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Synchronization conditions for chaotic nonlinear continuous neural networks
Chaos, Solitons and Fractals 41 (2009) 2495–2501
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Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Synchronization conditions for chaotic nonlinear continuous neural networks Jiming Hu Engineering College, Zhejiang Ocean University, Zhoushan 316004, China
a r t i c l e
i n f o
Article history: Accepted 17 September 2008
Communicated by Prof. Ji-Huan He
a b s t r a c t This paper deals with the synchronization problem of a class of chaotic nonlinear neural networks. A feedback control gain matrix is derived to achieve the state synchronization of two identical nonlinear neural networks by using the Lyapunov stability theory, and the obtained criterion condition can be verified if a certain Hamiltonian matrix with no eigenvalues on the imaginary axis. The new sufficient condition can avoid solving an algebraic Riccati equation. The results are illustrated through one numerical example. Ó 2009 Published by Elsevier Ltd.
1. Introduction Recently, many efforts have mainly been devoted to the stability analysis and periodic oscillations of different kinds of neural networks [1–8]. It has been shown that such neural networks can exhibit complicated dynamics and even chaotic behavior if the parameters and time delays are appropriately chosen for the neural networks [9]. Up to now, there have been some studies in the synchronization of this class of chaotic neural networks with or without delays [10–15]. Cao et al. [10] analyzed synchronization of almost all kinds of coupled identical neural networks based on a simple adaptive feedback scheme. Zhou et al. [11] investigated lag synchronization of coupled chaotic delayed neural networks without noise perturbation by using adaptive feedback control techniques. Gu et al. [12] investigated complete synchronization of star-shaped complex networks using linear stability analysis. In [13], the authors discussed asymptotic synchronization of a class of neural networks with reaction-diffusion terms and time-varying delays. More recently, based on the Lyapunov functional method and Hermitian matrices theory, the authors [14] derived a synchronization criterion for coupled delayed neural networks, and Cheng et al. [15] applied the method into the synchronization for a class of neural networks with time-varying delays. As a continuation of their previous published results, in this paper, a sufficient condition for the exponential synchronization of a class of chaotic nonlinear neural networks with time-varying delays is further exploited. The criteria are presented by employing the Lyapunov stability method and Hermitian matrices theory. A numerical example illustrates the applicability of the proposed approach. 1.1. Notations T 1 In the sequel, we denote A ; A the transpose of, inverse of any square matrix A, respectively. We use A > 0; ðA < 0Þ to denote a positive- (negative-) definite matrix A; and I is used to denote the n n identity matrix. The vector norm is taken to be Euclidian, denoted by k k. diagðÞ denotes a block diagonal matrix. kðAÞ denotes the eigenvalue of a square matrix A. Rn and Rmn denote, respectively, the n-dimensional Euclidean space, and the set of all m n real matrices.
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Published by Elsevier Ltd. doi:10.1016/j.chaos.2008.09.026
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2. Synchronization problem formulation Based on the drive-response concept, the unidirectional coupled nonlinear neural networks are described by the following equations:
x_ i ðtÞ ¼ ci ðxi ðtÞÞ þ
n X
Dij fj ðxj ðtÞÞ þ
n X Dsij fj ðxj ðt sj ðtÞÞÞ þ J i ;
j¼1
z_ i ðtÞ ¼ ci ðzi ðtÞÞ þ
n X
ð1Þ
j¼1
Dij fj ðzj ðtÞÞ þ
n X Dsij fj ðzj ðt sj ðtÞÞÞ þ J i þ ui ðtÞ;
j¼1
ð2Þ
j¼1
for i ¼ 1; 2; . . . ; n; where n P 2 denotes the number of neurons in the network, xi is the state variable associated with the ith neuron. D ¼ ðDij Þnn , Ds ¼ ðDsij Þnn indicate the interconnection strength among neurons without and with time-varying delay sj ðtÞ P 0, respectively. The function fi describes the manner in which the neurons respond to each other. While Ji is an external constant input; and ui ðtÞ is unidirectional coupled term which is considered as the control input and will be appropriately designed to obtain a certain control objective. Furthermore, it is assumed that sðtÞ ¼ ðs1 ðtÞ; s2 ðtÞ; . . . ; sn ðtÞÞT , s ¼ maxðsj ðtÞÞ and r ¼ maxðs_ j ðtÞÞ < 1 for j ¼ 1; . . . ; n and t P 0, where s and r are constants, and the systems (1) and (2) possess initial conditions xi ðtÞ ¼ /i ðtÞ 2 C½s ; 0; RÞ and zi ðtÞ ¼ ui ðtÞ 2 C½s ; 0; RÞ, where C½s ; 0; RÞ denotes the set of all continuous functions from ½s ; 0 to R. We further assume that the functions ai ðÞ and fj ðÞ satisfies the following conditions. (H1) Each function ci : R ! R is locally Lipschitz and nondecreasing, and there exists a positive real ai such that c0i ðxÞ P ai for any x 2 R at which ci is differentiable. Let A ¼ diagðai Þ; i ¼ 1; 2; . . . ; n. (H2) Each fj : R ! R is monotonic nondecreasing and globally Lipschitz, i.e. there exists a positive real kj > 0 such that
0 6 ðfj ðxÞ fj ðyÞÞ=ðx yÞ 6 kj ;
j ¼ 1; 2; . . . ; N
for any x; y 2 R; x–y. Definition 1. The system (1) and the uncontrolled system (2) (i.e. u 0 in (2)) are said to be exponentially synchronized, if there exist constants bðaÞ P 1 and a > 1 such that
jxi ðtÞ ~xi ðtÞj 6 bðaÞ sup j/i ðsÞ ui ðsÞjeat 8t P 0;
i ¼ 1; 2; . . . ; n:
ð3Þ
s 6s60
Constant a said to be the degree of exponential synchronization. Exponential synchronization problem: The exponential synchronization problem considered here is to determine the control input ui associated with the state-feedback for the purpose of exponentially synchronizing the two identical chaotic nonlinear neural networks (1) and (2) with the same system’s parameters except the differences in initial conditions.
3. Some criteria for exponential synchronization 3.1. Controller design Let us define the synchronization error signal ei ðtÞ ¼ xi ðtÞ zi ðtÞ, where xi ðtÞ and zi ðtÞ are the ith state variable of the drive and response neural networks, respectively. Therefore, the error dynamics between (1) and (2) can be expressed by
e_ i ðtÞ ¼ bi ðei ðtÞÞ þ
n X
Dij g j ðej ðtÞÞ þ
j¼1
n X Dsij g j ðej ðt sj ðtÞÞÞ ui ðtÞ;
ð4Þ
j¼1
for i ¼ 1; 2; . . . ; n, where
bi ðei ðtÞÞ ¼ ai ðxi ðtÞÞ ai ðzi ðtÞÞ; g j ðej ðtÞÞ ¼ fj ðxj ðtÞÞ fj ðzj ðtÞÞ; g j ðej ðt sj ðtÞÞÞ ¼ fj ðxj ðt sj ðtÞÞÞ fj ðzj ðt sj ðtÞÞÞ: Here, from (H1) and (H2), we can obtain that the function g i ðÞ satisfies 0 6 ei ðtÞg i ðei ðtÞÞ 6 ki e2i ðtÞ, and according to the Lebourg Theorem [17], there exist ci P ai such that bi ðei ðtÞÞ ¼ ci ei ðtÞ; i ¼ 1; 2; . . . ; N. If the state variables of the drive system are used to drive the response system, then the control input vector with state feedback is designed as follows:
J. Hu / Chaos, Solitons and Fractals 41 (2009) 2495–2501
2 6 6 4
2P n
3
x1j ðxj ðtÞ zj ðtÞÞ 7
2 6 j¼1 u1 ðtÞ x11 7 6 7 6 6 . 7 .. 7 ¼ 6 .. 7¼6 . . 5 6 7 4 . . 7 6 n 5 4 P xn1 un ðtÞ xnj ðxj ðtÞ zj ðtÞÞ 3
32 3 x1 ðtÞ z1 ðtÞ x1n 7 6 7 .. .. 76 .. 7 ¼ XeðtÞ; . . 54 . 5 xnn xn ðtÞ zn ðtÞ
2497
ð5Þ
j¼1
where eðtÞ ¼ ðe1 ðtÞ; . . . ; en ðtÞÞT and X ¼ ðxij Þnn 2 Rnn is the gain matrix to be determined for synchronizing both a drive system and response system. Furthermore, if a new error ^ei ðtÞ is defined by ^ei ðtÞ ¼ eat ei ðtÞ, then the dynamics of (4) can be transformed into the following form:
^e_ i ðtÞ ¼ ci ^ei ðtÞ þ a^ei ðtÞ þ
n X
Dij Gj ð^ej ðtÞÞ þ
n n X X Dsij Gj ð^ej ðt sj ðtÞÞÞ xij ^ej ðtÞ;
j¼1
j¼1
ð6Þ
j¼1
where
Gj ð^ej ðtÞÞ ¼ eat g j ðej ðtÞÞ; Gj ð^ej ðt sj ðtÞÞÞ ¼ eat g j ðej ðt sj ðtÞÞÞ: For further deriving the exponential synchronization condition on the control law, the following Lemma are needed. Lemma 1. Define a 2n 2n Hamiltonian matrix
"
P¼
e C ðH1 þ H2 Þ nIn
# EET ; e C
e ¼ diagðci aÞ þ X, i ¼ 1; 2; . . . ; n, E ¼ ½D Ds . where n is sufficiently small and positive constant, In is a n n identity matrix C e is a stable matrix and Hamiltonian matrix P has no eigenH1 ¼ diagðk2j Þ and H2 ¼ diagðe2as k2j =ð1 r ÞÞ, j ¼ 1; 2; . . . ; n: If C values on the imaginary axis, then the algebraic Riccati equation (ARE)
e TP P C e þ PEET P þ ðH1 þ H2 Þ þ nIn ¼ 0 C
ð7Þ
has a symmetric and positive definite solution P for a given a > 0: Remark 1. The proof is an immediate consequence of the Lemma 4 in the work of Doyle et al. [16], and therefore it is omitted here. e is stable if and only if all of its eigenvalues have negative real parts. All eigenvalues of C e Remark 2. A real matrix C defined in Lemma 1 can be arbitrarily assigned by appropriately choosing the controller gain matrix X. Especially, if we e are choose the gain matrix as a diagonal matrix X ¼ diagðxi Þ and xi > a ci , i ¼ 1; 2; . . . ; n, then the eigenvalues of C e is a stable matrix. ðxi þ ci aÞ < 0, i ¼ 1; 2; . . . ; n, which implies that C 3.2. Exponential synchronization The exponential synchronization problem of systems (1) and (2) can be solved if the controller gain matrix is suitably designed. The exponential synchronization condition is established in the following main theorems. Theorem 1. For these drive-response neural networks (1) and (2) which satisfy assumption (H1)–(H2), if the controller gain e is a stable matrix and Hamiltonian matrix P defined in Lemma 1 for a given a > 0 matrix X in (5) is suitably designed such that C has no eigenvalues on the imaginary axis, then the networks (1) and (2) are synchronized exponentially with a degree a at least. Proof. Step 1: Transform (6) into a compact form as follows:
e ^eðtÞ þ DGð^eðtÞÞ þ Ds Gð^eðt sðtÞÞÞ; ^e_ ðtÞ ¼ C where
e ¼ diagðci aÞ þ X; C Gð^eðtÞÞ ¼ ðG1 ð^e1 ðtÞÞ; G2 ð^e2 ðtÞÞ; . . . ; Gn ð^en ðtÞÞÞT ; Gð^eðt sðtÞÞÞ ¼ ðG1 ð^e1 ðt sðtÞÞÞ; G2 ð^e2 ðt sðtÞÞÞ; . . . ; Gn ð^en ðt sðtÞÞÞÞT :
ð8Þ
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J. Hu / Chaos, Solitons and Fractals 41 (2009) 2495–2501
e is stable and the Hamiltonian matrix P has no eigenvalues on the imaginary axis. According to Lemma the algeSince C braic Riccati equation (ARE) in (7) has a symmetric and positive definite solution P. Here since there exists reaction-diffusion terms, for convenience we need to suppose P as a positive diagonally matrix. To confirm that the origin of (8) is globally asymptotically convergent, a continuous Lyapunov functional VðtÞ is defined as follows:
VðtÞ ¼ ^eT ðtÞP^eðtÞ þ
n Z t e2as X G2 ð^ej ðs; xÞÞds: 1 r j¼1 tsj ðtÞ j
ð9Þ
It can be easily verified that VðtÞ is a nonnegative function over ½s ; þ1Þ and radially unbounded, i.e. VðtÞ ! þ1 as ^eðtÞ ! þ1. Using the definition of g j ðej ðtÞÞ, Gj ð^ej ðtÞÞ and the assumption ðH2 Þ yields
jg j ðej ðtÞÞj 6 kj jej ðtÞj; jGj ð^ej ðtÞÞj ¼ jeat Gj ðej ðtÞÞj 6 kj jeat ej ðtÞj ¼ kj j^ej ðtÞj: Step 2: Evaluating the time derivative of V along the trajectory of (8), and the following proofs compared with those in [6] is largely identical but with minor differences. Subsequently, we can obtain
eTP PC e ^eðtÞ þ ^eT ðtÞPDDT P^eðtÞ þ ^eT ðtÞPDs ðDs ÞT P^eðtÞ þ ^eT ðtÞH1 ^eðtÞ þ ^eT ðtÞH2 ^eðtÞ _ VðtÞ 6 ^eT ðtÞ C
ð10Þ
_ According to Lyapunov theory, the last inequality VðtÞ 6 nk^e2 ðtÞk indicates VðtÞ converges to zero asymptotically as well as ^eðtÞ ¼ 0 is asymptotically convergent. So we conclude eðtÞ converges to zero exponentially with a rate of a, i.e. j xi ðtÞ zi ðtÞ j6 bðaÞsups ðsÞ6s60 j /i ðsÞ ui ðsÞjeat is satisfied. The proof is thus completed. h Remark 3. As described in Ref. [15], in order to obtain the gain matrix X in the proposed controller (5), a computational procedure is proposed as follows: Step 1: Given a positive constant a and an arbitrarily sufficiently small positive constant a, choose a suitable gain matrix X e is a stable matrix. by using any eigenvalue assignment technique such that C Step 2: Construct the Hamiltonian matrix P in Lemma 1 and using the matlab eig function to check if P has no eigenvalues on the imaginary axis. If so, then the procedure goes to Step 4. Otherwise, the procedure continues to Step 3. e Þg more negative by selecting a new gain matrix X and go back to Step 2. Step 3: Tune the value of Remax fki ð C Step 4: Obtain the state-feedback controller (5). By constructing another Lyapunov functional, we can obtain the following theorem. Theorem 2. Under the assumptions (H1) and (H2), if the controller gain matrix X in (5) is suitably designed and there exist a positive real number r > 0, a positive definite matrix R and a positive diagonal matrix P ¼ diagðp1 ; p2 ; . . . ; pn Þ, such that the matrix s T 1 s 1 2PðA þ XÞK 1 þ PD þ DT P þ rð1 rÞ ðD Þ R D þ rPRP are negative definite, then the two chaotic nonlinear neural networks defined by (1) and (2) are synchronized exponentially with a degree a at least. Proof. To confirm that the origin of (6) is globally asymptotically convergent, a continuous Lyapunov functional VðtÞ is defined as follows:
VðtÞ ¼ 2
n X
pi
i¼1
Z
^ei ðtÞ
Gi ðsÞds þ
0
1 rð1 rÞ
Z
t
GT ð^eðsÞÞðDs ÞT R1 Ds Gð^eðsÞÞds:
ð11Þ
tsðtÞ
Evaluating the derivative of V along the solutions of (6), it follows that
_ VðtÞ ¼2
N X i¼1
þ 62
(
i¼1
pi
n X j¼1
d 1 dt rð1 rÞ
N X
þ
( pi Gi ð^ei ðtÞÞ ci ^ei ðtÞ þ a^ei ðtÞ þ Z
t
Dij Gj ð^ej ðtÞÞþ )
n n X X Dsij Gj ð^ej ðt sj ðtÞÞÞ xij ^ej ðtÞ j¼1
)
j¼1
GT ð^eðsÞÞðDs ÞT R1 Ds Gð^eðsÞÞds
tsðtÞ
ai þ a 2 Gi ð^ei ðtÞÞ þ 2GT ð^eðtÞÞPDGð^eðtÞÞ þ 2GT ð^eðtÞÞPDs Gð^eðt sðtÞÞÞ 2GT ð^eðtÞÞPXK 1 Gð^eðtÞÞ ki
1 1 GT ð^eðtÞÞðDs ÞT R1 Ds Gð^eðtÞÞ GT ð^eðt sðtÞÞÞðDs ÞT R1 Ds Gð^eðt sðtÞÞÞds: rð1 rÞ r
ð12Þ
Since R is a positive definite matrix, there must exist a nonsingular matrix S of appropriate dimension such that R ¼ ST S, then the third term in (12) can be estimated as
J. Hu / Chaos, Solitons and Fractals 41 (2009) 2495–2501
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h iT 2GT ð^eðtÞÞPDs Gð^eðt sðtÞÞÞ ¼ r1=2 SPGð^eðtÞÞ r1=2 ðS1 ÞT Ds Gð^eðt sðtÞÞÞ h i 1 r 1=2 SPGð^eðtÞÞ r 1=2 ðS1 ÞT Ds Gð^eðt sðtÞÞÞ þ rGT ð^eðtÞÞPRPGð^eðtÞÞ þ GT ð^eðt sðtÞÞÞ r ðDs ÞT R1 Ds Gð^eðt sðtÞÞÞds 1 6 þrGT ð^eðtÞÞPRPGð^eðtÞÞ þ GT ð^eðt sðtÞÞÞðDs ÞT R1 Ds Gð^eðt sðtÞÞÞds: r Substituting above inequality into (12), we have
_ VðtÞ 6 GT ð^eðtÞÞ 2PðA þ XÞK 1 þ 2aPK 1 þ PD þ DT P þ
1 ðDs ÞT R1 Ds þ rPRP Gð^eðtÞÞ; rð1 rÞ
ð13Þ
s T 1 s 1 If 2PðA þ XÞK 1 þ PD þ DT P þ rð1 rÞ ðD Þ R D þ rPRP are negative definite, there must exist small positive e such that s T 1 s 1 _ 2PðA þ XÞK 1 þ 2aPK 1 þ PD þ DT P þ rð1 rÞ ðD Þ R D þ rPRP. It follows from (13) that VðtÞ < 0, which implies that V is
strictly decreasing and converges to 0 when t ! 1; as well as ^eðtÞ ¼ 0 is asymptotically convergent. So we conclude eðtÞ converges to zero exponentially with a rate of a; i.e. j xi ðtÞ zi ðtÞ j6 bðaÞsups 6s60 j /i ðsÞ ui ðsÞ j eat is satisfied. The proof is thus completed. h Taking R ¼ P ¼ I and r ¼ 1, it can be easily to obtain the following corollary. Corollary 1. Under the assumptions (H1) and (H2), if the controller gain matrix X in (5) is suitably designed such that the matrix 2ðA þ XÞK 1 þ D þ DT þ ð11rÞ ðDs ÞT Ds þ I are negative definite, then the two chaotic nonlinear neural networks defined by (1) and (2) are synchronized exponentially with a degree a at least. 4. An illustrative example To demonstrate the validity of the exponential synchronization condition, an example is given in this section and the dynamic behaviours are shown for the cellular neural networks with time-varying delays and the Hopfield neural networks with time-varying delays, respectively. Example 1. A two-dimensional nonlinear neural networks with time-varying delays is given as follows:
x_ i ðtÞ ¼ ci ðxi ðtÞÞ þ
2 X
Dij fj ðxj ðtÞÞ þ
j¼1
where c1 ðvÞ ¼ 2v sinðvÞ;
D ¼ ðDij Þ22 ¼
2 X Dsij fj ðxj ðt sj ðtÞÞÞ þ J i ; i ¼ 1; 2;
ð14Þ
j¼1
c2 ðvÞ ¼ 2v þ cosðvÞ,
2:1
0:12
5:1
3:2
;
J 1 ¼ J 2 ¼ 0 and
Ds ¼ ðDsij Þ22 ¼
1:6 0:1 0:2
2:4
:
The delays s1 ðtÞ ¼ s2 ðtÞ ¼ s3 ðtÞð1 e Þ=ð1 þ e Þ are time-varying and satisfy 0 6 sj ðtÞ 6 1 ¼ s , 0 6 s_ j ðtÞ 6 0:5; j ¼ 1; 2: Noting that while the nonlinear functions are taking as c1 ðvÞ ¼ c2 ðvÞ ¼ v, the neural network reduces to a general chaotic neural networks. And we can see c01 ðvÞ ¼ 2 cosðvÞ P 1, c02 ðvÞ ¼ 2 þ sinðvÞ P 1, so (H1) is satisfied. Now consider the Hopfield neural networks, i.e., taking the activation function as g 1 ðvÞ ¼ g 2 ðvÞ ¼ tanhðvÞ. So the system satisfies assumption (H2) with k1 ¼ k2 ¼ 1. Fig. 1 shows the chaotic attractor plot with the initial condition /ðs; xÞ ¼ ½1; 1:5T . Fig. 2 shows the power spectral plot of this system. And by using Jacobia method, the Lyapunov exponents are obtained as follows: t
k1 ¼ 0:3037;
t
k2 ¼ 0:3608:
As we can see, from the phase plot, power spectral plot and the Lyapunov exponents, the system (14) is a chaotic system. To achieve synchronization, the response system is designed as follows:
z_ i ðtÞ ¼ ci ðzi ðtÞÞ þ
2 X
Dij fj ðzj ðtÞÞ þ
j¼1
2 X Dsij fj ðzj ðt sj ðtÞÞÞ þ J i þ ui ðtÞ;
i ¼ 1; 2;
ð15Þ
j¼1
If we choose the synchronization degree a ¼ 0:15 and the controller gain matrix is chosen as X ¼ diagðxi Þ; i ¼ 1; 2; with x1 ¼ 0:4; x2 ¼ 0:3; it is easily found that the matrix Hamiltonian matrix P, then the three sub-matrices of P are obtained as follows:
H1 ¼
1 0 ; 0 1
H2 ¼ ðe2as =ð1 rÞÞ
1 0 6:9944 10:5340 ; EET ¼ : 0 1 10:5340 42:0500
e is stable and the eigenvalues of Hamiltonian matrix P are as follows: On the other hand, the matrix C
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J. Hu / Chaos, Solitons and Fractals 41 (2009) 2495–2501
5 4 3 2
x
2
1 0 -1 -2 -3 -4 -5 -1
-0.5
0
0.5
1
x1 Fig. 1. Phase plot of x1 x2 .
25
Power Spectral
20
15
10
5
0
-5 0
20
40
60
80
100
120
Frequency Fig. 2. Power spectral plot.
1.8
|e |
1.6
|e2|
1
1.4
errors
1.2 1 0.8 0.6 0.4 0.2 0 -0.2
0
50
100
150
t/s Fig. 3. Synchronized errors when x1 ¼ 0:4; x2 ¼ 0:3.
200
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J. Hu / Chaos, Solitons and Fractals 41 (2009) 2495–2501
0.8 0.7
|e1|
0.6
|e2|
errors
0.5 0.4 0.3 0.2 0.10 -0.1 -0.2 0
50
100
150
200
t/s Fig. 4. Synchronized errors when x1 ¼ 1:2; x2 ¼ 1:3.
k1 ¼ 1:1755 þ 3:7135i; k2 ¼ 1:1755 3:7135i; k3 ¼ 1:0897 þ 12:3398i; k4 ¼ 1:0897 12:3398i; i.e., the Hamiltonian matrix P has no eigenvalues on the imaginary axis for the sufficiently small constant e ¼ 104 . Hence, from Theorem 1, the two chaotic nonlinear neural networks defined by (14) and (15) are synchronized exponentially. In this case, Fig. 3 shows the synchronized errors between the states of system (14) and those of system (15). If the controller gain matrix is chosen as X ¼ diagðxi Þ; i ¼ 1; 2, with x1 ¼ 1:2; x2 ¼ 1:3, the synchronized errors between the states of system (14) and those of system (15) are shown in Fig. 4. It can be seen that choosing the controller gain matrix properly, the synchronization speed may be very fast.
5. Conclusions In this paper, two sufficient conditions to guarantee the globally exponential synchronization for a class of chaotic nonlinear neural networks with time-varying delays are presented. This class of neural networks generalize a large class of delayed neural networks, such as Hopfield neural networks and cellular neural networks. Our results have been shown to be more extensive and easier to verify than those reported previously. A simulation example is given to show the effectiveness of the proposed method and criterion. References [1] Qi H, Qi L. Deriving sufficient conditions for global asymptotical stability of delayed neural networks via nonsmooth analysis. IEEE Trans Neural Networks 2004;15:99–109. [2] Liu HL, Chen GH. Delay-dependent stability for neural networks with time-varying delay. Chaos, Solitons & Fractals 2007;33(1):171–7. [3] Ju H Park. On global stability criterion of neural networks with continuously distributed delays. Chaos, Solitons & Fractals 2008;37(2):444–9. [4] Zhao ZJ. Global exponential robust periodicity and stability of interval neural networks with both variable and unbounded delays. Chaos, Solitons & Fractals 2008;36(1):91–7. [5] Lou X, Cui B. Boundedness and exponential stability for nonautonomous RCNNs with distributed delays. Comput Math Appl 2007;54(4):589–98. [6] Huang ZT, Yang QG, Luo XS. Exponential stability of impulsive neural networks with time-varying delays. Chaos, Solitons & Fractals 2008;35(4):770–80. [7] Lou XY, Cui BT. Stochastic exponential stability for Markovian jumping BAM neural networks with time-varying delays. IEEE Trans Syst Man Cybernet B 2007;37(3):713–9. [8] Qu Q. Global robust exponential stability of delayed neural networks: an LMI approach. Chaos, Solitons & Fractals 2007;32(5):1742–8. [9] Gilli M. Strange attractors in delayed cellular neural networks. IEEE Trans Circuits Syst I 1993;40(11):849–53. [10] Cao JD, Lu JQ. Adaptive synchronization of neural networks with or without time-varying delays. Chaos 2006;16(1):013133-1–3-6. [11] Zhou J, Chen TP, Xiang L. Chaotic lag synchronization of coupled delayed neural networks and its applications in secure communication. Circuits Syst Signal Process 2005;24(5):599–613. [12] Lou XY, Cui BT. Synchronization of competitive neural networks with different time scales. Physica A 2007;380:563–76. [13] Lou XY, Cui BT. Asymptotic synchronization of a class of neural networks with reaction-diffusion terms and time-varying delays. Comput Math Appl 2006;52(6-7):897–904. [14] Cheng CJ, Liao TL, Hwang CC. Exponential synchronization of a class of chaotic neural networks. Chaos, Solitons & Fractals 2005;24:197–206. [15] Cheng CJ, Liao TL, Yan JJ, Hwang CC. Exponential synchronization of a class of neural networks with time-varying delays. IEEE Trans Syst Man Cybernet B: Cybernetics 2006;36(1):209–15. [16] Doyle JC, Glover K, Khargonekar PP, Francis BA. Statespace solutions to standard H2 and H1 control problems. IEEE Trans Automat Control 1989;34:831–47. [17] Clarke FH. Optimization and nonsmooth analysis. New york: Wiley; 1983.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Synchronization conditions for chaotic nonlinear continuous neural networks Jiming Hu Engineering College, Zhejiang Ocean University, Zhoushan 316004, China
a r t i c l e
i n f o
Article history: Accepted 17 September 2008
Communicated by Prof. Ji-Huan He
a b s t r a c t This paper deals with the synchronization problem of a class of chaotic nonlinear neural networks. A feedback control gain matrix is derived to achieve the state synchronization of two identical nonlinear neural networks by using the Lyapunov stability theory, and the obtained criterion condition can be verified if a certain Hamiltonian matrix with no eigenvalues on the imaginary axis. The new sufficient condition can avoid solving an algebraic Riccati equation. The results are illustrated through one numerical example. Ó 2009 Published by Elsevier Ltd.
1. Introduction Recently, many efforts have mainly been devoted to the stability analysis and periodic oscillations of different kinds of neural networks [1–8]. It has been shown that such neural networks can exhibit complicated dynamics and even chaotic behavior if the parameters and time delays are appropriately chosen for the neural networks [9]. Up to now, there have been some studies in the synchronization of this class of chaotic neural networks with or without delays [10–15]. Cao et al. [10] analyzed synchronization of almost all kinds of coupled identical neural networks based on a simple adaptive feedback scheme. Zhou et al. [11] investigated lag synchronization of coupled chaotic delayed neural networks without noise perturbation by using adaptive feedback control techniques. Gu et al. [12] investigated complete synchronization of star-shaped complex networks using linear stability analysis. In [13], the authors discussed asymptotic synchronization of a class of neural networks with reaction-diffusion terms and time-varying delays. More recently, based on the Lyapunov functional method and Hermitian matrices theory, the authors [14] derived a synchronization criterion for coupled delayed neural networks, and Cheng et al. [15] applied the method into the synchronization for a class of neural networks with time-varying delays. As a continuation of their previous published results, in this paper, a sufficient condition for the exponential synchronization of a class of chaotic nonlinear neural networks with time-varying delays is further exploited. The criteria are presented by employing the Lyapunov stability method and Hermitian matrices theory. A numerical example illustrates the applicability of the proposed approach. 1.1. Notations T 1 In the sequel, we denote A ; A the transpose of, inverse of any square matrix A, respectively. We use A > 0; ðA < 0Þ to denote a positive- (negative-) definite matrix A; and I is used to denote the n n identity matrix. The vector norm is taken to be Euclidian, denoted by k k. diagðÞ denotes a block diagonal matrix. kðAÞ denotes the eigenvalue of a square matrix A. Rn and Rmn denote, respectively, the n-dimensional Euclidean space, and the set of all m n real matrices.
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Published by Elsevier Ltd. doi:10.1016/j.chaos.2008.09.026
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2. Synchronization problem formulation Based on the drive-response concept, the unidirectional coupled nonlinear neural networks are described by the following equations:
x_ i ðtÞ ¼ ci ðxi ðtÞÞ þ
n X
Dij fj ðxj ðtÞÞ þ
n X Dsij fj ðxj ðt sj ðtÞÞÞ þ J i ;
j¼1
z_ i ðtÞ ¼ ci ðzi ðtÞÞ þ
n X
ð1Þ
j¼1
Dij fj ðzj ðtÞÞ þ
n X Dsij fj ðzj ðt sj ðtÞÞÞ þ J i þ ui ðtÞ;
j¼1
ð2Þ
j¼1
for i ¼ 1; 2; . . . ; n; where n P 2 denotes the number of neurons in the network, xi is the state variable associated with the ith neuron. D ¼ ðDij Þnn , Ds ¼ ðDsij Þnn indicate the interconnection strength among neurons without and with time-varying delay sj ðtÞ P 0, respectively. The function fi describes the manner in which the neurons respond to each other. While Ji is an external constant input; and ui ðtÞ is unidirectional coupled term which is considered as the control input and will be appropriately designed to obtain a certain control objective. Furthermore, it is assumed that sðtÞ ¼ ðs1 ðtÞ; s2 ðtÞ; . . . ; sn ðtÞÞT , s ¼ maxðsj ðtÞÞ and r ¼ maxðs_ j ðtÞÞ < 1 for j ¼ 1; . . . ; n and t P 0, where s and r are constants, and the systems (1) and (2) possess initial conditions xi ðtÞ ¼ /i ðtÞ 2 C½s ; 0; RÞ and zi ðtÞ ¼ ui ðtÞ 2 C½s ; 0; RÞ, where C½s ; 0; RÞ denotes the set of all continuous functions from ½s ; 0 to R. We further assume that the functions ai ðÞ and fj ðÞ satisfies the following conditions. (H1) Each function ci : R ! R is locally Lipschitz and nondecreasing, and there exists a positive real ai such that c0i ðxÞ P ai for any x 2 R at which ci is differentiable. Let A ¼ diagðai Þ; i ¼ 1; 2; . . . ; n. (H2) Each fj : R ! R is monotonic nondecreasing and globally Lipschitz, i.e. there exists a positive real kj > 0 such that
0 6 ðfj ðxÞ fj ðyÞÞ=ðx yÞ 6 kj ;
j ¼ 1; 2; . . . ; N
for any x; y 2 R; x–y. Definition 1. The system (1) and the uncontrolled system (2) (i.e. u 0 in (2)) are said to be exponentially synchronized, if there exist constants bðaÞ P 1 and a > 1 such that
jxi ðtÞ ~xi ðtÞj 6 bðaÞ sup j/i ðsÞ ui ðsÞjeat 8t P 0;
i ¼ 1; 2; . . . ; n:
ð3Þ
s 6s60
Constant a said to be the degree of exponential synchronization. Exponential synchronization problem: The exponential synchronization problem considered here is to determine the control input ui associated with the state-feedback for the purpose of exponentially synchronizing the two identical chaotic nonlinear neural networks (1) and (2) with the same system’s parameters except the differences in initial conditions.
3. Some criteria for exponential synchronization 3.1. Controller design Let us define the synchronization error signal ei ðtÞ ¼ xi ðtÞ zi ðtÞ, where xi ðtÞ and zi ðtÞ are the ith state variable of the drive and response neural networks, respectively. Therefore, the error dynamics between (1) and (2) can be expressed by
e_ i ðtÞ ¼ bi ðei ðtÞÞ þ
n X
Dij g j ðej ðtÞÞ þ
j¼1
n X Dsij g j ðej ðt sj ðtÞÞÞ ui ðtÞ;
ð4Þ
j¼1
for i ¼ 1; 2; . . . ; n, where
bi ðei ðtÞÞ ¼ ai ðxi ðtÞÞ ai ðzi ðtÞÞ; g j ðej ðtÞÞ ¼ fj ðxj ðtÞÞ fj ðzj ðtÞÞ; g j ðej ðt sj ðtÞÞÞ ¼ fj ðxj ðt sj ðtÞÞÞ fj ðzj ðt sj ðtÞÞÞ: Here, from (H1) and (H2), we can obtain that the function g i ðÞ satisfies 0 6 ei ðtÞg i ðei ðtÞÞ 6 ki e2i ðtÞ, and according to the Lebourg Theorem [17], there exist ci P ai such that bi ðei ðtÞÞ ¼ ci ei ðtÞ; i ¼ 1; 2; . . . ; N. If the state variables of the drive system are used to drive the response system, then the control input vector with state feedback is designed as follows:
J. Hu / Chaos, Solitons and Fractals 41 (2009) 2495–2501
2 6 6 4
2P n
3
x1j ðxj ðtÞ zj ðtÞÞ 7
2 6 j¼1 u1 ðtÞ x11 7 6 7 6 6 . 7 .. 7 ¼ 6 .. 7¼6 . . 5 6 7 4 . . 7 6 n 5 4 P xn1 un ðtÞ xnj ðxj ðtÞ zj ðtÞÞ 3
32 3 x1 ðtÞ z1 ðtÞ x1n 7 6 7 .. .. 76 .. 7 ¼ XeðtÞ; . . 54 . 5 xnn xn ðtÞ zn ðtÞ
2497
ð5Þ
j¼1
where eðtÞ ¼ ðe1 ðtÞ; . . . ; en ðtÞÞT and X ¼ ðxij Þnn 2 Rnn is the gain matrix to be determined for synchronizing both a drive system and response system. Furthermore, if a new error ^ei ðtÞ is defined by ^ei ðtÞ ¼ eat ei ðtÞ, then the dynamics of (4) can be transformed into the following form:
^e_ i ðtÞ ¼ ci ^ei ðtÞ þ a^ei ðtÞ þ
n X
Dij Gj ð^ej ðtÞÞ þ
n n X X Dsij Gj ð^ej ðt sj ðtÞÞÞ xij ^ej ðtÞ;
j¼1
j¼1
ð6Þ
j¼1
where
Gj ð^ej ðtÞÞ ¼ eat g j ðej ðtÞÞ; Gj ð^ej ðt sj ðtÞÞÞ ¼ eat g j ðej ðt sj ðtÞÞÞ: For further deriving the exponential synchronization condition on the control law, the following Lemma are needed. Lemma 1. Define a 2n 2n Hamiltonian matrix
"
P¼
e C ðH1 þ H2 Þ nIn
# EET ; e C
e ¼ diagðci aÞ þ X, i ¼ 1; 2; . . . ; n, E ¼ ½D Ds . where n is sufficiently small and positive constant, In is a n n identity matrix C e is a stable matrix and Hamiltonian matrix P has no eigenH1 ¼ diagðk2j Þ and H2 ¼ diagðe2as k2j =ð1 r ÞÞ, j ¼ 1; 2; . . . ; n: If C values on the imaginary axis, then the algebraic Riccati equation (ARE)
e TP P C e þ PEET P þ ðH1 þ H2 Þ þ nIn ¼ 0 C
ð7Þ
has a symmetric and positive definite solution P for a given a > 0: Remark 1. The proof is an immediate consequence of the Lemma 4 in the work of Doyle et al. [16], and therefore it is omitted here. e is stable if and only if all of its eigenvalues have negative real parts. All eigenvalues of C e Remark 2. A real matrix C defined in Lemma 1 can be arbitrarily assigned by appropriately choosing the controller gain matrix X. Especially, if we e are choose the gain matrix as a diagonal matrix X ¼ diagðxi Þ and xi > a ci , i ¼ 1; 2; . . . ; n, then the eigenvalues of C e is a stable matrix. ðxi þ ci aÞ < 0, i ¼ 1; 2; . . . ; n, which implies that C 3.2. Exponential synchronization The exponential synchronization problem of systems (1) and (2) can be solved if the controller gain matrix is suitably designed. The exponential synchronization condition is established in the following main theorems. Theorem 1. For these drive-response neural networks (1) and (2) which satisfy assumption (H1)–(H2), if the controller gain e is a stable matrix and Hamiltonian matrix P defined in Lemma 1 for a given a > 0 matrix X in (5) is suitably designed such that C has no eigenvalues on the imaginary axis, then the networks (1) and (2) are synchronized exponentially with a degree a at least. Proof. Step 1: Transform (6) into a compact form as follows:
e ^eðtÞ þ DGð^eðtÞÞ þ Ds Gð^eðt sðtÞÞÞ; ^e_ ðtÞ ¼ C where
e ¼ diagðci aÞ þ X; C Gð^eðtÞÞ ¼ ðG1 ð^e1 ðtÞÞ; G2 ð^e2 ðtÞÞ; . . . ; Gn ð^en ðtÞÞÞT ; Gð^eðt sðtÞÞÞ ¼ ðG1 ð^e1 ðt sðtÞÞÞ; G2 ð^e2 ðt sðtÞÞÞ; . . . ; Gn ð^en ðt sðtÞÞÞÞT :
ð8Þ
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J. Hu / Chaos, Solitons and Fractals 41 (2009) 2495–2501
e is stable and the Hamiltonian matrix P has no eigenvalues on the imaginary axis. According to Lemma the algeSince C braic Riccati equation (ARE) in (7) has a symmetric and positive definite solution P. Here since there exists reaction-diffusion terms, for convenience we need to suppose P as a positive diagonally matrix. To confirm that the origin of (8) is globally asymptotically convergent, a continuous Lyapunov functional VðtÞ is defined as follows:
VðtÞ ¼ ^eT ðtÞP^eðtÞ þ
n Z t e2as X G2 ð^ej ðs; xÞÞds: 1 r j¼1 tsj ðtÞ j
ð9Þ
It can be easily verified that VðtÞ is a nonnegative function over ½s ; þ1Þ and radially unbounded, i.e. VðtÞ ! þ1 as ^eðtÞ ! þ1. Using the definition of g j ðej ðtÞÞ, Gj ð^ej ðtÞÞ and the assumption ðH2 Þ yields
jg j ðej ðtÞÞj 6 kj jej ðtÞj; jGj ð^ej ðtÞÞj ¼ jeat Gj ðej ðtÞÞj 6 kj jeat ej ðtÞj ¼ kj j^ej ðtÞj: Step 2: Evaluating the time derivative of V along the trajectory of (8), and the following proofs compared with those in [6] is largely identical but with minor differences. Subsequently, we can obtain
eTP PC e ^eðtÞ þ ^eT ðtÞPDDT P^eðtÞ þ ^eT ðtÞPDs ðDs ÞT P^eðtÞ þ ^eT ðtÞH1 ^eðtÞ þ ^eT ðtÞH2 ^eðtÞ _ VðtÞ 6 ^eT ðtÞ C
ð10Þ
_ According to Lyapunov theory, the last inequality VðtÞ 6 nk^e2 ðtÞk indicates VðtÞ converges to zero asymptotically as well as ^eðtÞ ¼ 0 is asymptotically convergent. So we conclude eðtÞ converges to zero exponentially with a rate of a, i.e. j xi ðtÞ zi ðtÞ j6 bðaÞsups ðsÞ6s60 j /i ðsÞ ui ðsÞjeat is satisfied. The proof is thus completed. h Remark 3. As described in Ref. [15], in order to obtain the gain matrix X in the proposed controller (5), a computational procedure is proposed as follows: Step 1: Given a positive constant a and an arbitrarily sufficiently small positive constant a, choose a suitable gain matrix X e is a stable matrix. by using any eigenvalue assignment technique such that C Step 2: Construct the Hamiltonian matrix P in Lemma 1 and using the matlab eig function to check if P has no eigenvalues on the imaginary axis. If so, then the procedure goes to Step 4. Otherwise, the procedure continues to Step 3. e Þg more negative by selecting a new gain matrix X and go back to Step 2. Step 3: Tune the value of Remax fki ð C Step 4: Obtain the state-feedback controller (5). By constructing another Lyapunov functional, we can obtain the following theorem. Theorem 2. Under the assumptions (H1) and (H2), if the controller gain matrix X in (5) is suitably designed and there exist a positive real number r > 0, a positive definite matrix R and a positive diagonal matrix P ¼ diagðp1 ; p2 ; . . . ; pn Þ, such that the matrix s T 1 s 1 2PðA þ XÞK 1 þ PD þ DT P þ rð1 rÞ ðD Þ R D þ rPRP are negative definite, then the two chaotic nonlinear neural networks defined by (1) and (2) are synchronized exponentially with a degree a at least. Proof. To confirm that the origin of (6) is globally asymptotically convergent, a continuous Lyapunov functional VðtÞ is defined as follows:
VðtÞ ¼ 2
n X
pi
i¼1
Z
^ei ðtÞ
Gi ðsÞds þ
0
1 rð1 rÞ
Z
t
GT ð^eðsÞÞðDs ÞT R1 Ds Gð^eðsÞÞds:
ð11Þ
tsðtÞ
Evaluating the derivative of V along the solutions of (6), it follows that
_ VðtÞ ¼2
N X i¼1
þ 62
(
i¼1
pi
n X j¼1
d 1 dt rð1 rÞ
N X
þ
( pi Gi ð^ei ðtÞÞ ci ^ei ðtÞ þ a^ei ðtÞ þ Z
t
Dij Gj ð^ej ðtÞÞþ )
n n X X Dsij Gj ð^ej ðt sj ðtÞÞÞ xij ^ej ðtÞ j¼1
)
j¼1
GT ð^eðsÞÞðDs ÞT R1 Ds Gð^eðsÞÞds
tsðtÞ
ai þ a 2 Gi ð^ei ðtÞÞ þ 2GT ð^eðtÞÞPDGð^eðtÞÞ þ 2GT ð^eðtÞÞPDs Gð^eðt sðtÞÞÞ 2GT ð^eðtÞÞPXK 1 Gð^eðtÞÞ ki
1 1 GT ð^eðtÞÞðDs ÞT R1 Ds Gð^eðtÞÞ GT ð^eðt sðtÞÞÞðDs ÞT R1 Ds Gð^eðt sðtÞÞÞds: rð1 rÞ r
ð12Þ
Since R is a positive definite matrix, there must exist a nonsingular matrix S of appropriate dimension such that R ¼ ST S, then the third term in (12) can be estimated as
J. Hu / Chaos, Solitons and Fractals 41 (2009) 2495–2501
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h iT 2GT ð^eðtÞÞPDs Gð^eðt sðtÞÞÞ ¼ r1=2 SPGð^eðtÞÞ r1=2 ðS1 ÞT Ds Gð^eðt sðtÞÞÞ h i 1 r 1=2 SPGð^eðtÞÞ r 1=2 ðS1 ÞT Ds Gð^eðt sðtÞÞÞ þ rGT ð^eðtÞÞPRPGð^eðtÞÞ þ GT ð^eðt sðtÞÞÞ r ðDs ÞT R1 Ds Gð^eðt sðtÞÞÞds 1 6 þrGT ð^eðtÞÞPRPGð^eðtÞÞ þ GT ð^eðt sðtÞÞÞðDs ÞT R1 Ds Gð^eðt sðtÞÞÞds: r Substituting above inequality into (12), we have
_ VðtÞ 6 GT ð^eðtÞÞ 2PðA þ XÞK 1 þ 2aPK 1 þ PD þ DT P þ
1 ðDs ÞT R1 Ds þ rPRP Gð^eðtÞÞ; rð1 rÞ
ð13Þ
s T 1 s 1 If 2PðA þ XÞK 1 þ PD þ DT P þ rð1 rÞ ðD Þ R D þ rPRP are negative definite, there must exist small positive e such that s T 1 s 1 _ 2PðA þ XÞK 1 þ 2aPK 1 þ PD þ DT P þ rð1 rÞ ðD Þ R D þ rPRP. It follows from (13) that VðtÞ < 0, which implies that V is
strictly decreasing and converges to 0 when t ! 1; as well as ^eðtÞ ¼ 0 is asymptotically convergent. So we conclude eðtÞ converges to zero exponentially with a rate of a; i.e. j xi ðtÞ zi ðtÞ j6 bðaÞsups 6s60 j /i ðsÞ ui ðsÞ j eat is satisfied. The proof is thus completed. h Taking R ¼ P ¼ I and r ¼ 1, it can be easily to obtain the following corollary. Corollary 1. Under the assumptions (H1) and (H2), if the controller gain matrix X in (5) is suitably designed such that the matrix 2ðA þ XÞK 1 þ D þ DT þ ð11rÞ ðDs ÞT Ds þ I are negative definite, then the two chaotic nonlinear neural networks defined by (1) and (2) are synchronized exponentially with a degree a at least. 4. An illustrative example To demonstrate the validity of the exponential synchronization condition, an example is given in this section and the dynamic behaviours are shown for the cellular neural networks with time-varying delays and the Hopfield neural networks with time-varying delays, respectively. Example 1. A two-dimensional nonlinear neural networks with time-varying delays is given as follows:
x_ i ðtÞ ¼ ci ðxi ðtÞÞ þ
2 X
Dij fj ðxj ðtÞÞ þ
j¼1
where c1 ðvÞ ¼ 2v sinðvÞ;
D ¼ ðDij Þ22 ¼
2 X Dsij fj ðxj ðt sj ðtÞÞÞ þ J i ; i ¼ 1; 2;
ð14Þ
j¼1
c2 ðvÞ ¼ 2v þ cosðvÞ,
2:1
0:12
5:1
3:2
;
J 1 ¼ J 2 ¼ 0 and
Ds ¼ ðDsij Þ22 ¼
1:6 0:1 0:2
2:4
:
The delays s1 ðtÞ ¼ s2 ðtÞ ¼ s3 ðtÞð1 e Þ=ð1 þ e Þ are time-varying and satisfy 0 6 sj ðtÞ 6 1 ¼ s , 0 6 s_ j ðtÞ 6 0:5; j ¼ 1; 2: Noting that while the nonlinear functions are taking as c1 ðvÞ ¼ c2 ðvÞ ¼ v, the neural network reduces to a general chaotic neural networks. And we can see c01 ðvÞ ¼ 2 cosðvÞ P 1, c02 ðvÞ ¼ 2 þ sinðvÞ P 1, so (H1) is satisfied. Now consider the Hopfield neural networks, i.e., taking the activation function as g 1 ðvÞ ¼ g 2 ðvÞ ¼ tanhðvÞ. So the system satisfies assumption (H2) with k1 ¼ k2 ¼ 1. Fig. 1 shows the chaotic attractor plot with the initial condition /ðs; xÞ ¼ ½1; 1:5T . Fig. 2 shows the power spectral plot of this system. And by using Jacobia method, the Lyapunov exponents are obtained as follows: t
k1 ¼ 0:3037;
t
k2 ¼ 0:3608:
As we can see, from the phase plot, power spectral plot and the Lyapunov exponents, the system (14) is a chaotic system. To achieve synchronization, the response system is designed as follows:
z_ i ðtÞ ¼ ci ðzi ðtÞÞ þ
2 X
Dij fj ðzj ðtÞÞ þ
j¼1
2 X Dsij fj ðzj ðt sj ðtÞÞÞ þ J i þ ui ðtÞ;
i ¼ 1; 2;
ð15Þ
j¼1
If we choose the synchronization degree a ¼ 0:15 and the controller gain matrix is chosen as X ¼ diagðxi Þ; i ¼ 1; 2; with x1 ¼ 0:4; x2 ¼ 0:3; it is easily found that the matrix Hamiltonian matrix P, then the three sub-matrices of P are obtained as follows:
H1 ¼
1 0 ; 0 1
H2 ¼ ðe2as =ð1 rÞÞ
1 0 6:9944 10:5340 ; EET ¼ : 0 1 10:5340 42:0500
e is stable and the eigenvalues of Hamiltonian matrix P are as follows: On the other hand, the matrix C
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J. Hu / Chaos, Solitons and Fractals 41 (2009) 2495–2501
5 4 3 2
x
2
1 0 -1 -2 -3 -4 -5 -1
-0.5
0
0.5
1
x1 Fig. 1. Phase plot of x1 x2 .
25
Power Spectral
20
15
10
5
0
-5 0
20
40
60
80
100
120
Frequency Fig. 2. Power spectral plot.
1.8
|e |
1.6
|e2|
1
1.4
errors
1.2 1 0.8 0.6 0.4 0.2 0 -0.2
0
50
100
150
t/s Fig. 3. Synchronized errors when x1 ¼ 0:4; x2 ¼ 0:3.
200
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J. Hu / Chaos, Solitons and Fractals 41 (2009) 2495–2501
0.8 0.7
|e1|
0.6
|e2|
errors
0.5 0.4 0.3 0.2 0.10 -0.1 -0.2 0
50
100
150
200
t/s Fig. 4. Synchronized errors when x1 ¼ 1:2; x2 ¼ 1:3.
k1 ¼ 1:1755 þ 3:7135i; k2 ¼ 1:1755 3:7135i; k3 ¼ 1:0897 þ 12:3398i; k4 ¼ 1:0897 12:3398i; i.e., the Hamiltonian matrix P has no eigenvalues on the imaginary axis for the sufficiently small constant e ¼ 104 . Hence, from Theorem 1, the two chaotic nonlinear neural networks defined by (14) and (15) are synchronized exponentially. In this case, Fig. 3 shows the synchronized errors between the states of system (14) and those of system (15). If the controller gain matrix is chosen as X ¼ diagðxi Þ; i ¼ 1; 2, with x1 ¼ 1:2; x2 ¼ 1:3, the synchronized errors between the states of system (14) and those of system (15) are shown in Fig. 4. It can be seen that choosing the controller gain matrix properly, the synchronization speed may be very fast.
5. Conclusions In this paper, two sufficient conditions to guarantee the globally exponential synchronization for a class of chaotic nonlinear neural networks with time-varying delays are presented. This class of neural networks generalize a large class of delayed neural networks, such as Hopfield neural networks and cellular neural networks. Our results have been shown to be more extensive and easier to verify than those reported previously. A simulation example is given to show the effectiveness of the proposed method and criterion. References [1] Qi H, Qi L. Deriving sufficient conditions for global asymptotical stability of delayed neural networks via nonsmooth analysis. IEEE Trans Neural Networks 2004;15:99–109. [2] Liu HL, Chen GH. Delay-dependent stability for neural networks with time-varying delay. Chaos, Solitons & Fractals 2007;33(1):171–7. [3] Ju H Park. On global stability criterion of neural networks with continuously distributed delays. Chaos, Solitons & Fractals 2008;37(2):444–9. [4] Zhao ZJ. Global exponential robust periodicity and stability of interval neural networks with both variable and unbounded delays. Chaos, Solitons & Fractals 2008;36(1):91–7. [5] Lou X, Cui B. Boundedness and exponential stability for nonautonomous RCNNs with distributed delays. Comput Math Appl 2007;54(4):589–98. [6] Huang ZT, Yang QG, Luo XS. Exponential stability of impulsive neural networks with time-varying delays. Chaos, Solitons & Fractals 2008;35(4):770–80. [7] Lou XY, Cui BT. Stochastic exponential stability for Markovian jumping BAM neural networks with time-varying delays. IEEE Trans Syst Man Cybernet B 2007;37(3):713–9. [8] Qu Q. Global robust exponential stability of delayed neural networks: an LMI approach. Chaos, Solitons & Fractals 2007;32(5):1742–8. [9] Gilli M. Strange attractors in delayed cellular neural networks. IEEE Trans Circuits Syst I 1993;40(11):849–53. [10] Cao JD, Lu JQ. Adaptive synchronization of neural networks with or without time-varying delays. Chaos 2006;16(1):013133-1–3-6. [11] Zhou J, Chen TP, Xiang L. Chaotic lag synchronization of coupled delayed neural networks and its applications in secure communication. Circuits Syst Signal Process 2005;24(5):599–613. [12] Lou XY, Cui BT. Synchronization of competitive neural networks with different time scales. Physica A 2007;380:563–76. [13] Lou XY, Cui BT. Asymptotic synchronization of a class of neural networks with reaction-diffusion terms and time-varying delays. Comput Math Appl 2006;52(6-7):897–904. [14] Cheng CJ, Liao TL, Hwang CC. Exponential synchronization of a class of chaotic neural networks. Chaos, Solitons & Fractals 2005;24:197–206. [15] Cheng CJ, Liao TL, Yan JJ, Hwang CC. Exponential synchronization of a class of neural networks with time-varying delays. IEEE Trans Syst Man Cybernet B: Cybernetics 2006;36(1):209–15. [16] Doyle JC, Glover K, Khargonekar PP, Francis BA. Statespace solutions to standard H2 and H1 control problems. IEEE Trans Automat Control 1989;34:831–47. [17] Clarke FH. Optimization and nonsmooth analysis. New york: Wiley; 1983.