Physics Letters A 364 (2007) 277–285

www.elsevier.com/locate/pla

Adaptive lag synchronization of unknown chaotic delayed neural networkswith noise perturbation ✩

Yonghui Sun, Jinde Cao ∗

Department of Mathematics, Southeast University, Nanjing 210096, China

Received 9 August 2006; received in revised form 9 November 2006; accepted 6 December 2006

Available online 14 December 2006

Communicated by J. Flouquet

Abstract

In this Letter, the adaptive lag synchronization issue of unknown chaotic delayed neural networks with noise perturbation is considered in detail.Via adaptive feedback control techniques, the suitable parameters update laws are proposed, then several sufficient conditions are derived to ensurelag synchronization of unknown delayed neural networks with or without noise perturbation. Moreover, some appropriate comparisons are madeto contrast to some of existing results about complete synchronization of chaotic systems with or without noise perturbation. Then, a numericalexample with its computer simulations is provided to illustrate and verify the effectiveness of the proposed adaptive scheme. Finally, the proposedadaptive feedback scheme is applied to the secure communication.© 2006 Elsevier B.V. All rights reserved.

Keywords: Adaptive lag synchronization; Noise perturbation; Mean square asymptotic stability; LaSalle invariance principle; Chaotic delayed neural networks;Adaptive feedback control

1. Introduction

Chaos synchronization has played a significant role in the field of nonlinear science for its many potential applications [1,2].Since the pioneering works by Pecora and Carroll in 1990 [3], lots of synchronization strategies have been developed to synchronizetwo identical or different chaotic systems, such as feedback control [4–6], impulsive control [7,8], adaptive control [9–11], etc.Recently, it has been revealed that if the network’s parameters and time delays are appropriately chosen, the delayed neural networks(DNNs) can exhibit some complicated dynamics and even chaotic behaviors [12–14], in addition to the stability and periodicoscillations investigated previously in [15–18,40,41]. Hence, there has been a great deal of activity studying the synchronization ofchaotic DNNs.

There are several types of chaos synchronization notations such as complete synchronization (CS), which means coincidenceof states of interacting systems, i.e., y(t) → x(t), t → ∞; lag synchronization (LS) appears as a coincidence of shifted-in-timestates of two systems y(t) → x(t − σ), t → ∞, with a propagation delay σ > 0, [19,20]; generalized synchronization (GS) isdefined as the presence of some functional relation between the states of drive and response, i.e., y(t) → Φ(x(t)), t → ∞, [21–23];anticipated synchronization (AS) [24], etc., and other results see [25]. To be noted that the propagation delay may exist in theremote communication systems, which might not be equal to the time delay, so lag synchronization has become a hot subject ofresearch nowadays. In this research field, some results have been reported. Shahverdiev et al. [20] investigated lag synchronizationbetween unidirectionally coupled Ikeda systems with time-delay via feedback control techniques; Li et al. [26] considered the lag

✩ This work was jointly supported by the National Natural Science Foundation of China under Grant 60574043, and the Natural Science Foundation of JiangsuProvince of China under Grant BK2006093.

* Corresponding author.E-mail addresses: jdcao@seu.edu.cn, jdcaoseu@gmail.com (J. Cao).

0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2006.12.019

278 Y. Sun, J. Cao / Physics Letters A 364 (2007) 277–285

synchronization issue of coupled time-delayed systems with chaos, then applied proposed lag synchronization strategies towardsthe secure communication. Zhou et al. [27] investigated lag synchronization of coupled chaotic delayed neural networks withoutnoise perturbation by using adaptive feedback control techniques. For the other results reported in the literature, see, e.g., Ref. [28].However, almost all the regimes of lag synchronization above-mentioned are limited to parameters of the system known in prior,which are hardly the case in real-life applications.

Moreover, it should be pointed that more and more numerical results show that noise plays an important role in chaos syn-chronization in different ways. In [29], it was reported that the external noise added to two uncoupled chaotic circuits does inducesynchronization. In [30], it was concluded that the noise-induced synchronization can be induced by a nonzero mean of the signaland not by its stochastic character. In [31], the authors found that noise plays a very constructive role in the enhancement of phasesynchronization of weakly coupled chaotic oscillators, and other results [32–34]. Hence, developing the corresponding theoreticalresults on chaos synchronization with noise perturbation to support the experimental and numerical results has become an inter-esting topic. Some theoretical results have just been developed. Lin and He [35] discussed the complete synchronization problembetween unidirectionally coupled Chua’s circuits within stochastic perturbation via feedback control techniques. Lin and Chen [36]took white noise to enhance synchronization of coupled chaotic systems. However, to the best of our knowledge, the problem oflag synchronization of unknown chaotic DNNs with fully unknown parameters still remain open, not to mention the adaptive lagsynchronization of unknown chaotic systems with noise perturbation.

With the above motivations, our main aim in this Letter is to investigate lag synchronization of unknown chaotic DNNs withnoise perturbation. Via adaptive feedback control techniques, the suitable parameters update laws are proposed, several analyticalresults are developed to ensure adaptive lag synchronization of unknown chaotic DNNs with noise perturbation. It is worth pointingthat the adaptive synchronization scheme can also be applied to the complete synchronization of unknown chaotic systems withor without noise perturbation, some comparisons are provided to illustrate that our results improve and general some of existingresults.

The rest of this Letter is organized as follows. In Section 2, the unknown chaotic DNNs model is presented, then the lag syn-chronization problem is described. In Section 3, the adaptive synchronization scheme and the parameters update laws ensuring lagsynchronization are developed. In Section 4, an illustrated example with its simulations is provided to demonstrate the effectivenessof the derived analytical results. In Section 5, the proposed adaptive scheme is applied to the secure communication. Finally, inSection 6, the Letter is completed with a conclusion and some discussions.

2. Notations and preliminaries

Throughout this Letter, we need the following notations: for any symmetric matrix A, A > 0 means A is a positive definitematrix; E{·} stands for the mathematical expectation operator; ‖x‖2 is used to denote a vector norm defined by ‖x‖2 = ∑n

i=1 x2i ;

‘T’ represents the transpose of a matrix or a vector; I is an identical matrix; ω(t) is an m-dimensional Brownian motion.The unknown DNNs model is described by

(1)dxi(t) =[−cixi(t) +

n∑j=1

aijfj

(xj (t)

) +n∑

j=1

bijfj

(xj (t − τj )

)]dt, i = 1,2, . . . , n,

or in compact form

(2)dx(t) = [−Cx(t) + Af(x(t)

) + Bf(xτ (t)

)]dt,

where x(t) = (x1(t), x2(t), . . . , xn(t))T ∈ R

n is the state vector associated with the neurons; C = diag(c1, c2, . . . , cn) > 0 is anunknown matrix; A = (aij )n×n and B = (bij )n×n are both unknown weight matrix and delayed weight matrix, respectively;f and g are activation functions, f (x(t)) = (f1(x1(t)), f2(x2(t)), . . . , fn(xn(t)))

T ∈ Rn, f (xτ (t)) = (f1(x1(t − τ1)), f2(x2(t −

τ2)), . . . , fn(xn(t − τn)))T ∈ R

n, where τ > 0 is the transmission delay.The initial condition associated with system (2) is given in the following form:

x(s) = ξ(s), −τ � s � 0,

for any ξ ∈ L2F0

([−τ,0],Rn), where L2

F0([−τ,0],R

n) is the family of all F0-measurable C([−τ,0];Rn)-valued random variables

satisfying that sup−τ�s�0 E|ξ(s)|2 < ∞, and C([−τ,0];Rn) denotes the family of all continuous R

n-valued functions ξ(s) on[−τ,0] with the norm ‖ξ‖ = sup−τ�s�0 |ξ(s)|.

In order to realize lag synchronization, the noise-perturbed response system is given as

(3)dy(t) = [−Cy(t) + Af(y(t)

) + Bf(yτ (t)

) + u]dt + H

(t, y(t) − x(t − σ), yτ (t) − xτ (t − σ)

)dω(t),

where C, A and B are the estimations of the unknown matrices C,A and B , respectively. ω(t) is a n-dimensional Brownian motionsatisfying E{dω(t)} = 0 and E{[dω(t)]2} = dt . σ > 0 is the propagation delay. u is a control input.

Y. Sun, J. Cao / Physics Letters A 364 (2007) 277–285 279

Remark 1. For the reason of time-delay τ , if the noise-perturbed response DNNs (3) can be synchronized with the unknown DNNs(2) with the propagation delay σ , the synchronization manifold must turn to be x(t) = y(t − σ). Moreover, inspired by the ideasin [37], the control input in the response system is taken as u = k(y(t) − x(t − σ)), where k varies with the synchronization errory(t) − x(t − σ).

Throughout this Letter, the following assumptions are needed:

(A1). fi(x) satisfies the Lipschitz condition. That is, for each i = 1,2, . . . , n, there is a constant βi > 0 such that∣∣fi(x) − fi(y)∣∣ � βi |x − y|, ∀x, y ∈ R.

(A2). H(t, x, y) satisfies the Lipschitz condition. Moreover, there exist constant matrices of appropriate dimensions G1,G2 suchthat

trace[HT(t, x, y)H(t, x, y)

]� ‖G1x‖2 + ‖G2y‖2, ∀(t, x, y) ∈ R

+ × Rn × R

n.

(A3). f (0) ≡ 0, σ(t,0,0) ≡ 0.

Let e(t) = y(t) − x(t − σ) be the lag synchronization error, then yields the error system

de(t) = [−Ce(t) + A[f

(y(t)

) − f(x(t − σ)

)] + B[f

(yτ (t)

) − f(xτ (t − σ)

)] − (C − C)y(t)

(4)+ (A − A)f(y(t)

) + (B − B)f(yτ (t)

)]dt + H

(t, e(t), eτ (t)

)dω(t) + ke(t) dt,

denote that C = C − C = diag(c1, c2, . . . , cn) = diag(c1 − c1, c2 − c2, . . . , cn − cn), A = A − A = (aij )n×n = (aij − aij )n×n andB = B − B = (bij )n×n = (bij − bij )n×n, then error system (4) can be rewritten as

de(t) = [−Ce(t) + Ag(e(t)

) + Bg(eτ (t)

) − Cy(t) + Af(y(t)

) + Bf(yτ (t)

)]dt

(5)+ H(t, e(t), eτ (t)

)dω(t) + ke(t) dt,

where

g(e(t)

) = [f1

(y1(t)

) − f1(x1(t − σ1)

), f2

(y2(t)

) − f2(x2(t − σ2)

), . . . , fn

(yn(t)

) − fn

(xn(t − σn)

)]T,

g(eτ (t)

) = [f1

(y1(t − τ1)

) − f1(x1(t − τ1 − σ1)

), f2

(y2(t − τ2)

) − f2(x2(t − τ2 − σ2)

), . . . ,

fn

(yn(t − τn)

) − fn

(xn(t − τn − σn)

)]T.

Under the assumptions (A1) and (A3), it is easy to get

(6)∣∣gi

(ei(t)

)∣∣ = ∣∣fi

(yi(t)

) − fi

(xi(t − σi)

)∣∣ � βi

∣∣ei(t)∣∣,

and g(0) = 0. Hence, together with (A2), it follows from [38] that error system (5) admits a trivial solution e(0) ≡ 0.

Definition 1. The noise-perturbed response system (3) and the drive system (2) can be synchronized with propagation delay σ , ifthe trivial solution of the error system (5) is asymptotically stable in mean square with the initial conditions x0 and y0, respectively,i.e.,

(7)limt→∞E

∥∥e(t)∥∥2 = lim

t→∞E∥∥y(t, y0) − x(t − σ,x0)

∥∥2 = 0.

3. Main results

In this section, by using adaptive feedback control techniques, several theoretical results are developed to realize lag synchro-nization between the drive and the noise-perturbed response systems (2) and (3), several appropriate comparisons are provided toillustrate the effectiveness of our results.

Before stating our main results, we need a few more notations. Let C1,2(R+ × Rn;R

+) denote the family of all nonnegativefunctions V (t, x) on R

+ ×Rn which are continuously twice differentiable in x and once differentiable in t . For each V ∈ C1,2(R+ ×

Rn;R

+), define an operator L associated with the error system (5) acting on V by

LV(t, e(t)

) = Vt

(t, e(t)

) + Ve

(t, e(t)

)[−Ce(t) + Ag(e(t)

) + Bg(e(t − τ)

) − Cy(t) + Af(y(t)

) + Bf(yτ (t)

) + ke(t)]

(8)+ 1

2trace

[HT(

t, e(t), eτ (t))Vee

(t, e(t)

)H

(t, e(t), eτ (t)

)],

280 Y. Sun, J. Cao / Physics Letters A 364 (2007) 277–285

where

Vt

(t, e(t)

) = ∂V (t, e(t))

∂t, Ve

(t, e(t)

) =(

∂V (t, e(t))

∂e1, . . . ,

∂V (t, e(t))

∂en

), Vee

(t, e(t)

) =(

∂2V (t, e(t))

∂ei∂ej

)n×n

.

Now, we begin to state our main results.

Theorem 1. Under the assumptions (A1)–(A3), the noise-perturbed response system (3) can be synchronized with the unknowndrive chaotic DNNs (2) with propagation delay σ , if the time-varying coupling strength k = diag(k1, k2, . . . , kn) with the updatelaw is chosen as:

(9)ki = −αie2i (t),

and the parameters update laws of matrices C, A and B are chosen as:

(10)˙ci = γiei(t)yi(t), ˙aij = −αij ei(t)fj

(yj (t)

),

˙bij = −βij ei(t)fj

(yj (t − τj )

),

in which αi > 0, γi > 0, αij > 0 and βij > 0 (i, j = 1,2, . . . , n) are arbitrary constants, respectively.

Proof. Construct the following nonnegative function as

V(t, e(t)

) = 1

2eT(t)e(t) +

t∫t−τ

eT(s)Qe(s) ds + 1

2

n∑i=1

[1

αi

(ki + li )2 + 1

γi

c2i +

n∑j=1

1

αij

a2ij +

n∑j=1

1

βij

b2ij

],

where Q is a positive definite matrix and li are constants to be determined, respectively.By Itô-differential rule [38], the stochastic derivative of V along the trajectory of error system (5) can be obtained as follows

(11)dV(t, e(t)

) = LV(t, e(t)

)dt + Ve

(t, e(t)

)H

(t, e(t), eτ (t)

)dω(t),

where operator L is given as follows

LV(t, e(t)

) = Vt

(t, e(t)

) + Ve

(t, e(t)

)[−Ce(t) + Ag(e(t)

) + Bg(e(t − τ)

) − Cy(t) + Af(y(t)

) + Bf(yτ (t)

) + ke(t)]

+ 1

2trace

[HT(

t, e(t), eτ (t))Vee

(t, e(t)

)H

(t, e(t), eτ (t)

)]= eT(t)

[−Ce(t) + Ag(e(t)

) + Bg(e(t − τ)

) − Cy(t) + Af(y(t)

) + Bf(yτ (t)

) + ke(t)]

+ eT(t)Qe(t) − eT(t − τ)Qe(t − τ) −n∑

i=1

(ki + li )e2i (t) +

n∑i=1

ciei(t)yi(t) −n∑

i=1

n∑j=1

αij ei(t)fj

(yj (t)

)

−n∑

i=1

n∑j=1

βij ei(t)fj

(yj (t − τj )

) + 1

2trace

[HT(

t, e(t), eτ (t))H

(t, e(t), eτ (t)

)]= eT(t)

[−Ce(t) + Ag(e(t)

) + Bg(e(t − τ)

)] + eT(t)Qe(t) − eT(t − τ)Qe(t − τ)

(12)−n∑

i=1

lie2i (t) + 1

2trace

[HT(

t, e(t), eτ (t))H

(t, e(t), eτ (t)

)].

By condition (6) and the elementary inequality

(13)eT(t)Ag(e(t)

)� 1

2eT(t)ATAe(t) + 1

2gT(

e(t))g(e(t)

)� 1

2eT(t)ATAe(t) + 1

2eT(t)ΣTΣe(t),

(14)eT(t)Bg(eτ (t)

)� 1

2eT(t)BTBe(t) + 1

2gT(

eτ (t))g(eτ (t)

)� 1

2eT(t)BTBe(t) + 1

2eT(t − τ)ΣTΣe(t − τ),

where Σ = diag(β1, β2, . . . , βn), and from assumption (A2), it holds

(15)trace[HT(

t, e(t), eτ (t))H

(t, e(t), eτ (t)

)]� eT(t)GT

1 G1e(t) + eT(t − τ)GT2 G2e(t − τ).

Substituting (13), (14) and (15) into (12), it can be derived that

LV(t, e(t)

)� −eT(t)

[L + C − 1

2ATA − 1

2BTB − 1

2ΣTΣ − 1

2GT

1 G1 − Q

]e(t)

(16)+ eT(t − τ)

[1ΣTΣ + 1

GT2 G2 − Q

]e(t − τ),

2 2

Y. Sun, J. Cao / Physics Letters A 364 (2007) 277–285 281

the constant diagonal matrix L can be chosen as L = −C + ΣTΣ + I + 12 {λmax(A

TA) + λmax(BTB) + λmax(G

T1 G1) +

λmax(GT2 G2)}I , and the matrix Q satisfies that Q = 1

2ΣTΣ + 12GT

2 G2, where L = diag(l1, l2, . . . , ln), then it yields

(17)LV(t, e(t)

)� −eT(t)e(t).

Based on the LaSalle invariance principle of stochastic differential equation, which was proposed in [39] and also was used in [35],it yields e(t) → 0, then we derive E‖e(t; ξ)‖2 → 0 and at the same time, C → C∗, A → A∗ and B → B∗, where C∗,A∗ and B∗are constants matrices, respectively. This completes the proof. �Remark 2. By using the adaptive feedback idea in [37] lag synchronization of unknown chaotic DNNs (2) with noise perturbationis realized. The adaptive synchronization scheme is simple to implement in practice. The simulation results provided in Section 4can be found to match the theoretical results perfectly.

Remark 3. It is worth pointing out that in order to achieve synchronization, the variable feedback strength will be automaticallyadapted to a suitable strength depending on the initial values, which is different from the traditional linear feedback in [4–6] and[35], where the feedback strength is fixed, thus the strength must be maximal, which means a kind of waste in practice.

Remark 4. The main value of this Letter is the combination of noise perturbation with lag synchronization of unknown chaoticDNNs via adaptive feedback techniques. The model considered is more general and the synchronization condition does not needany restrictions, the derived results generalize some of existing results, some of them can be seen as the special cases of our results.

Moreover, when the matrices A,B and C in drive system (2) are known in prior, we can derive the following results via the sameadaptive feedback techniques.

Corollary 1. Under the assumptions (A1)–(A3), if the time-varying coupling strength k = diag(k1, k2, . . . , kn) with the update lawis chosen as:

(18)ki = −αie2i (t),

then the noise-perturbed response system (3) and the known drive chaotic DNNs can be synchronized with propagation delay σ ,where αi > 0 (i = 1,2, . . . , n) are arbitrary constants.

Proof. Construct the following non-negative function as

V = 1

2eT(t)e(t) +

t∫t−τ

eT(s)Qe(s) ds +n∑

i=1

1

2αi

(ki + li )2.

The proof is similar to that of Theorem 1. Here it is omitted. �When the noise perturbation is removed from system (3), it yields the noiseless response system

(19)dy(t) = [−Cy(t) + Af(y(t)

) + Bf(yτ (t)

) + u]dt,

which can lead to the following results.

Theorem 2. Under the assumptions (A1) and (A3), the noiseless response system (19) and the unknown drive chaotic DNNs (2)can be synchronized with the propagation delay σ , if the time-varying coupling strength k = diag(k1, k2, . . . , kn) with the updatelaw is taken as:

(20)ki = −αie2i (t),

and the parameters update laws of matrices C, A and B are chosen as

(21)˙ci = γiei(t)yi(t), ˙aij = −αij ei(t)fj

(yj (t)

),

˙bij = −βij ei(t)fj

(yj (t − τj )

),

where αi > 0, γi > 0, αij > 0 and βij > 0 (i, j = 1,2, . . . , n) are arbitrary constants, respectively.

Remark 5. In [27], the issue of lag synchronization of coupled chaotic delayed neural networks is investigated by using the adaptivefeedback techniques, while the model they considered are known in prior, which is hardly the case in real-life application, so theresults derived in this Letter are more general than those in [27].

282 Y. Sun, J. Cao / Physics Letters A 364 (2007) 277–285

Similarly, when the matrices A,B and C in drive system (2) are known in prior, we get

Corollary 2. Under the assumptions (A1) and (A3), if the time-varying coupling strength k = diag(k1, k2, . . . , kn) with the updatelaw is chosen as:

(22)ki = −αie2i (t),

then the noiseless response system (19) can be synchronized with the known drive chaotic DNNs with propagation delay σ , whereαi > 0 (i = 1,2, . . . , n) are arbitrary constants.

Remark 6. To be noted that the exponential lag synchronization results derived in [27] must need the coefficients satisfy the givenconditions, while Corollary 2 does not have these restrictions, hence in some sense that the results derived here generalize andimprove the results in [27].

Remark 7. It is worth pointing out that the adaptive scheme used in this Letter can also be extended to investigate the completesynchronization problem of the known or unknown chaotic DNNs with or without noise perturbation, only need to let σ = 0 in thedefinition of the lag synchronization error, here they are omitted.

Remark 8. In [35], Lin and He investigated complete synchronization of noise-perturbed Chua’s system via feedback controltechniques. However, the feedback gains in their paper are fixed and must be the maximal, which means a kind of waste in practice.The feedback gains in the extended complete synchronization results from Corollary 1 have the on-line learning ability, and aremore general then theirs.

Remark 9. In [10], Cao and Lu considered the adaptive synchronization of neural networks with or without time-varying delay viaadaptive feedback scheme. Though the time delay considered in their paper is time-varying, the extended complete synchronizationresults from Corollary 2 can be equal to Theorem 1 in their paper, only need to modify the nonnegative function in the course ofproof.

4. Illustrative example

In this section, we employ an example to illustrate the effectiveness of the obtained results.

Example. Consider the following chaotic DNNs [12]:

(23)dx(t) = [−Cx(t) + Ag(x(t)

) + Bg(x(t − τ)

)]dt,

where f (x) = tanh(x),

C =[

1 00 1

], A =

[2.0 −0.1

−5.0 4.5

], B =

[−1.5 −0.1−0.2 −4

].

The noise-perturbed response system is designed as follows

(24)dy(t) = [−Cy(t) + Af(y(t)

) + Bf(yτ (t)

) + u]dt + H

(t, y(t) − x(t − σ), yτ (t) − xτ (t − σ)

)dω(t),

without loss of generality, we take

C =[

c1 00 c2

], A =

[a11 −0.15 a22

], B =

[b11 −0.1

−0.2 −4

],

where the noise intensity

H(t, e(t), eτ (t)

) =[

a1e1(t) + b1eτ1(t) 00 a2e2(t) + b2eτ2(t)

]

and ω(t) is a 2-dimensional Brownian motion satisfying E{dω(t)} = 0 and E{[dω(t)]2} = dt , following from (A2), it is easy toget G1 = √

2 diag(|a1|, |a2|) and G2 = √2 diag(|b1|, |b2|).

In the simulations, the Euler–Maruyama numerical scheme is used to simulate the drive-response systems (23) and (24). Theinitial conditions of the unknown chaotic DNNs (23) and the response system (24) are taken as [x1(t), x2(t)]T = [0.4,0.6]T and[y1(t), y2(t)]T = [0.2,0.3]T, for −1 � t � 0, respectively. Some initial parameters are given as follows: T = 200 and time stepsize is δt = 0.02. Let the initial conditions of the feedback strength and the unknown parameters as follows [k1(0), k2(0)]T =[1,1]T and [c11(0), c22(0), a11(0), a22(0), b11(0)]T = [3,1.5,−6,3.5,−8]T, respectively, and we choose [α1, α2]T = [0.5,0.5]T,[γ1, γ2, α11, α22, β11]T = [1.5,0.2,6,1,4]T.

Y. Sun, J. Cao / Physics Letters A 364 (2007) 277–285 283

Fig. 1. Chaotic dynamic of response system with noise perturbation.

Fig. 2. Lag synchronization between (23) and (24). Fig. 3. The state response of error system.

The simulation results are as follows: Fig. 1 shows the chaotic behavior of the noise-perturbed response system (24) in phasespace. Fig. 2 depicts the lag synchronization between unknown chaotic DNNs (23) and the response system (24) with propagationdelay σ = 0.8. Fig. 3 is the state response of the error system between (23) and (24). Figs. 4–5 depict the adaptive parameters ofk1, k2, c11, c22 and a11, a22, b11, respectively.

5. Applications

In this section, the adaptive synchronization scheme proposed above is applied towards the chaotic secure communication withnoise perturbation, the transmitter and the receiver are designed as follows:Transmitter:

(25)dx(t) = [−Cx(t) + Af(x(t)

) + Bf(xτ (t)

) + M(t)]dt,

where M(t) = [0,m(t)]T, and m(t) is the transmitted signal, C,A,B and f (x) are the same as those in (23).Receiver:

(26)dy(t) = [−(C − k)y(t) + Af(y(t)

) + Bf(yτ (t)

) + S(t)]dt + H

(t, y(t) − x(t − σ), yτ (t) − xτ (t − σ)

)dω(t),

where S(t) = [−k1x1(t −σ), s(t)]T , and s(t) = −k2x2(t −σ)+m(t −σ) is the chaotic transmitted signal, which does not affect thechaotic phenomenon of the system (26), and ω(t) is a 2-dimensional Brownian motion satisfying E{dω(t)} = 0 and E{[dω(t)]2} =dt , so the transmitted signal can be recovered by r(t) = s(t) + k2y2(t), the parameters defined are the same as that in (24).

284 Y. Sun, J. Cao / Physics Letters A 364 (2007) 277–285

Fig. 4. Adaptive parameters k1, k2, c11 and c22. Fig. 5. Adaptive parameters a11, a22 and b11.

Fig. 6. The transmitted signal m(t). Fig. 7. The chaotic transmitted signal s(t).

In the simulations, the transmitted signal m(t) = ε sin(0.5t), where ε = 0.1. For convenience, A, B, C are known in prior and thepropagation delay is set σ = 0, the other parameters are chosen the same as that of in the illustrate example. Fig. 6 depicts the signaltransmitted signal m(t), and Fig. 7 shows the chaotic signal s(t). Fig. 8 represents the recovered signal r(t) with noise perturbation.Fig. 9 depicts the error between transmitted signal m(t) and the recovered signal r(t). From the following simulations, one can findthat the encrypted signal can be exactly recovered under the adaptive feedback controller with noise perturbation, the developedresults and the simulations can be found a perfect agreement.

6. Conclusions

In this Letter, based on the invariance principle of stochastic differential equations, with the adaptive feedback control schemeand the suitable parameters update laws, the adaptive lag synchronization problem of unknown chaotic delayed neural networkswith noise perturbation has been tackled. Several results guaranteeing the adaptive lag synchronization were derived. It shouldbe noted that the adaptive synchronization scheme can be extended to the complete synchronization of delayed neural networkswith or without noise perturbation, some appropriate comparisons were provided to illustrate that the developed results improve andgeneralize some of existing results. At last, a numerical example is provided to illustrate the effectiveness of the derived results, thenthe adaptive synchronization scheme was applied towards the secure communication, which could be found a perfect agreementbetween the analytical results and the simulations.

Y. Sun, J. Cao / Physics Letters A 364 (2007) 277–285 285

Fig. 8. The recovered signal r(t).Fig. 9. Error between m(t) and r(t).

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