. In addition, Li, et al. [16] further reported the synchronization of stochastic perturbed chaotic neural networks via a nonlinearly delayed feedback controller. At the same time, the adaptive synchronization issues of chaotic neural networks with delay have also drawn much research interest, and many synchronization results on these topics have been reported in the literatures [2, 6, 8]. For instance, He and Cao [6] studied the adaptive synchronization problem in a class of chaotic neural networks by utilizing a simple state-feedback controller. Later, Cai, et al. [2] considered the adaptive synchronization of delayed chaotic neural networks by employing a nonlinear state-feedback controller. Wang, et al. [10] presented a complex dynamical network with coupling time-varying delays and further obtained a new criterion for cluster synchronization by using an adaptive controller. This result has been restricted to the systems in [10]. Very recently, Zhu, et al. [17] presented stochastic neural networks of neutral-type with mixed time-delays and also obtained the adaptive synchronization conditions. However, these results in [10,17] cannot be applied to general chaotic delayed neural networks. Motivated by these, we will put more attention into the adaptive synchronization of general chaotic delayed neural networks via a new feedback controller. To the best of

3 No. 4 LUO Yujing, et al.: Adaptive synchronization of chaotic neural networks 445 our knowledge, up to now, this design of controller is different from others. Based on the Lyapunov s stability theory and the linear matrix inequality (LMI) technique, some sufficient conditions are obtained by using error feedback and adaptive control schemes. The rest of this paper is organized as follows. In Section 1, the delayed neural network model is formulated, and some necessary definitions and hypotheses are given. In Section 2, the main theoretical results are presented, and the numerical simulation results are shown in Section 3. In Section 4, conclusions of this paper are given. 1 Neural networks model and preliminaries The neural networks with delay considered in this paper are described by the following differential equations: ẋ(t) = Cx(t) + Af(x(t)) + Bf(x(t τ)) + I, (1) where τ 0 is the transmission delay; x(t) = (x 1 (t), x 2 (t),, x n (t)) T R n is the vector associated with the neurons at time t, and n corresponds to the number of neurons in the networks; C = diag(c 1, c 2,, c n ) > 0 indicates the self-feedback matrix; A = (a ij ) n n and B = (b ij ) n n represent the weight matrix and the delayed connectivity weight matrix, namely, a ij, b ij denote the strengths of connectivity between the cell i and j at time t and time t τ, respectively; I = (I 1, I 2,, I n ) T R n is a constant external input vector; (f 1 (x 1 (t)), f 2 (x 2 (t)),, f n (x n (t))) T R n denotes the activation functions of the neurons. The initial conditions of system (1) are given by x i (t) = φ i (t) C([ τ, 0], R), where C([ τ, 0], R) denotes the sets of all continuous functions from [ τ, 0] to R. It has been shown that networks (1) can exhibit the chaotic behavior if the networks parameters and time delay are appropriately chosen [14]. In this paper, the synchronization of such networks will be investigated. We consider model (1) as the drive system. The response system is given by the following equation: ẏ(t) = Cy(t) + Af(y(t)) + Bf(y(t τ)) + I + U(t), (2) where C, A, and B are matrices which are the same as in (1), U(t) = [u 1 (t), u 2 (t),, u n (t)] T is the controller. Let e(t) = y(t) x(t) be the synchronization error. Then, the dynamical system of the synchronization error can be written as follows: ė(t) = Ce(t) + Ag(e(t)) + Bg(e(t τ)) + U(t), (3) where g(e(t)) = f(x(t) + e(t)) f(x(t)). Throughout this paper, the activation functions f i ( ), i = 1, 2,, n satisfy the following assumption:

4 446 Communication on Applied Mathematics and Computation Vol. 31 H1 For i 1, 2,, n, the neuron activation functions f i ( ) satisfy where v i and v + i are some constants. v i f i(x) f i (y) x y v + i, (4) In addition, one can check that the functions g i ( ) satisfy the following condition: v i g i(x) g i (y) v + i (5) x y for all x, y R, x y, i = 1, 2,, n. Next, we first introduce the definition of synchronization for the two coupled neural networks (1) and (2), and then present a lemma, which is needed to prove our main result. Definition 1 [1] System (1) and (2) are said to be globally synchronized by the adaptive controller if there holds lim x i(t) y i (t) = 0, i = 1, 2,, n, t where denotes the Euclidean norm on R n. Lemma 1 [18] For any vectors a, b R n, the inequality holds, in which X is any matrix with X > 0. ±2a T Xb a T Xa + b T Xb 2 Synchronization criteria For simplicity, we denote ( v V 1 = diag(v1 v+ 1,, v n v n + ), V 2 = diag 1 + v 1 +,, v n + v + ) n. 2 2 In this section, we will prove that the adaptive synchronization between system (1) and system (2) can be reached by adding the adaptive controller based on the Lyapunov s stability theory and the LMI method. Theorem 1 Under the assumption H1, the two coupled delayed neural networks (1) and (2) can be synchronized, if there exist three n n positive definite diagonal matrices P =diag(p 1, p 2,, p n )>0, U 1 =diag(u 11, u 12,, u 1n )>0, U 2 =diag(u 21, u 22,, u 2n )>0, two n n positive definite matrices Q > 0, H > 0 and suitable positive scalars α 1, α 2, such that the following matrix inequality holds: Z 1 0 Z 2 PB M = Z 3 0 V 2 U 2 Z 4 0 < 0, (6) Z 5

5 No. 4 LUO Yujing, et al.: Adaptive synchronization of chaotic neural networks 447 where Z 1 = 2PC + Q + 2P V 1 U 1 2α 1 P, Z 2 = PA + V 2 U 1, Z 3 = Q 2α 2 P V 1 U 2, Z 4 = H U 1, Z 5 = H U 2. In addition, the adaptive feedback controller is designed as U(t) = L(y(t) x(t)) + K(y(t τ) x(t τ)), (7) where the matrices L = diag(l 1, l 2,, l n ), K = diag(k 1, k 2,, k n ) > 0, and the feedback strength matrices K and L are updated by the following laws: where ε i and σ i (i = 1, 2,, n) are arbitrary nonzero constants. Proof Construct the Lyapunov function as V (t) = e T (t)pe(t) + + n t t τ p i (l i + α 1 ) 2 ε 2 i l i = ε 2 ie 2 i(t), (8) k i = σi 2 e2 i (t τ), (9) e T (s)(q + P)e(s)ds + + n p i (k i + α 2 ) 2 σ 2 i t t τ n g T (e(s))hg(e(s))ds ki 3 p i σi 2. (10) Calculating the derivative of (10) along the trajectories of (3), we obtain V = 2e T (t)pė(t) + e T (t)(q + P)e(t) e T (t τ)(q + P)e(t τ) + g T (e(t))hg(e(t)) n g T l i + α 1 n k i + α 2 n ki (e(t τ))hg(e(t τ)) + 2 p i l ε 2 i + 2 p i k 2 i σi 2 i + 2 p i k σi 2 i = 2e T (t)p[ Ce(t) + Ag(e(t)) + Bg(e(t τ)) + U(t)] + e T (t)(q + P)e(t) e T (t τ)(q + P)e(t τ) + g T (e(t))hg(e(t)) g T (e(t τ))hg(e(t τ)) 2e T (t)ple(t) 2α 1 e T (t)pe(t) 2e T (t τ)pke(t τ) 2α 2 e T (t τ)pe(t τ) 2e T (t τ)k T PKe(t τ) = 2e T (t)pce(t) + 2e T (t)pag(e(t)) + 2e T (t)pbg(e(t τ)) + 2e T (t)pke(t τ) +e T (t)(q + P)e(t) e T (t τ)(q + P)e(t τ) + g T (e(t))hg(e(t)) g T (e(t τ))hg(e(t τ)) 2α 1 e T (t)pe(t) 2α 2 e T (t τ)pe(t τ) 2e T (t τ)pke(t τ) 2e T (t τ)k T PKe(t τ). (11) From Lemma 1, we have 2e T (t)pke(t τ) e T (t)pe(t) + e T (t τ)k T PKe(t τ), (12) 2e T (t τ)pke(t τ) e T (t τ)pe(t τ) + e T (t τ)k T PKe(t τ). (13)

6 448 Communication on Applied Mathematics and Computation Vol. 31 On the other hand, from H1 it follows that [ e(t) ] T [ U1 V 1 U 1 V 2 g(e(t)) U 1 V 2 U 1 ][ e(t) g(e(t)) ] 0. (14) Similarly, one has [ e(t τ) ] T [ V1 U 2 V 2 U 2 ][ e(t τ) ] g(e(t τ)) V 2 U 2 U 2 g(e(t τ)) 0, (15) where U 1 > 0, U 2 > 0 are diagonal matrices. Substituting (12) (15) into (11), we derive V e T (t)( 2PC + Q + 2P 2α 1 P)e(t) + 2e T (t)pag(e(t)) + 2e T (t)pbg(e(t τ)) +e(t τ) T ( Q 2α 2 P)e(t τ) + g T (e(t))hg(e(t)) g T (e(t τ))hg(e(t τ)) [ ] T [ ][ ] e(t) V1 U 1 V 2 U 1 e(t) g(e(t)) [ e(t τ) g(e(t τ)) V 2 U 1 U 1 g(e(t)) ] T [ ][ V1 U 2 V 2 U 2 e(t τ) V 2 U 2 U 2 g(e(t τ)) = e T (t)( 2PC + Q + 2P 2α 1 P V 1 U 1 )e(t) + 2e T (t)(pa + V 2 U 1 )g(e(t)) +2e T (t)pbg(e(t τ)) + e(t τ) T ( Q 2α 2 P V 1 U 2 )e(t τ) +2e T (t τ)v 2 U 2 g(e(t τ)) + g T (e(t))(h U 1 )g(e(t)) g T (e(t τ))(h + U 2 )g(e(t τ)) = (e T (t), e T (t τ), g T (e(t)), g T (e(t τ)))m(e T (t), e T (t τ), g T (e(t)), g T (e(t τ))) T. (16) Hence, we can induce V < 0. Therefore, applying the Lyapunov s stability theory for the differential delay equations, we can conclude that the zero solution of system (3) is stable. That is, the synchronization between delayed chaotic neural networks (1) and (2) is proved. Remark 1 In this paper, we give a new Lyapunov function (10). Remark 2 Li, et al. [16] investigated the synchronization of delayed neural networks via a delayed state feedback controller Ge(t) + G 1 e(t τ(t)). However, in the paper, we consider the adaptive synchronization of chaotic delayed neural networks by using the adaptive controller (7). He and Cao [6] gave an adaptive controller ǫe(t) with the adaptive law ǫ 2 i = α i e 2 i (t). In comparison with [6], we give a new delayed adaptive feedback controller (7) which also includes a new adaptive law (9). In [19], the state feedback controller u(t) = K(y(t) x(t)), with the adaptive law k i = ϕ i e 2 i (t) + ϕ id i e i (t)e i (t τ) was proposed. Lately, Xu, et al. [20] gave an adaptive feedback controller u i = α i (t)(e i (t) + g(e i (t))) ]

7 No. 4 LUO Yujing, et al.: Adaptive synchronization of chaotic neural networks 449 with the adaptive law α i (t) = θ i (e i (t) + g(e i (t τ))) T (e i (t) + g(e i (t τ))). It is noticed that Zheng, et al. [19-20] investigated the adaptive synchronization of neutraltype networks. In comparison with [19-20], in this paper, we consider the adaptive synchronization of delayed chaotic neural networks by utilizing a new adaptive controller (7) and adaptive law (8) and (9). 3 Numerical example and simulation In this section, we give an example so as to demonstrate the effectiveness of the theoretical result. Example 1 Consider the following delayed neural networks: ẋ(t) = Cx(t) + Af(x(t)) + Bf(x(t τ)) + I, (17) where ( 1 0 ) ( ) ( ) ( 0 ) C = 0 1, A = 5 3.2, B = , I = 0, f(x(t)) = tanh(x). For the drive system (17), we construct a corresponding response system as follows: ẏ(t) = Cy(t) + Af(y(t)) + Bf(y(t τ)) + I + U(t). (18) Let α 1 = α 2 = 10, using the Matlab LMI toolbox, we can obtain the following feasible solutions to the LMI (6): ( ) ( ) P =, Q =, ( ) ( ) H =, U 1 =, ( ) U 2 = In the following, we take l 1 (0) = 3, l 2 (0) = 3, k 1 (0) = 0.5, k 2 (0) = 0.5, ε 1 = 0.5, ε 2 = 0.5, σ 1 = 0.3, and σ 2 = 0.3. Figure 1 shows that delayed neural networks (17) actually have a double-scroll-like chaotic attractor with initial values x 1 (t) = 0.2, x 2 (t) = 1.3. Figure 2 shows the evolution of the adaptive coupling strength l 1, l 2. Figure 3 shows

8 450 Communication on Applied Mathematics and Computation Vol. 31 the evolution of the adaptive coupling strength k 1, k 2. The evolutions of variables x 1 (t), y 1 (t) and x 2 (t), y 2 (t) of coupled neural networks (17) and (18) are given in Figs. 4 and 5, respectively. Figure 6 shows that the synchronization errors e 1 (t), e 2 (t) converge to zero. The simulation results show that by using the proposed adaptive scheme, the complete synchronization of two coupled chaotic neural networks can be achieved. Fig. 1 Chaotic attractors of delayed neural network (17) Fig. 2 Evolution of adaptive coupling strength l 1, l 2 Fig. 3 Evolution of adaptive coupling strength k 1, k 2

9 No. 4 LUO Yujing, et al.: Adaptive synchronization of chaotic neural networks 451 Fig. 4 Evolution of variables x 1(t), y 1 (t) of coupled neural networks (17) and (18) Fig. 5 Evolution of variables x 2(t), y 2 (t) of coupled neural networks (17) and (18) Fig. 6 Evolution of synchronization errors e 1(t), e 2(t) 4 Conclusions A new adaptive feedback controller has been designed to achieve the synchronization for the delayed neural networks. By constructing a new Lyapunov function and combining the Lyapunov s stability theory and the LMI technique, we obtain the adaptive synchronization

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