- Email: [email protected]
Dynamics of Cohen–Grossberg neural networks with time-varying delays
Physics Letters A 354 (2006) 414–422 www.elsevier.com/locate/pla
Dynamics of Cohen–Grossberg neural networks with time-varying delays ✩ Haijun Jiang a,b,∗ , Jinde Cao a , Zhidong Teng b a Department of Mathematics, Southeast University, Nanjing 210096, China b College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
Received 20 April 2005; received in revised form 31 January 2006; accepted 31 January 2006 Available online 9 February 2006 Communicated by A.P. Fordy
Abstract This Letter studies the Cohen–Grossberg networks dynamical systems with variable coefficients and time-varying delays. By applying the Young inequality technique, Dini derivative and introducing many real parameters, and estimating the upper bound of solutions of the system, a series of new and useful criteria on the boundedness, global exponential stability are established. The results obtained in this Letter extend and generalize the corresponding results existing in previous literature. © 2006 Elsevier B.V. All rights reserved. Keywords: Cohen–Grossberg neural networks; Time-varying delay; Boundedness; Global exponential stability; Equilibrium point
1. Introduction In recent years, the dynamical characteristic such as stability and periodicity of Hopfield network, cellular neural network and bidirectional associative memory neural network play an important rule in the pattern recognition, associative memory, and combinatorial optimization (see [1–13]). Among them, the Cohen–Grossberg neural network [14] is an important one, which can be described by the set of ordinary differential equations: n dxi (t) cij gj xj (t) , = −ai xi (t) bi xi (t) − dt j =1
i = 1, 2, . . . , n, ✩
(1)
This work was supported by the National Natural Science Foundation of China under Grants 60574043, 60373067 and 10361004, and the Natural Science Foundation of Jiangsu Province, China under Grant BK2003053, the Postdoctoral Science Foundation of Jiangsu Province, China under Grant 1660631111, The Major Project of the Ministry of Education and the Doctoral Foundation of Xinjiang University. * Corresponding author. E-mail addresses: [email protected] (H. Jiang), [email protected] (J. Cao). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.01.078
where, n 2 is the number of neurons in the network; xi (t) denotes the state variable of the ith neuron at time t ; gj (xj (t)) denotes the activation function of the j th neuron at time t ; the feedback matrix C = (cij )n×n indicates the strength of the neuron interconnections within the network; ai (xi (t)) represents an amplification function; bi (xi (t)) is an appropriately behaved function such that the solutions of model (1) remain bounded. In reality, due to the finite switching speeds of neurons and amplifiers, time delays inevitably exist in biological and artificial neural networks and thus should be incorporated into the model. In [15], the authors observed that time delays could induce instability, causing sustained oscillations which may be harmful to a system by experiment and numerical analysis. For the Cohen–Grossberg system (1), in [16], the authors also introduced delays by considering the following system of delayed differential equations: n K dxi (t) k cij gj xj (t − τk ) , = −ai xi (t) bi xi (t) − dt k=0 j =1
i = 1, 2, . . . , n,
(2)
where the time delays τk (k = 0, 1, . . . , K) are arrange such that 0 = τ0 < τ1 < · · · < τK . In [17–22], the authors further
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
investigated the following modified Cohen–Grossberg system: n dxi (t) cij gj xj (t) = −ai xi (t) bi xi (t) − dt j =1 n dij gj xj (t − τij ) + Ii , − j =1
i = 1, 2, . . . , n,
2. Model description
j =1
(4)
However, as we well know, the nonautonomous phenomenon often occurs in many realistic systems. Particularly, when we consider a long-time dynamical behavior of a system, the parameters of the system usually will arise change along with time. In addition, in many applications, the property of periodic oscillatory solutions of a neural networks also is great interest. Therefore, the research on the nonautonomous neural networks with delays is very important in like manner. In this Letter, like in Cao and Liang [24], Li [25], we will continuous to consider a modified Cohen–Grossberg neural network with variable coefficients and time-varying delays. n dxi (t) cij (t)gj xj (t) = −ai xi (t) bi xi (t) − dt j =1 n dij (t)gj xj t − τij (t) + Ii (t) , − j =1
i = 1, 2, . . . , n,
establish new criteria on the boundedness for general nonautonomous system (5). And the conditions ensuring the exponential stability of the Cohen–Grossberg neural network are obtained in Section 4. In Section 5, two examples are given to illustrate the results obtained in this Letter. In Section 6, we will give some concluding remarks of the results.
(3)
where D = (dij )n×n indicates the strength of the neuron interconnections within the network with time delay parameters τij . In [23], the authors discussed the following Cohen–Grossberg neural networks with time-varying delays: n dxi (t) cij gj xj (t) = −ai xi (t) bi xi (t) − dt j =1 n dij gj xj t − τij (t) + Ii , − i = 1, 2, . . . , n.
415
(5)
where i = 1, 2, . . . , n; cij (t), dij (t) and Ii (t) (i, j = 1, 2, . . . , n) are continuous and bounded functions defined on R+ . The main purpose of this Letter is to study the dynamic behavior of the general Cohen–Grossberg neural networks system (5). Different from Cao and Liang [24], in this Letter we will introduce a new research method. That is, by applying the Young inequality technique, Dini derivative and introducing many real parameters, estimate directly the upper bound of solutions of system (5). We will establish a series of new and useful criteria, which are completely different from the corresponding results given in Cao and Liang [24] on the boundedness and global exponential stability. We will see that the results obtained in this Letter will extend and generalize the corresponding results existing in [16–25]. The rest of this Letter is organized as follows. In Section 2, we will give a description for system (5). In Section 3, we will
In system (5), the integer n corresponds to the number of units in a neural network; xi corresponds to the state of the ith unit at time t ; gj (xj (t)) denotes the activation function of the j th unit at time t ; cij (t), dij (t) denotes the strength of the j th unit on the ith unit at time t and t −τij (t), respectively; Ii (t) denotes the external bias on the ith unit at time t ; τij (t) are delays caused during the switching and transmission processes; and ai (xi (t)) represents an amplification function; bi (xi (t)) is an appropriately behaved function such that the solution of model (5) remain bounded. In this Letter, for system (5) we introduce the following assumptions. (H1 ) Each function ai (u) is bounded, positive and locally Lipschitz continuous. Furthermore, 0 < α i ai (u) α i < +∞ for all u ∈ R = (−∞, +∞) and i = 1, 2, . . . , n. (H2 ) Each functions bi (u) is locally Lipschitz continuous and there exists βi > 0 such that ubi (u) βi u2 for u ∈ R, i = 1, 2, . . . , n. (H2 ) Each functions bi (u) ∈ C 1 (R, R) and b˙i (u) βi > 0; both bi (·) and bi−1 (·) are locally Lipschitz continuous. (H3 ) Each functions gj (u) (j = 1, 2, . . . , n) satisfy the Lipschitz condition, i.e., there are constants μj > 0 such that gj (u1 ) − gj (u2 ) μj |u1 − u2 | for all u1 , u2 ∈ R. (H3 ) Each functions gj (u) (j = 1, 2, . . . , n) is bounded and satisfies the Lipschitz condition with Lipschitz constant μj > 0. (H4 ) Functions τij (t) (i, j = 1, 2, . . . , n) are non-negative, bounded and continuous defined on R+ = [0, +∞). Let τ = sup{τij (t): t ∈ [0, +∞), i, j = 1, 2, . . . , n} and r > 1 be a given constant. We denoted by C[−τ, 0] the Banach space of continuous functions φ(θ) = (φ1 (θ ), φ2 (θ ), . . . , φn (θ )) : [−τ, 0] → R n with the norm r 1/r . φ = sup φ(θ), where φ(θ) = max φi (θ ) −τ θ0
1in
In this Letter we always assume that all solutions of system (5) satisfy the following initial conditions xi (θ ) = φi (θ ),
θ ∈ [−τ, 0], i = 1, 2, . . . , n,
(6)
where φ = (φ1 , φ2 , . . . , φn ) ∈ C[−τ, 0]. It is well known that by the fundamental theory of functional differential equations [26], system (5) has a unique solution x(t) = (x1 (t), x2 (t), . . . , xn (t)) satisfying the initial condition (6). Definition 1. A solution x(t) = (x1 (t), x2 (t), . . . , xn (t)) of system (5) is said to be bounded on R+ , if there is a constant B > 0
416
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
such that xi (t) B
for all t ∈ R+ , i = 1, 2, . . . , n.
When cij (t) = cij , dij (t) = dij , Ii (t) = Ii (i, j = 1, 2, . . . , n), where cij , dij and Ii are constants, then system (5) is transformed into system (4). For system (4), making use of the approach in Wang and Zou [17], it is easy to obtain the following lemma. (H2 ),
(H3 )
and (H4 ) hold, then for every Lemma 1. If (H1 ), input Ii , there exist an equilibrium point x ∗ = (x1∗ , x2∗ , . . . , xn∗ ) for system (4). Let y(t) = x(t) − x ∗ , substitute x(t) = y(t) + x ∗ in (7) and we have dyi (t) ∗ = −ai yi (t) + xi bi yi (t) + xi∗ − bi xi∗ dt −
n j =1
−
n
cij gj yj (t) + xj∗ − gj xj∗ ∗ ∗ . dij gj yj t − τij (t) + xj − gj xj
(7)
Lemma 3. Assume that u(t) is a differentiable function defined on R+ . Then for any t ∈ R+ the Dini upper right derivative D + |u(t)| of function |u(t)| exists and has the expression as follows 1 D + u(t) = lim sup u(t + h) − u(t) = σ u(t) u(t), ˙ h→0+ h where
σ u(t) =
and u(t) ˙ =
1, −1, 0,
du(t) dt .
3. Boundedness In this section, we consider the boundedness for system (5). We have the following results. Theorem 1. Under assumptions (H1 )–(H4 ), all solutions of system (5) are defined and are bounded on R+ if there are constants hij , lij , h∗ij , lij∗ ∈ R, ωi > 0 (i, j = 1, 2, . . . , n), r > 1 and σ > 0 such that
j =1
Denote Ai (yi (t)) = ai (yi (t)+x ∗ ), Bi (yi (t)) = bi (yi (t)+xi∗ )− bi (xi∗ ), fj (yj (t)) = gj (yj (t) + xj∗ ) − gj (xj∗ ), then system (7) becomes dyi (t) = −Ai yi (t) Bi yi (t) dt n n cij fj yj (t) − dij fj yj t − τij (t) . − j =1
By assumption (H3 ), we have fj (u) μj |u|.
(9)
In the following, the Young inequality and Dini upper right derivative will be used to prove the main results of this Letter, particularly, the boundedness of solutions and global exponential stability of system (1). On the Young inequality we have the following lemma (see [12,27]). Lemma 2. Assume that a 0, b 0, p > 1, q > 1 with
1 p
+
= 1. Then we have inequality
ab
n
r−hij r−lij ωj α i cij (t) r−1 μjr−1
j =1
−
n
h l ωj α i μjij cij (t) ij
j =1
− (r − 1)
n
∗
∗
ij r−hij r−l ωj α i dij (t) r−1 μ r−1
j
j =1
−
n
h∗ l∗ ωj α i μjij dij (t) ij > σ
(11)
j =1
Definition 2. System (4) or (8) is globally exponentially stable, if there are constants > 0 and M 1 such that
y(t) = x(t) − x ∗ M φ − x ∗ exp(−t) for all t 0. (10)
1 q
rωi α i βi − (r − 1)
j =1
(8)
if u(t) > 0 or u(t) = 0 and u(t) ˙ > 0, if u(t) < 0 or u(t) = 0 and u(t) ˙ < 0, if u(t) = 0 and u(t) ˙ =0
1 p 1 q a + b . p q
And on the Dini upper right derivative we have the following lemma (see [12]).
for all t 0 and i = 1, 2, . . . , n. Proof. Let x(t) = (x1 (t), x2 (t), . . . , xn (t)) be any solution of system (5) with initial function φ ∈ C[−τ, 0] at t = 0. Let further xi (t) = ωi ui (t) (i = 1, 2, . . . , n), then system (5) transforms into the following form: n 1 dui (t) cij (t)gj ωj uj (t) = − ai (ωi ui ) bi ωi ui (t) − dt ωi j =1 n dij (t)gj ωj uj t − τij (t) + Ii (t) , − j =1
where i = 1, 2, . . . , n. Calculating the upper right derivative D + |ui (t)|r , we get r r−1 dui (t) D + ui (t) = r ui (t) σ ui (t) dt r−1 1 = r ui (t) σ ui (t) − ai (ωi ui ) bi ωi ui (t) ωi
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
−
n j =1
−
n
cij (t)gj ωj uj (t) dij (t)gj ωj uj t − τij (t) + Ii (t)
417
We further have r r D + ui (t) − rα i βi ui (t)
+ (r − 1)
r −rα i βi ui (t) n ωj j =1
+r
ωi
n ωj j =1
ωi
+
r−1 α i cij (t)ui (t) μj uj (t)
j =1
r−1 α i dij (t)ui (t) μj uj t − τij (t)
r−1 1 α i Ii (t)ui (t) ωi
+r
n r−1 1 α i cij (t)gj (0)ui (t) ωi
+
n r−1 1 α i dij (t)gj (0)ui (t) . ωi
=r
j =1
ωi
+ (r − 1) + (r − 1)
+
n ωj j =1
ωi
ωi
n ωj j =1
ωi
=r
j =1
ωi
−1 ui (t)r + M ui (t)
r−hij r−lij r α i cij (t) r−1 μ r−1 ui (t)
+
ωi
ωi
j
h∗ l ∗ r α i dij (t) ij μ ij uj t − τij (t) . j
h∗ l ∗ α i dij (t) ij μjij
j =1
r max uj (s) (13)
t−τ st
t 0, i = 1, 2, . . . , n . We choose a sufficiently large constant p such that −rωi α i βi + (r − 1)
n
r−hij r−lij ωj α i cij (t) r−1 μjr−1
j =1
∗
∗
ωi
h l r α i cij (t) ij μjij uj (t)
for all t 0 and i = 1, 2, . . . , n, where n ∗ cij (t) + dij (t) gj (0) : M = sup α i Ii (t) +
j
r−hij r−lij r α i dij (t) r−1 μ r−1 ui (t)
ωi
n ωj j =1
∗ ∗
ij r−hij r−l r r−1 r r−1 r−1 ui (t) α i dij (t) μ
n ωj j =1
+
j
j =1
n ωj j =1
j
h∗ l ∗ r 1/r × dij (t) ij μjij uj t − τij (t) (r − 1)
+
h l r α i cij (t) ij μ ij uj (t)
n ωj
j
r−1 α i dij (t)ui (t) μj uj t − τij (t)
n ωj
∗
∗
r−hij r−lij ωj α i dij (t) r−1 μ r−1
∗
and r
n
ωi
r−hij r−lij α i cij (t) r−1 μjr−1
j =1
j
j =1
n ωj j =1
h l r 1/r × cij (t) ij μjij uj (t) (r − 1)
h∗ l ∗ r α i dij (t) ij μjij uj t − τij (t)
1 −rωi α i βi ωi
ij r−hij r−l r r−1 r α i cij (t) r−1 μ r−1 ui (t)
n ωj
n ωj
j
j =1
(12)
r−1 ωj α i cij (t)ui (t) μj uj (t) ωi n ωj
ωi
∗
∗
r−hij r−lij r α i dij (t) r−1 μ r−1 ui (t)
ωi n α i cij (t)gj (0) +r Ii (t) + ωi j =1 n dij (t)gj (0) ui (t)r−1 +
Estimating the right of (12) by using Young inequality. Since
j =1
n ωj
j =1
j =1
r
ωi
r−hij r−lij r α i cij (t) r−1 μjr−1 ui (t)
h l r α i cij (t) ij μjij uj (t)
j =1
j =1
n
n ωj
+ (r − 1)
+r
+r
ωi
j =1
j =1
+r
n ωj
+ dij (t) +
n j =1
r−h∗ ij r−1
∗ r−lij r−1
μj
+ M ∗ p −1
hij lij h∗ lij∗ r α i ωj cij (t) μj + dij (t) ij μj p <0
(14)
418
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
and φ < min {ωi }p. 1in
Then, we can obtain xi (t) < ωi p for all t 0 and i = 1, 2, . . . , n. In fact, if it is not true, then there exist some i and time t1 > 0 such that |xi (t1 )| = ωi p, D + |xi (t1 )|r 0, and |xj (t)| ωj p for all −τ < t t1 and j = 1, 2, . . . , n. From this we further obtain |ui (t1 )| = p, D + |ui (t1 )|r 0 and |uj (t)| p for all −τ t t1 and j = 1, 2, . . . , n. However, from (13) and (14) we get r D + ui (t1 ) 1 < −rωi α i βi ωi n
r−hij r−lij + (r − 1) ωj α i cij (t) r−1 μjr−1 j =1 ∗
ij r−hij r−l + dij (t) r−1 μjr−1 + M ∗ p −1 ∗
+
n
(H6 ) Each functions gij l (u) (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m) satisfy the Lipschitz condition, i.e., there are constants μij l > 0 such that gij l (u1 ) − gij l (u2 ) μij l |u1 − u2 | for all u1 , u2 ∈ R. (H7 ) Functions τij l (t) (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m) are non-negative, bounded and continuous defined on R+ = [0, +∞).
hij lij h∗ lij∗ r ij α i ωj cij (t) μj + dij (t) μj p < 0.
j =1
This is a contradiction. So, |ui (t)| < p for all t 0. Hence, |xi (t)| < ωi p for all t 0. Thus, the solution x(t) = (x1 (t), x2 (t), . . . , xn (t)) of system (5) is defined and is bounded on R+ . This completes the proof. 2 Remark 1. The condition (11) is quite general, because it includes many known assumptions as its special cases. For example, if in the condition (11) we choose the constants hij = lij = 1 and r = 2, then (11) will transform into the following form. (H5 ) There are constants ωi > 0 (i = 1, 2, . . . , n) and σ > 0 such that n n ωj μj α i cij (t) − ωj α i μj dij (t) > σ ωi α i βi − j =1
In system (15), the integer n corresponds to the number of units in a neural network and the integer m corresponds to the number of neural axons, that is, signals that emit from the ith unit have m pathways to the j th unit; xi corresponds to the state of the ith unit at time t ; gij l (xj (t)) denotes the activation function of the j th unit at time t ; cij l (t) denotes the strength of the j th unit on the ith unit at time t − τij l (t); Ii (t) denotes the external bias on the ith unit at time t ; τij l (t) corresponds to the transmission delay of the ith unit along the l axon of the j th unit at time t and is a non-negative function; and ai (xi (t)) represents an amplification function; bi (xi (t)) is an appropriately behaved function such that the solution of model (15) remain bounded. For system (15), the assumptions (H3 ) and (H4 ) are transformed into the following form.
j =1
for all t ∈ R+ and i = 1, 2, . . . , n.
Moreover, we have the following result similar to Theorem 1. Theorem 2. Under assumptions (H1 )–(H2 ), (H6 ) and (H7 ), all solutions of system (15) are defined and are bounded on R+ if there exist constants hij l , lij l ∈ R, ωi > 0 (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m), r > 1 and σ > 0 such that rωi α i β i − (r − 1)
n m
r−lij l r−hij l cij l (t) r−1 ωj α i μijr−1 l
l=1 j =1
−
n m
h l ωj α i μijijll cij l (t) ij l > σ
(16)
l=1 j =1
for all t 0 and i = 1, 2, . . . , n.
As a consequence of Theorem 1, we have the following corollary.
The proof of Theorem 2 is similar to Theorem 1. Here, we omit it. As a consequence of Theorem 2, we have the following result similar to Corollary 1.
Corollary 1. Assume that (H1 )–(H5 ) hold. Then all solutions of system (5) are defined and are bounded on R+ .
Corollary 2. Under assume that (H1 ), (H2 ), (H6 ) and (H7 ), if there are constants ωi > 0 (i = 1, 2, . . . , n) and σ > 0 such that
A general case of system (5) is the following modified Cohen–Grossberg neural networks with variable coefficients and time-varying delay dxi (t) = −ai xi (t) bi xi (t) dt n m − cij l (t)gij l xj t − τij l (t) + Ii (t) , l=1 j =1
i = 1, 2, . . . , n.
(15)
ωi α i βi −
n m
ωj α i μij l cij l (t) > σ
l=1 j =1
for all t 0 and i = 1, 2, . . . , n. Then all solutions of system (15) are defined and are bounded on R+ . Remark 2. In Cao and Liang [24], the authors established a series of results on the boundedness for systems (15), for example, see Theorem 1 in Cao and Liang [24]. We see that in these results the delay functions τij (t) (i, j = 1, 2, . . . , n, l =
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
1, 2, . . . , m) given in the systems are continuously differentiable on t ∈ R+ and satisfy the conditions inft∈R+ {1− τ˙ij (t)} > 0, where τ˙ij (t) express the derivative of τij (t) with respect to time t . Furthermore, functions 1 − τ˙ij (t) appear in the main assumptions. However, in this Letter, we do not require that the delay functions τij (t) and τij l (t) (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m) given in systems (5) and (15) are continuously differentiable and satisfy the conditions inft∈R+ {1 − τ˙ij (t)} > 0 and inft∈R+ {1 − τ˙ij l (t)} > 0. Particularly, we only require that the delay functions are non-negative, bounded and continuous on R+ . In addition, functions 1 − τ˙ij (t) and 1 − τ˙ij l (t) do not appear in the main assumptions. Remark 3. For system (5), when ai (xi (t)) ≡ 1, bi (xi (t)) = bi (t)xi (t) (in which bi (t) is not only differentiable but also bounded on interval [−τ, +∞), and its maximal lower bound is denoted as βi > 0), then system (5) transform into cellular neural networks system with variable coefficients and timevarying delays. In this case, we can see that the condition given in Theorem 2 is less conservative and less restrictive than those given in [28,29]. In addition, the results in [28,29] are special cases of ours.
In this section, we will obtain some criteria for global exponential stability of (4) or (8). Moreover, the uniqueness of the equilibrium point follows directly its global exponential stability. Theorem 3. Under assumptions (H1 ), (H2 ), (H3 ) and (H4 ), system (4) or (8) is globally exponentially stable if there exist constants hij , lij , h∗ij , lij∗ ∈ R, ωi > 0 (i, j = 1, 2, . . . , n), r > 1 and σ > 0 such that rωi α i βi − (r − 1)
n
−
ωj α i |cij |
l ωj α i μjij |cij |hij
r−hij r−1
− (r − 1)
j =1
−
n
r−lij
μjr−1
n
ωj α i |dij |
r−h∗ ij r−1
∗ r−lij r−1
∗
l
ωj α i μjij |cij |hij − (r − 1)
j =1
− erτ
n
1 dzi (t) cij fj ωj zj (t) = − Ai ωi zi (t) Bi ωi zi (t) − dt ωi j =1 n − (18) dij fj ωj zj t − τij (t) .
n
∗ r−lij
ωj α i μjr−1 |dij |
r−h∗ ij r−1
j =1
∗ lij
∗ 1 ωj α i μj |dij |hij − rωi > σ 2
for all t 0 and i = 1, 2, . . . , n. Let pi (t) = ert |zi (t)|r . Calculating the Dini upper right derivative D + pi (t) of pi (t), from (18) we get D + pi (t) r−1 r dzi (t) ert r zi (t) σ zi (t) + r zi (t) dt r−1 1 rt Bi ωi zi (t) r zi (t) σ zi (t) − Ai ωi zi (t) =e ωi n n cij fj ωj zj (t) − dij fj ωj zj t − τij (t) − r + r zi (t)
j =1
r r ert −rα i βi zi (t) + r zi (t) +r
n ωj j =1
+r
ωi
n ωj
ωi
r−1 α i |cij |μj zi (t) zj (t) r−1 α i |dij |μj zi (t) zj t − τij (t) .
(19)
Estimating the right of (19) by using Young inequality, we have r r D + pi (t) ert −rα βi zi (t) + r zi (t) i
n ωj j =1
n ωj j =1
Proof. Let yi (t) = ωi zi (t) (i = 1, 2, . . . , n), then system (8) is transformed into the following form.
r−hij r−1
j =1 n
+
for all t 0 and i = 1, 2, . . . , n.
j =1
−
(17)
j =1
r−lij
ωj α i μjr−1 |cij |
j =1 n
+ (r − 1)
ωj α i μjij |dij |hij > σ
n
μj
j =1 l∗
rωi α i βi − (r − 1)
j =1
j =1 n
By the condition (17), we can choose a constant > 0 such that
j =1
4. Stability results
419
ωi
+ (r − 1) +
j =1
ωi
r−hij r−1
r−lij r μjr−1 zi (t)
r l α i |cij |hij μjij zj (t) n ωj j =1
n ωj
ωi
α i |cij |
ωi
α i |dij |
h∗ij
α i |dij |
r−h∗ ij r−1
l∗ μjij zj t
∗ r−lij r−1
μj
zi (t)r
r − τij (t)
r = r( − α i βi )ert zi (t) + (r − 1)
n ωj j =1
ωi
α i |cij |
r−hij r−1
r−lij r μjr−1 ert zi (t)
420
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
+
n ωj
ωi
j =1
+ (r − 1)
r l α i |cij |hij μjij ert zj (t) n ωj j =1
+
n ωj
ωi
j =1
ωi
α i |dij | ∗
∗ r−lij r−1
r−h∗ ij r−1
μj
for all t 0. Hence,
zi (t) M φ − x ∗ e−t
r ert zi (t)
for all t 0. Furthermore, we finally have n
yi (t) nωM φ − x ∗ e−t
∗ lij
α i |dij |hij μj erτij (t) er(t−τij (t))
where, ω = max1in {ωi }. This shows that, from Definition 2, system (9) is globally exponentially stable. This completes the proof. 2
r × zj t − τij (t) r( − α i βi )pi (t) n r−lij r−hij ωj α i |cij | r−1 μjr−1 pi (t) + (r − 1) ωi
In the condition (17), if we choose the constants hij l = lij l = 1 and r = 2, as a special cases of Theorem 3, we have the following corollary.
j =1
+
n ωj
ωi
j =1
+ (r − 1)
l
α i |cij |hij μjij pj (t) n ωj j =1
+
n j =1
ωi
α i |dij |
∗ r−lij r−1
r−h∗ ij r−1
μj
Corollary 3. Under assumptions (H1 ), (H2 ), (H3 ) and (H4 ), system (4) or (8) is globally exponentially stable if there exist constants ωi > 0 (i = 1, 2, . . . , n) and σ > 0 such that
pi (t)
∗ lij
∗ ωj α i |dij |hij μj erτ sup pj (s). ωi t−τ st
(20)
Choose the constant M 1 such that M r > (min1in {ωir })−1 , then we have r |yi (0)|r |φi (0) − xi∗ |r = pi (0) = zi (0) = r ωi ωir
r φ − x ∗ r M r φ − x ∗ . r min{ωi } From this we further can obtain pi (t) < M r φ − x ∗ r for all t > 0 and i = 1, 2, . . . , n. In fact, if it is not true, then there exist some i and t1 > 0 such that pi (t1 ) = M r φ − x ∗ r , D + pi (t1 ) 0 and pj (t) M r φ − x ∗ r for all −τ t t1 and j = 1, 2, . . . , n. However, from (20), we can obtain 1 + D pi (t1 ) r( − α i βi )ωi ωi + (r − 1)
n
ωj αi |cij |
r−hij r−1
r−lij r−1
μj
j =1
+
n
l
ωj αi |cij |hij μjij
j =1
+ (r − 1)
n
ωj αi |dij |
+
j =1
h∗ij
ωj αi |dij |
∗ r−lij r−1
r−h∗ ij r−1
j =1 n
for all t 0,
i=1
l∗ μjij erτ
M
n
ωj μj α i |cij | −
j =1
n
ωj α i μj |dij | > σ
j =1
for all t ∈ R+ and i = 1, 2, . . . , n. For system (15), when cij l (t) = cij l , Ii (t) = Ii , where cij l , Ii (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m) are constants, then the system (15) is transformed into the following form. dxi (t) = −ai xi (t) bi xi (t) dt n m − cij l gij l xj t − τij l (t) + Ii , l=1 j =1
i = 1, 2, . . . , n.
(21)
For system (21), we have the following result. Theorem 4. Under assumptions (H1 ), (H2 ) and (H7 ), further, each functions gij l (u) (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m) is bounded and satisfies the Lipschitz condition with Lipschitz constant μij l > 0, system (21) is globally exponentially stable if there exist constants hij l , lij l ∈ R, ωi > 0 (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m), r > 1 and σ > 0 such that rωi α i β i − (r − 1)
μj
ωi α i βi −
n m
r−lij l
ωj α i μijr−1 l |cij l |
r−hij l r−1
l=1 j =1
r
φ−x
∗ r
r 1 − σ M r φ − x ∗ < 0. 2 This is a contradiction. So, pi (t) M r φ − x ∗ r for all t 0. That is, r
r ert zi (t) M r φ − x ∗
−
m
n
l
ωj α i μijijll |cij l |hij l > σ
l=1 j =1
for all t 0 and i = 1, 2, . . . , n. The proof of Theorem 4 is similar to Theorem 3, here we omit it. As a special cases of Theorem 4, we further have following corollary.
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
Corollary 4. Under assumptions (H1 ), (H2 ) and (H7 ), further, each function gij l (u) (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m) is bounded and satisfies the Lipschitz condition with Lipschitz constant μij l > 0, system (21) is globally exponentially stable if there exist constants ωi > 0 (i = 1, 2, . . . , n) and σ > 0 such that ωi α i βi −
n m
421
for all t 0. Thus, by Theorem 1 all solutions of system (22) are bounded. However, since the derivatives τ˙ij (t) have not the determinate sign on R+ , we see that for system (22) the assumptions given in Cao and Liang [24] do not satisfy. Therefore, the results obtained in Cao and Liang [24] cannot be applied to system (22).
ωj α i μij l |cij l | > σ
l=1 j =1
for all t 0 and i = 1, 2, . . . , n. Remark 4. From Theorems 3 and 4 we can see that the results obtained in this Letter improve and extend the correspondent to results given in prior literatures. The corresponding remarks are similar to Remark 2. Here, we omit it. 5. Two examples Example 1. Consider the following 2-dimensional Cohen– Grossberg neural networks with variable coefficients and timevarying delays system: dx1 (t) = − 3 + cos x1 (t) 18x1 (t) − (1 + cos t)f x1 (t) dt − (1 − sin t)f x2 (t) 1 − (1 + sin t)f x1 t − sin t − 1 3 1 − sin t −1 + sin t , − (1 − cos t)f x2 t − e 4 dx2 (t) = − 3 + sin x2 (t) 17x2 (t) − (1 + sin t)f x1 (t) dt − (1 − cos t)f x2 (t) 1 − (1 − cos t)f x1 t − sin t − 1 3 1 − sin t −1 + cos t , − (1 + sin t)f x2 t − e 4 (22) where f (x) = 0.5(|x + 1| − |x − 1|). This system satisfies all assumptions in this Letter with α 1 = 2, α 2 = 2, α 1 = 4, α 2 = 4, μj = 1 (j = 1, 2), β1 = 18, β2 = 17. Further, we choose ωi = hij = lij = 1 (i, j = 1, 2) and r = 2 in the condition (11), then we see that ω1 α 1 β1 − α 1 |c11 |ω1 μ1 − α 1 |c12 |ω2 μ2 − α 1 |d11 |ω1 μ1 − α 1 |d12 |ω2 μ2 = 36 − 4|1 + cos t| − 4|1 − sin t| − 4|1 + sin t| − 4|1 − cos t| > 4 > 0, ω2 α 2 β2 − α 2 |c21 |ω1 μ21 − α 2 |c22 |ω2 μ2 − α 2 |d21 |ω1 μ1 − α 2 |d22 |ω2 μ2 = 34 − 4|1 + sin t| − 4|1 + cos t| − 4|1 − cos t| − 4|1 + sin t| > 2 > 0.
Example 2. Consider the following 2-dimensional Cohen– Grossberg neural networks with delays dx1 (t) = − 8 + sin x1 (t) 7x1 (t) − tanh x1 (t) − 2 tanh x2 (t) dt 1 − tanh x1 t − sin t − 1 3 1 − tanh x2 t − e− sin t − 1 + 2 , 4 dx2 (t) = − 5 + cos x2 (t) 10x2 (t) − 2 tanh x1 (t) − tanh x2 (t) dt 1 − sin t −1 − tanh x1 t − e 4 1 − 2 tanh x2 t − sin t − 1 + 3 . (23) 3 This system satisfies all assumptions in this Letter with α 1 = 7, α 2 = 4, α 1 = 9, α 2 = 6, ωj = 1, μij = 1 (i, j = 1, 2), β1 = 7, β2 = 10, then α 1 β1 − α 1 |c11 |μ1 − α 1 |c12 |μ2 − α 1 |d11 |μ1 − α 1 |d12 |μ2 = 49 − 9 − 18 − 9 − 9 = 4 > 0, α 2 β2 − α 2 |c21 |μ1 − α 2 |c22 |μ2 − α 2 |d21 |μ1 − α 2 |d22 |μ2 = 40 − 12 − 6 − 6 − 12 = 4 > 0. From Corollary 3 we know the solutions (23) are globally exponentially stable. 6. Conclusions In this Letter, we introduce the new research method, that is, by introducing ingeniously many real parameters hij , lij , h∗ij , lij∗ , ωi and r, applying Young inequality technique and Dini derivative, directly estimate the above bound of solutions of the systems. We obtain a series of new criteria on the boundedness, globally exponentially stability for the systems (5) and (4). Particularly, for the delay functions τij (t) and τij l (t), we only require that they are non-negative, bounded and continuous on R+ . The results obtained in this Letter improve and extend many previous works, and they are easy to check and apply in practice. References [1] J.J. Hopfield, Proc. Natl. Acad. Sci. Biol. 81 (1984) 3088. [2] L.O. Chua, L. Yang, IEEE Trans. Circuits Systems 35 (1988) 1257. [3] L.O. Chua, L. Yang, IEEE Trans. Circuits Systems 35 (1988) 1273.
422
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
[4] B. Kosko, Neural Networks and Fuzzy Systems—A Dynamical System Approach Machine Intelligence, Prentice–Hall, Englewood Cliffs, NJ, 1992, p. 38. [5] B. Kosko, IEEE Trans. Systems Man Cybernet. 18 (1988) 49. [6] K. Gopalsamy, X.Z. He, IEEE Trans. Neural Networks 5 (1994) 998. [7] M. Forti, A. Tesi, IEEE Trans. Circuits Systems I 42 (1995) 354. [8] A. Michel, J.A. Farrell, W. Porod, IEEE Trans. Circuits Systems 36 (1989) 229. [9] S. Arik, V. Tavsanoglu, IEEE Trans. Circuits Systems I 47 (2000) 571. [10] X. Liao, G. Chen, E. Sanchez, Neural Networks 15 (2002) 855. [11] X. Liang, J. Wang, IEEE Trans. Circuits Systems I 47 (2000) 1258. [12] J. Cao, Phys. Lett. A 307 (2003) 136. [13] J. Cao, J. Wang, IEEE Trans. Circuits Systems I 50 (2003) 34. [14] M.A. Cohen, S. Grossberg, IEEE Trans. Systems Man Cybernet. 13 (1983) 815. [15] C. Marcus, R. Westervelt, Phys. Rev. A 39 (1989) 347. [16] H. Ye, A.N. Michel, K. Wang, Phys. Rev. E 51 (1995) 2611.
[17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
L. Wang, X. Zou, Neural Networks 15 (2002) 415. L. Wang, X. Zou, Physica D 170 (2002) 163. W. Lu, T. Chen, Neural Comput. 15 (2003) 1173. T. Chen, L. Rong, Phys. Lett. A 317 (2003) 436. J. Cao, X. Li, Physica D 212 (1–2) (2005) 54. K. Yuan, J. Cao, IEEE Trans. Circuits Systems I 52 (9) (2005) 1854. C. Hwang, C. Cheng, T. Li, Phys. Lett. A 319 (2003) 157. J. Cao, J. Liang, J. Math. Anal. Appl. 296 (2004) 665. Y. Li, Chaos Solitons Fractals 20 (2004) 459. J. Hale, Theory of Functional Differential Equations, Springer, New York, 1977. [27] G. Hardy, H. Littlewood, G. Polya, Inequality, Cambridge Univ. Press, Cambridge, 1934. [28] H. Jiang, Z. Li, Z. Teng, Phys. Lett. A 306 (2003) 313. [29] J. Liang, J. Cao, Phys. Lett. A 318 (2003) 53.
Dynamics of Cohen–Grossberg neural networks with time-varying delays ✩ Haijun Jiang a,b,∗ , Jinde Cao a , Zhidong Teng b a Department of Mathematics, Southeast University, Nanjing 210096, China b College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
Received 20 April 2005; received in revised form 31 January 2006; accepted 31 January 2006 Available online 9 February 2006 Communicated by A.P. Fordy
Abstract This Letter studies the Cohen–Grossberg networks dynamical systems with variable coefficients and time-varying delays. By applying the Young inequality technique, Dini derivative and introducing many real parameters, and estimating the upper bound of solutions of the system, a series of new and useful criteria on the boundedness, global exponential stability are established. The results obtained in this Letter extend and generalize the corresponding results existing in previous literature. © 2006 Elsevier B.V. All rights reserved. Keywords: Cohen–Grossberg neural networks; Time-varying delay; Boundedness; Global exponential stability; Equilibrium point
1. Introduction In recent years, the dynamical characteristic such as stability and periodicity of Hopfield network, cellular neural network and bidirectional associative memory neural network play an important rule in the pattern recognition, associative memory, and combinatorial optimization (see [1–13]). Among them, the Cohen–Grossberg neural network [14] is an important one, which can be described by the set of ordinary differential equations: n dxi (t) cij gj xj (t) , = −ai xi (t) bi xi (t) − dt j =1
i = 1, 2, . . . , n, ✩
(1)
This work was supported by the National Natural Science Foundation of China under Grants 60574043, 60373067 and 10361004, and the Natural Science Foundation of Jiangsu Province, China under Grant BK2003053, the Postdoctoral Science Foundation of Jiangsu Province, China under Grant 1660631111, The Major Project of the Ministry of Education and the Doctoral Foundation of Xinjiang University. * Corresponding author. E-mail addresses: [email protected] (H. Jiang), [email protected] (J. Cao). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.01.078
where, n 2 is the number of neurons in the network; xi (t) denotes the state variable of the ith neuron at time t ; gj (xj (t)) denotes the activation function of the j th neuron at time t ; the feedback matrix C = (cij )n×n indicates the strength of the neuron interconnections within the network; ai (xi (t)) represents an amplification function; bi (xi (t)) is an appropriately behaved function such that the solutions of model (1) remain bounded. In reality, due to the finite switching speeds of neurons and amplifiers, time delays inevitably exist in biological and artificial neural networks and thus should be incorporated into the model. In [15], the authors observed that time delays could induce instability, causing sustained oscillations which may be harmful to a system by experiment and numerical analysis. For the Cohen–Grossberg system (1), in [16], the authors also introduced delays by considering the following system of delayed differential equations: n K dxi (t) k cij gj xj (t − τk ) , = −ai xi (t) bi xi (t) − dt k=0 j =1
i = 1, 2, . . . , n,
(2)
where the time delays τk (k = 0, 1, . . . , K) are arrange such that 0 = τ0 < τ1 < · · · < τK . In [17–22], the authors further
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
investigated the following modified Cohen–Grossberg system: n dxi (t) cij gj xj (t) = −ai xi (t) bi xi (t) − dt j =1 n dij gj xj (t − τij ) + Ii , − j =1
i = 1, 2, . . . , n,
2. Model description
j =1
(4)
However, as we well know, the nonautonomous phenomenon often occurs in many realistic systems. Particularly, when we consider a long-time dynamical behavior of a system, the parameters of the system usually will arise change along with time. In addition, in many applications, the property of periodic oscillatory solutions of a neural networks also is great interest. Therefore, the research on the nonautonomous neural networks with delays is very important in like manner. In this Letter, like in Cao and Liang [24], Li [25], we will continuous to consider a modified Cohen–Grossberg neural network with variable coefficients and time-varying delays. n dxi (t) cij (t)gj xj (t) = −ai xi (t) bi xi (t) − dt j =1 n dij (t)gj xj t − τij (t) + Ii (t) , − j =1
i = 1, 2, . . . , n,
establish new criteria on the boundedness for general nonautonomous system (5). And the conditions ensuring the exponential stability of the Cohen–Grossberg neural network are obtained in Section 4. In Section 5, two examples are given to illustrate the results obtained in this Letter. In Section 6, we will give some concluding remarks of the results.
(3)
where D = (dij )n×n indicates the strength of the neuron interconnections within the network with time delay parameters τij . In [23], the authors discussed the following Cohen–Grossberg neural networks with time-varying delays: n dxi (t) cij gj xj (t) = −ai xi (t) bi xi (t) − dt j =1 n dij gj xj t − τij (t) + Ii , − i = 1, 2, . . . , n.
415
(5)
where i = 1, 2, . . . , n; cij (t), dij (t) and Ii (t) (i, j = 1, 2, . . . , n) are continuous and bounded functions defined on R+ . The main purpose of this Letter is to study the dynamic behavior of the general Cohen–Grossberg neural networks system (5). Different from Cao and Liang [24], in this Letter we will introduce a new research method. That is, by applying the Young inequality technique, Dini derivative and introducing many real parameters, estimate directly the upper bound of solutions of system (5). We will establish a series of new and useful criteria, which are completely different from the corresponding results given in Cao and Liang [24] on the boundedness and global exponential stability. We will see that the results obtained in this Letter will extend and generalize the corresponding results existing in [16–25]. The rest of this Letter is organized as follows. In Section 2, we will give a description for system (5). In Section 3, we will
In system (5), the integer n corresponds to the number of units in a neural network; xi corresponds to the state of the ith unit at time t ; gj (xj (t)) denotes the activation function of the j th unit at time t ; cij (t), dij (t) denotes the strength of the j th unit on the ith unit at time t and t −τij (t), respectively; Ii (t) denotes the external bias on the ith unit at time t ; τij (t) are delays caused during the switching and transmission processes; and ai (xi (t)) represents an amplification function; bi (xi (t)) is an appropriately behaved function such that the solution of model (5) remain bounded. In this Letter, for system (5) we introduce the following assumptions. (H1 ) Each function ai (u) is bounded, positive and locally Lipschitz continuous. Furthermore, 0 < α i ai (u) α i < +∞ for all u ∈ R = (−∞, +∞) and i = 1, 2, . . . , n. (H2 ) Each functions bi (u) is locally Lipschitz continuous and there exists βi > 0 such that ubi (u) βi u2 for u ∈ R, i = 1, 2, . . . , n. (H2 ) Each functions bi (u) ∈ C 1 (R, R) and b˙i (u) βi > 0; both bi (·) and bi−1 (·) are locally Lipschitz continuous. (H3 ) Each functions gj (u) (j = 1, 2, . . . , n) satisfy the Lipschitz condition, i.e., there are constants μj > 0 such that gj (u1 ) − gj (u2 ) μj |u1 − u2 | for all u1 , u2 ∈ R. (H3 ) Each functions gj (u) (j = 1, 2, . . . , n) is bounded and satisfies the Lipschitz condition with Lipschitz constant μj > 0. (H4 ) Functions τij (t) (i, j = 1, 2, . . . , n) are non-negative, bounded and continuous defined on R+ = [0, +∞). Let τ = sup{τij (t): t ∈ [0, +∞), i, j = 1, 2, . . . , n} and r > 1 be a given constant. We denoted by C[−τ, 0] the Banach space of continuous functions φ(θ) = (φ1 (θ ), φ2 (θ ), . . . , φn (θ )) : [−τ, 0] → R n with the norm r 1/r . φ = sup φ(θ), where φ(θ) = max φi (θ ) −τ θ0
1in
In this Letter we always assume that all solutions of system (5) satisfy the following initial conditions xi (θ ) = φi (θ ),
θ ∈ [−τ, 0], i = 1, 2, . . . , n,
(6)
where φ = (φ1 , φ2 , . . . , φn ) ∈ C[−τ, 0]. It is well known that by the fundamental theory of functional differential equations [26], system (5) has a unique solution x(t) = (x1 (t), x2 (t), . . . , xn (t)) satisfying the initial condition (6). Definition 1. A solution x(t) = (x1 (t), x2 (t), . . . , xn (t)) of system (5) is said to be bounded on R+ , if there is a constant B > 0
416
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
such that xi (t) B
for all t ∈ R+ , i = 1, 2, . . . , n.
When cij (t) = cij , dij (t) = dij , Ii (t) = Ii (i, j = 1, 2, . . . , n), where cij , dij and Ii are constants, then system (5) is transformed into system (4). For system (4), making use of the approach in Wang and Zou [17], it is easy to obtain the following lemma. (H2 ),
(H3 )
and (H4 ) hold, then for every Lemma 1. If (H1 ), input Ii , there exist an equilibrium point x ∗ = (x1∗ , x2∗ , . . . , xn∗ ) for system (4). Let y(t) = x(t) − x ∗ , substitute x(t) = y(t) + x ∗ in (7) and we have dyi (t) ∗ = −ai yi (t) + xi bi yi (t) + xi∗ − bi xi∗ dt −
n j =1
−
n
cij gj yj (t) + xj∗ − gj xj∗ ∗ ∗ . dij gj yj t − τij (t) + xj − gj xj
(7)
Lemma 3. Assume that u(t) is a differentiable function defined on R+ . Then for any t ∈ R+ the Dini upper right derivative D + |u(t)| of function |u(t)| exists and has the expression as follows 1 D + u(t) = lim sup u(t + h) − u(t) = σ u(t) u(t), ˙ h→0+ h where
σ u(t) =
and u(t) ˙ =
1, −1, 0,
du(t) dt .
3. Boundedness In this section, we consider the boundedness for system (5). We have the following results. Theorem 1. Under assumptions (H1 )–(H4 ), all solutions of system (5) are defined and are bounded on R+ if there are constants hij , lij , h∗ij , lij∗ ∈ R, ωi > 0 (i, j = 1, 2, . . . , n), r > 1 and σ > 0 such that
j =1
Denote Ai (yi (t)) = ai (yi (t)+x ∗ ), Bi (yi (t)) = bi (yi (t)+xi∗ )− bi (xi∗ ), fj (yj (t)) = gj (yj (t) + xj∗ ) − gj (xj∗ ), then system (7) becomes dyi (t) = −Ai yi (t) Bi yi (t) dt n n cij fj yj (t) − dij fj yj t − τij (t) . − j =1
By assumption (H3 ), we have fj (u) μj |u|.
(9)
In the following, the Young inequality and Dini upper right derivative will be used to prove the main results of this Letter, particularly, the boundedness of solutions and global exponential stability of system (1). On the Young inequality we have the following lemma (see [12,27]). Lemma 2. Assume that a 0, b 0, p > 1, q > 1 with
1 p
+
= 1. Then we have inequality
ab
n
r−hij r−lij ωj α i cij (t) r−1 μjr−1
j =1
−
n
h l ωj α i μjij cij (t) ij
j =1
− (r − 1)
n
∗
∗
ij r−hij r−l ωj α i dij (t) r−1 μ r−1
j
j =1
−
n
h∗ l∗ ωj α i μjij dij (t) ij > σ
(11)
j =1
Definition 2. System (4) or (8) is globally exponentially stable, if there are constants > 0 and M 1 such that
y(t) = x(t) − x ∗ M φ − x ∗ exp(−t) for all t 0. (10)
1 q
rωi α i βi − (r − 1)
j =1
(8)
if u(t) > 0 or u(t) = 0 and u(t) ˙ > 0, if u(t) < 0 or u(t) = 0 and u(t) ˙ < 0, if u(t) = 0 and u(t) ˙ =0
1 p 1 q a + b . p q
And on the Dini upper right derivative we have the following lemma (see [12]).
for all t 0 and i = 1, 2, . . . , n. Proof. Let x(t) = (x1 (t), x2 (t), . . . , xn (t)) be any solution of system (5) with initial function φ ∈ C[−τ, 0] at t = 0. Let further xi (t) = ωi ui (t) (i = 1, 2, . . . , n), then system (5) transforms into the following form: n 1 dui (t) cij (t)gj ωj uj (t) = − ai (ωi ui ) bi ωi ui (t) − dt ωi j =1 n dij (t)gj ωj uj t − τij (t) + Ii (t) , − j =1
where i = 1, 2, . . . , n. Calculating the upper right derivative D + |ui (t)|r , we get r r−1 dui (t) D + ui (t) = r ui (t) σ ui (t) dt r−1 1 = r ui (t) σ ui (t) − ai (ωi ui ) bi ωi ui (t) ωi
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
−
n j =1
−
n
cij (t)gj ωj uj (t) dij (t)gj ωj uj t − τij (t) + Ii (t)
417
We further have r r D + ui (t) − rα i βi ui (t)
+ (r − 1)
r −rα i βi ui (t) n ωj j =1
+r
ωi
n ωj j =1
ωi
+
r−1 α i cij (t)ui (t) μj uj (t)
j =1
r−1 α i dij (t)ui (t) μj uj t − τij (t)
r−1 1 α i Ii (t)ui (t) ωi
+r
n r−1 1 α i cij (t)gj (0)ui (t) ωi
+
n r−1 1 α i dij (t)gj (0)ui (t) . ωi
=r
j =1
ωi
+ (r − 1) + (r − 1)
+
n ωj j =1
ωi
ωi
n ωj j =1
ωi
=r
j =1
ωi
−1 ui (t)r + M ui (t)
r−hij r−lij r α i cij (t) r−1 μ r−1 ui (t)
+
ωi
ωi
j
h∗ l ∗ r α i dij (t) ij μ ij uj t − τij (t) . j
h∗ l ∗ α i dij (t) ij μjij
j =1
r max uj (s) (13)
t−τ st
t 0, i = 1, 2, . . . , n . We choose a sufficiently large constant p such that −rωi α i βi + (r − 1)
n
r−hij r−lij ωj α i cij (t) r−1 μjr−1
j =1
∗
∗
ωi
h l r α i cij (t) ij μjij uj (t)
for all t 0 and i = 1, 2, . . . , n, where n ∗ cij (t) + dij (t) gj (0) : M = sup α i Ii (t) +
j
r−hij r−lij r α i dij (t) r−1 μ r−1 ui (t)
ωi
n ωj j =1
∗ ∗
ij r−hij r−l r r−1 r r−1 r−1 ui (t) α i dij (t) μ
n ωj j =1
+
j
j =1
n ωj j =1
j
h∗ l ∗ r 1/r × dij (t) ij μjij uj t − τij (t) (r − 1)
+
h l r α i cij (t) ij μ ij uj (t)
n ωj
j
r−1 α i dij (t)ui (t) μj uj t − τij (t)
n ωj
∗
∗
r−hij r−lij ωj α i dij (t) r−1 μ r−1
∗
and r
n
ωi
r−hij r−lij α i cij (t) r−1 μjr−1
j =1
j
j =1
n ωj j =1
h l r 1/r × cij (t) ij μjij uj (t) (r − 1)
h∗ l ∗ r α i dij (t) ij μjij uj t − τij (t)
1 −rωi α i βi ωi
ij r−hij r−l r r−1 r α i cij (t) r−1 μ r−1 ui (t)
n ωj
n ωj
j
j =1
(12)
r−1 ωj α i cij (t)ui (t) μj uj (t) ωi n ωj
ωi
∗
∗
r−hij r−lij r α i dij (t) r−1 μ r−1 ui (t)
ωi n α i cij (t)gj (0) +r Ii (t) + ωi j =1 n dij (t)gj (0) ui (t)r−1 +
Estimating the right of (12) by using Young inequality. Since
j =1
n ωj
j =1
j =1
r
ωi
r−hij r−lij r α i cij (t) r−1 μjr−1 ui (t)
h l r α i cij (t) ij μjij uj (t)
j =1
j =1
n
n ωj
+ (r − 1)
+r
+r
ωi
j =1
j =1
+r
n ωj
+ dij (t) +
n j =1
r−h∗ ij r−1
∗ r−lij r−1
μj
+ M ∗ p −1
hij lij h∗ lij∗ r α i ωj cij (t) μj + dij (t) ij μj p <0
(14)
418
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
and φ < min {ωi }p. 1in
Then, we can obtain xi (t) < ωi p for all t 0 and i = 1, 2, . . . , n. In fact, if it is not true, then there exist some i and time t1 > 0 such that |xi (t1 )| = ωi p, D + |xi (t1 )|r 0, and |xj (t)| ωj p for all −τ < t t1 and j = 1, 2, . . . , n. From this we further obtain |ui (t1 )| = p, D + |ui (t1 )|r 0 and |uj (t)| p for all −τ t t1 and j = 1, 2, . . . , n. However, from (13) and (14) we get r D + ui (t1 ) 1 < −rωi α i βi ωi n
r−hij r−lij + (r − 1) ωj α i cij (t) r−1 μjr−1 j =1 ∗
ij r−hij r−l + dij (t) r−1 μjr−1 + M ∗ p −1 ∗
+
n
(H6 ) Each functions gij l (u) (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m) satisfy the Lipschitz condition, i.e., there are constants μij l > 0 such that gij l (u1 ) − gij l (u2 ) μij l |u1 − u2 | for all u1 , u2 ∈ R. (H7 ) Functions τij l (t) (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m) are non-negative, bounded and continuous defined on R+ = [0, +∞).
hij lij h∗ lij∗ r ij α i ωj cij (t) μj + dij (t) μj p < 0.
j =1
This is a contradiction. So, |ui (t)| < p for all t 0. Hence, |xi (t)| < ωi p for all t 0. Thus, the solution x(t) = (x1 (t), x2 (t), . . . , xn (t)) of system (5) is defined and is bounded on R+ . This completes the proof. 2 Remark 1. The condition (11) is quite general, because it includes many known assumptions as its special cases. For example, if in the condition (11) we choose the constants hij = lij = 1 and r = 2, then (11) will transform into the following form. (H5 ) There are constants ωi > 0 (i = 1, 2, . . . , n) and σ > 0 such that n n ωj μj α i cij (t) − ωj α i μj dij (t) > σ ωi α i βi − j =1
In system (15), the integer n corresponds to the number of units in a neural network and the integer m corresponds to the number of neural axons, that is, signals that emit from the ith unit have m pathways to the j th unit; xi corresponds to the state of the ith unit at time t ; gij l (xj (t)) denotes the activation function of the j th unit at time t ; cij l (t) denotes the strength of the j th unit on the ith unit at time t − τij l (t); Ii (t) denotes the external bias on the ith unit at time t ; τij l (t) corresponds to the transmission delay of the ith unit along the l axon of the j th unit at time t and is a non-negative function; and ai (xi (t)) represents an amplification function; bi (xi (t)) is an appropriately behaved function such that the solution of model (15) remain bounded. For system (15), the assumptions (H3 ) and (H4 ) are transformed into the following form.
j =1
for all t ∈ R+ and i = 1, 2, . . . , n.
Moreover, we have the following result similar to Theorem 1. Theorem 2. Under assumptions (H1 )–(H2 ), (H6 ) and (H7 ), all solutions of system (15) are defined and are bounded on R+ if there exist constants hij l , lij l ∈ R, ωi > 0 (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m), r > 1 and σ > 0 such that rωi α i β i − (r − 1)
n m
r−lij l r−hij l cij l (t) r−1 ωj α i μijr−1 l
l=1 j =1
−
n m
h l ωj α i μijijll cij l (t) ij l > σ
(16)
l=1 j =1
for all t 0 and i = 1, 2, . . . , n.
As a consequence of Theorem 1, we have the following corollary.
The proof of Theorem 2 is similar to Theorem 1. Here, we omit it. As a consequence of Theorem 2, we have the following result similar to Corollary 1.
Corollary 1. Assume that (H1 )–(H5 ) hold. Then all solutions of system (5) are defined and are bounded on R+ .
Corollary 2. Under assume that (H1 ), (H2 ), (H6 ) and (H7 ), if there are constants ωi > 0 (i = 1, 2, . . . , n) and σ > 0 such that
A general case of system (5) is the following modified Cohen–Grossberg neural networks with variable coefficients and time-varying delay dxi (t) = −ai xi (t) bi xi (t) dt n m − cij l (t)gij l xj t − τij l (t) + Ii (t) , l=1 j =1
i = 1, 2, . . . , n.
(15)
ωi α i βi −
n m
ωj α i μij l cij l (t) > σ
l=1 j =1
for all t 0 and i = 1, 2, . . . , n. Then all solutions of system (15) are defined and are bounded on R+ . Remark 2. In Cao and Liang [24], the authors established a series of results on the boundedness for systems (15), for example, see Theorem 1 in Cao and Liang [24]. We see that in these results the delay functions τij (t) (i, j = 1, 2, . . . , n, l =
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
1, 2, . . . , m) given in the systems are continuously differentiable on t ∈ R+ and satisfy the conditions inft∈R+ {1− τ˙ij (t)} > 0, where τ˙ij (t) express the derivative of τij (t) with respect to time t . Furthermore, functions 1 − τ˙ij (t) appear in the main assumptions. However, in this Letter, we do not require that the delay functions τij (t) and τij l (t) (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m) given in systems (5) and (15) are continuously differentiable and satisfy the conditions inft∈R+ {1 − τ˙ij (t)} > 0 and inft∈R+ {1 − τ˙ij l (t)} > 0. Particularly, we only require that the delay functions are non-negative, bounded and continuous on R+ . In addition, functions 1 − τ˙ij (t) and 1 − τ˙ij l (t) do not appear in the main assumptions. Remark 3. For system (5), when ai (xi (t)) ≡ 1, bi (xi (t)) = bi (t)xi (t) (in which bi (t) is not only differentiable but also bounded on interval [−τ, +∞), and its maximal lower bound is denoted as βi > 0), then system (5) transform into cellular neural networks system with variable coefficients and timevarying delays. In this case, we can see that the condition given in Theorem 2 is less conservative and less restrictive than those given in [28,29]. In addition, the results in [28,29] are special cases of ours.
In this section, we will obtain some criteria for global exponential stability of (4) or (8). Moreover, the uniqueness of the equilibrium point follows directly its global exponential stability. Theorem 3. Under assumptions (H1 ), (H2 ), (H3 ) and (H4 ), system (4) or (8) is globally exponentially stable if there exist constants hij , lij , h∗ij , lij∗ ∈ R, ωi > 0 (i, j = 1, 2, . . . , n), r > 1 and σ > 0 such that rωi α i βi − (r − 1)
n
−
ωj α i |cij |
l ωj α i μjij |cij |hij
r−hij r−1
− (r − 1)
j =1
−
n
r−lij
μjr−1
n
ωj α i |dij |
r−h∗ ij r−1
∗ r−lij r−1
∗
l
ωj α i μjij |cij |hij − (r − 1)
j =1
− erτ
n
1 dzi (t) cij fj ωj zj (t) = − Ai ωi zi (t) Bi ωi zi (t) − dt ωi j =1 n − (18) dij fj ωj zj t − τij (t) .
n
∗ r−lij
ωj α i μjr−1 |dij |
r−h∗ ij r−1
j =1
∗ lij
∗ 1 ωj α i μj |dij |hij − rωi > σ 2
for all t 0 and i = 1, 2, . . . , n. Let pi (t) = ert |zi (t)|r . Calculating the Dini upper right derivative D + pi (t) of pi (t), from (18) we get D + pi (t) r−1 r dzi (t) ert r zi (t) σ zi (t) + r zi (t) dt r−1 1 rt Bi ωi zi (t) r zi (t) σ zi (t) − Ai ωi zi (t) =e ωi n n cij fj ωj zj (t) − dij fj ωj zj t − τij (t) − r + r zi (t)
j =1
r r ert −rα i βi zi (t) + r zi (t) +r
n ωj j =1
+r
ωi
n ωj
ωi
r−1 α i |cij |μj zi (t) zj (t) r−1 α i |dij |μj zi (t) zj t − τij (t) .
(19)
Estimating the right of (19) by using Young inequality, we have r r D + pi (t) ert −rα βi zi (t) + r zi (t) i
n ωj j =1
n ωj j =1
Proof. Let yi (t) = ωi zi (t) (i = 1, 2, . . . , n), then system (8) is transformed into the following form.
r−hij r−1
j =1 n
+
for all t 0 and i = 1, 2, . . . , n.
j =1
−
(17)
j =1
r−lij
ωj α i μjr−1 |cij |
j =1 n
+ (r − 1)
ωj α i μjij |dij |hij > σ
n
μj
j =1 l∗
rωi α i βi − (r − 1)
j =1
j =1 n
By the condition (17), we can choose a constant > 0 such that
j =1
4. Stability results
419
ωi
+ (r − 1) +
j =1
ωi
r−hij r−1
r−lij r μjr−1 zi (t)
r l α i |cij |hij μjij zj (t) n ωj j =1
n ωj
ωi
α i |cij |
ωi
α i |dij |
h∗ij
α i |dij |
r−h∗ ij r−1
l∗ μjij zj t
∗ r−lij r−1
μj
zi (t)r
r − τij (t)
r = r( − α i βi )ert zi (t) + (r − 1)
n ωj j =1
ωi
α i |cij |
r−hij r−1
r−lij r μjr−1 ert zi (t)
420
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
+
n ωj
ωi
j =1
+ (r − 1)
r l α i |cij |hij μjij ert zj (t) n ωj j =1
+
n ωj
ωi
j =1
ωi
α i |dij | ∗
∗ r−lij r−1
r−h∗ ij r−1
μj
for all t 0. Hence,
zi (t) M φ − x ∗ e−t
r ert zi (t)
for all t 0. Furthermore, we finally have n
yi (t) nωM φ − x ∗ e−t
∗ lij
α i |dij |hij μj erτij (t) er(t−τij (t))
where, ω = max1in {ωi }. This shows that, from Definition 2, system (9) is globally exponentially stable. This completes the proof. 2
r × zj t − τij (t) r( − α i βi )pi (t) n r−lij r−hij ωj α i |cij | r−1 μjr−1 pi (t) + (r − 1) ωi
In the condition (17), if we choose the constants hij l = lij l = 1 and r = 2, as a special cases of Theorem 3, we have the following corollary.
j =1
+
n ωj
ωi
j =1
+ (r − 1)
l
α i |cij |hij μjij pj (t) n ωj j =1
+
n j =1
ωi
α i |dij |
∗ r−lij r−1
r−h∗ ij r−1
μj
Corollary 3. Under assumptions (H1 ), (H2 ), (H3 ) and (H4 ), system (4) or (8) is globally exponentially stable if there exist constants ωi > 0 (i = 1, 2, . . . , n) and σ > 0 such that
pi (t)
∗ lij
∗ ωj α i |dij |hij μj erτ sup pj (s). ωi t−τ st
(20)
Choose the constant M 1 such that M r > (min1in {ωir })−1 , then we have r |yi (0)|r |φi (0) − xi∗ |r = pi (0) = zi (0) = r ωi ωir
r φ − x ∗ r M r φ − x ∗ . r min{ωi } From this we further can obtain pi (t) < M r φ − x ∗ r for all t > 0 and i = 1, 2, . . . , n. In fact, if it is not true, then there exist some i and t1 > 0 such that pi (t1 ) = M r φ − x ∗ r , D + pi (t1 ) 0 and pj (t) M r φ − x ∗ r for all −τ t t1 and j = 1, 2, . . . , n. However, from (20), we can obtain 1 + D pi (t1 ) r( − α i βi )ωi ωi + (r − 1)
n
ωj αi |cij |
r−hij r−1
r−lij r−1
μj
j =1
+
n
l
ωj αi |cij |hij μjij
j =1
+ (r − 1)
n
ωj αi |dij |
+
j =1
h∗ij
ωj αi |dij |
∗ r−lij r−1
r−h∗ ij r−1
j =1 n
for all t 0,
i=1
l∗ μjij erτ
M
n
ωj μj α i |cij | −
j =1
n
ωj α i μj |dij | > σ
j =1
for all t ∈ R+ and i = 1, 2, . . . , n. For system (15), when cij l (t) = cij l , Ii (t) = Ii , where cij l , Ii (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m) are constants, then the system (15) is transformed into the following form. dxi (t) = −ai xi (t) bi xi (t) dt n m − cij l gij l xj t − τij l (t) + Ii , l=1 j =1
i = 1, 2, . . . , n.
(21)
For system (21), we have the following result. Theorem 4. Under assumptions (H1 ), (H2 ) and (H7 ), further, each functions gij l (u) (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m) is bounded and satisfies the Lipschitz condition with Lipschitz constant μij l > 0, system (21) is globally exponentially stable if there exist constants hij l , lij l ∈ R, ωi > 0 (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m), r > 1 and σ > 0 such that rωi α i β i − (r − 1)
μj
ωi α i βi −
n m
r−lij l
ωj α i μijr−1 l |cij l |
r−hij l r−1
l=1 j =1
r
φ−x
∗ r
r 1 − σ M r φ − x ∗ < 0. 2 This is a contradiction. So, pi (t) M r φ − x ∗ r for all t 0. That is, r
r ert zi (t) M r φ − x ∗
−
m
n
l
ωj α i μijijll |cij l |hij l > σ
l=1 j =1
for all t 0 and i = 1, 2, . . . , n. The proof of Theorem 4 is similar to Theorem 3, here we omit it. As a special cases of Theorem 4, we further have following corollary.
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
Corollary 4. Under assumptions (H1 ), (H2 ) and (H7 ), further, each function gij l (u) (i, j = 1, 2, . . . , n, l = 1, 2, . . . , m) is bounded and satisfies the Lipschitz condition with Lipschitz constant μij l > 0, system (21) is globally exponentially stable if there exist constants ωi > 0 (i = 1, 2, . . . , n) and σ > 0 such that ωi α i βi −
n m
421
for all t 0. Thus, by Theorem 1 all solutions of system (22) are bounded. However, since the derivatives τ˙ij (t) have not the determinate sign on R+ , we see that for system (22) the assumptions given in Cao and Liang [24] do not satisfy. Therefore, the results obtained in Cao and Liang [24] cannot be applied to system (22).
ωj α i μij l |cij l | > σ
l=1 j =1
for all t 0 and i = 1, 2, . . . , n. Remark 4. From Theorems 3 and 4 we can see that the results obtained in this Letter improve and extend the correspondent to results given in prior literatures. The corresponding remarks are similar to Remark 2. Here, we omit it. 5. Two examples Example 1. Consider the following 2-dimensional Cohen– Grossberg neural networks with variable coefficients and timevarying delays system: dx1 (t) = − 3 + cos x1 (t) 18x1 (t) − (1 + cos t)f x1 (t) dt − (1 − sin t)f x2 (t) 1 − (1 + sin t)f x1 t − sin t − 1 3 1 − sin t −1 + sin t , − (1 − cos t)f x2 t − e 4 dx2 (t) = − 3 + sin x2 (t) 17x2 (t) − (1 + sin t)f x1 (t) dt − (1 − cos t)f x2 (t) 1 − (1 − cos t)f x1 t − sin t − 1 3 1 − sin t −1 + cos t , − (1 + sin t)f x2 t − e 4 (22) where f (x) = 0.5(|x + 1| − |x − 1|). This system satisfies all assumptions in this Letter with α 1 = 2, α 2 = 2, α 1 = 4, α 2 = 4, μj = 1 (j = 1, 2), β1 = 18, β2 = 17. Further, we choose ωi = hij = lij = 1 (i, j = 1, 2) and r = 2 in the condition (11), then we see that ω1 α 1 β1 − α 1 |c11 |ω1 μ1 − α 1 |c12 |ω2 μ2 − α 1 |d11 |ω1 μ1 − α 1 |d12 |ω2 μ2 = 36 − 4|1 + cos t| − 4|1 − sin t| − 4|1 + sin t| − 4|1 − cos t| > 4 > 0, ω2 α 2 β2 − α 2 |c21 |ω1 μ21 − α 2 |c22 |ω2 μ2 − α 2 |d21 |ω1 μ1 − α 2 |d22 |ω2 μ2 = 34 − 4|1 + sin t| − 4|1 + cos t| − 4|1 − cos t| − 4|1 + sin t| > 2 > 0.
Example 2. Consider the following 2-dimensional Cohen– Grossberg neural networks with delays dx1 (t) = − 8 + sin x1 (t) 7x1 (t) − tanh x1 (t) − 2 tanh x2 (t) dt 1 − tanh x1 t − sin t − 1 3 1 − tanh x2 t − e− sin t − 1 + 2 , 4 dx2 (t) = − 5 + cos x2 (t) 10x2 (t) − 2 tanh x1 (t) − tanh x2 (t) dt 1 − sin t −1 − tanh x1 t − e 4 1 − 2 tanh x2 t − sin t − 1 + 3 . (23) 3 This system satisfies all assumptions in this Letter with α 1 = 7, α 2 = 4, α 1 = 9, α 2 = 6, ωj = 1, μij = 1 (i, j = 1, 2), β1 = 7, β2 = 10, then α 1 β1 − α 1 |c11 |μ1 − α 1 |c12 |μ2 − α 1 |d11 |μ1 − α 1 |d12 |μ2 = 49 − 9 − 18 − 9 − 9 = 4 > 0, α 2 β2 − α 2 |c21 |μ1 − α 2 |c22 |μ2 − α 2 |d21 |μ1 − α 2 |d22 |μ2 = 40 − 12 − 6 − 6 − 12 = 4 > 0. From Corollary 3 we know the solutions (23) are globally exponentially stable. 6. Conclusions In this Letter, we introduce the new research method, that is, by introducing ingeniously many real parameters hij , lij , h∗ij , lij∗ , ωi and r, applying Young inequality technique and Dini derivative, directly estimate the above bound of solutions of the systems. We obtain a series of new criteria on the boundedness, globally exponentially stability for the systems (5) and (4). Particularly, for the delay functions τij (t) and τij l (t), we only require that they are non-negative, bounded and continuous on R+ . The results obtained in this Letter improve and extend many previous works, and they are easy to check and apply in practice. References [1] J.J. Hopfield, Proc. Natl. Acad. Sci. Biol. 81 (1984) 3088. [2] L.O. Chua, L. Yang, IEEE Trans. Circuits Systems 35 (1988) 1257. [3] L.O. Chua, L. Yang, IEEE Trans. Circuits Systems 35 (1988) 1273.
422
H. Jiang et al. / Physics Letters A 354 (2006) 414–422
[4] B. Kosko, Neural Networks and Fuzzy Systems—A Dynamical System Approach Machine Intelligence, Prentice–Hall, Englewood Cliffs, NJ, 1992, p. 38. [5] B. Kosko, IEEE Trans. Systems Man Cybernet. 18 (1988) 49. [6] K. Gopalsamy, X.Z. He, IEEE Trans. Neural Networks 5 (1994) 998. [7] M. Forti, A. Tesi, IEEE Trans. Circuits Systems I 42 (1995) 354. [8] A. Michel, J.A. Farrell, W. Porod, IEEE Trans. Circuits Systems 36 (1989) 229. [9] S. Arik, V. Tavsanoglu, IEEE Trans. Circuits Systems I 47 (2000) 571. [10] X. Liao, G. Chen, E. Sanchez, Neural Networks 15 (2002) 855. [11] X. Liang, J. Wang, IEEE Trans. Circuits Systems I 47 (2000) 1258. [12] J. Cao, Phys. Lett. A 307 (2003) 136. [13] J. Cao, J. Wang, IEEE Trans. Circuits Systems I 50 (2003) 34. [14] M.A. Cohen, S. Grossberg, IEEE Trans. Systems Man Cybernet. 13 (1983) 815. [15] C. Marcus, R. Westervelt, Phys. Rev. A 39 (1989) 347. [16] H. Ye, A.N. Michel, K. Wang, Phys. Rev. E 51 (1995) 2611.
[17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
L. Wang, X. Zou, Neural Networks 15 (2002) 415. L. Wang, X. Zou, Physica D 170 (2002) 163. W. Lu, T. Chen, Neural Comput. 15 (2003) 1173. T. Chen, L. Rong, Phys. Lett. A 317 (2003) 436. J. Cao, X. Li, Physica D 212 (1–2) (2005) 54. K. Yuan, J. Cao, IEEE Trans. Circuits Systems I 52 (9) (2005) 1854. C. Hwang, C. Cheng, T. Li, Phys. Lett. A 319 (2003) 157. J. Cao, J. Liang, J. Math. Anal. Appl. 296 (2004) 665. Y. Li, Chaos Solitons Fractals 20 (2004) 459. J. Hale, Theory of Functional Differential Equations, Springer, New York, 1977. [27] G. Hardy, H. Littlewood, G. Polya, Inequality, Cambridge Univ. Press, Cambridge, 1934. [28] H. Jiang, Z. Li, Z. Teng, Phys. Lett. A 306 (2003) 313. [29] J. Liang, J. Cao, Phys. Lett. A 318 (2003) 53.