Dynamical analysis of fuzzy cellular neural ... - Semantic Scholar
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JOURNAL OF COMPUTERS, VOL. 7, NO. 4, APRIL 2012
Dynamical analysis of fuzzy cellular neural networks with time-varying delays Qianhong Zhang1,∗ , Lihui Yang2 , Daixi Liao3 1. Guizhou Key Laboratory of Economics System Simulation, Guizhou College of Finance and Economics, Guiyang, Guizhou 550004, P.R.China 2. Department of Mathematics, Hunan City University, Yiyang, Hunan 413000, P. R. China 3. Basic Science Department, Hunan Institute of Technology, Hengyang, Hunan 421002, P. R. China Email: [email protected]; [email protected]; [email protected]
Abstract— In this paper, employing continuation theorem of the coincidence degree, and inequality technique, some sufficient conditions are derived to ensure global exponential stability and the existence of periodic solution for fuzzy cellular neural networks with time-varying delays. These results have important leading significance in the design and applications of globally stable neural networks. Moreover an example is given to illustrate the effectiveness and feasibility of results obtained. Index Terms— fuzzy cellular neural networks, periodic solution, global exponential stability, time-varying delays
fuzzy cellular neural networks: x0i (t)
= −ci (t)xi (t) + + +
n ^ j=1 n _ j=1 n ^ j=1
Cellular neural network is formed by many units named cells, and that cell contains linear and nonlinear circuit elements, which typically are linear capacitor, linear resistor, linear and nonlinear controlled source, and independent sources. Nowadays, cellular neural networks(CNNs) are widely used in signal and image processing, associative memories, pattern classification [1]–[6]. In the last decade, dynamic behaviors of CNNs have been intensively studied because of the successful hardware implementation and their widely application(see,for example, [4]–[26]). In this paper, we would like to integrate fuzzy operations into cellular neural networks. Speaking of fuzzy operations, Yang and Yang [27] first introduced fuzzy cellular neural networks (FCNNs) combining those operations with cellular neural networks. So far researchers have founded that FCNNs are useful in image processing, and some results have been reported on stability and periodicity of FCNNs [27]–[35]. However, to the best of our knowledge, few author investigated the stability of fuzzy cellular neural networks with time-varying delays. In this paper, we investigate the existence, and the global exponential stability of periodic solution for the following Corresponding author. Tel. +86 851 6902456. fax: +86 851 6902456. E-mail address: [email protected] (Q.Zhang). Manuscript received January 1, 2011; revised June 1, 2011; accepted July 1, 2011.
© 2012 ACADEMY PUBLISHER doi:10.4304/jcp.7.4.934-940
aij (t)fj (xj (t))
j=1
+ I. I NTRODUCTION
n X
αij (t)fj (xj (t − τij (t))) βij (t)fj (xj (t − τij (t))) + Ii (t) Tij (t)uj (t) +
n _
Hij (t)uj (t)
(1)
j=1
i = 1, 2, · · · , n. where n corresponds to the number of neurons in neural networks. For xi (t) is the activations of the ith neuron at time t. ci (t) denotes the rate with which the ith neuron will reset its potential to the resting state in isolationVwhen W disconnected from the network and external inputs; and denote the fuzzy AND and fuzzy OR operations. aij (t) denotes the strengths of connectivity between cell i and cell j at time t. αij (t), βij (t), Tij (t) and Hij (t) are elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template between cell i and j at time t. τij (t) corresponds to the time delay required in processing and transmitting a signal from the jth cell to the ith cell at time t. uj (t) and Ii (t) denote the external input, bias of the ith neurons at time t, respectively. fj (·) is signal transmission functions. Throughout the paper, we give the following assumptions (A1) |fj (x)| ≤ pj |x| + qj for all x ∈ R, j = 1, 2, · · · , n. where pj , qj are nonnegative constants. (A2) The signal transmission functions fj (·), (j = 1, 2, · · · , n) are Lipschtiz continuous on R with Lipschtiz constants pj , namely, for any x, y ∈ R |fj (x) − fj (y)| ≤ pj |x − y|, fj (0) = 0. Definition 1: If f (t) : R → R is a continuous function, then the upper right derivative of f (t) is defined as 1 D+ f (t) = lim+ sup (f (t + h) − f (t)). h h→0
JOURNAL OF COMPUTERS, VOL. 7, NO. 4, APRIL 2012
Let τ = max1≤i,j≤n supt≥0 {τij (t)}. For functions ϕi defined on [−τ, 0], i = we set Ψ = (ϕ1 , ϕ2 , · · · , ϕn )T . If (x1 (t), x2 (t), · · · , xn (t))T is an ω−periodic system (1.1), then we denote kΨ − xk =
continuous 1, 2, · · · , n, x(t) = solution of
n X ( sup |ϕi (t) − xi (t)|). i=1 −τ ≤t≤0
Assume that system (1) is supplemented with initial value xi (t) = ϕi (t),
−τ ≤ t ≤ 0.
Definition 2: The periodic solution x(t) = (x1 (t), x2 (t), · · · , xn (t))T is said to be globally exponentially stable. If there exist constants λ > 0 and M ≥ 1 such that for any solution x(t) = (x1 (t), x2 (t), · · · , xn (t))T of system (1) |xi (t) − xi (t)| ≤ M kΨ − xke−λt , t ≥ 0. (2) Lemma 1: (see [26]) If ρ(K) < 1 for matrix K = (kij )n×n ≥ 0, then (E − K)−1 ≥ 0, where E denotes the identity matrix of size n. Lemma 2:(see [27]) Suppose x and y are two states of system (1), then we have ^ n ^ n αij (t)fj (y) αij (t)fj (x) − j=1 j=1 ≤
n X
|αij (t)||fj (x) − fj (y)|,
j=1
and n _ βij (t)fj (x) j=1
−
≤
βij (t)fj (y) j=1 n _
n X
|βij (t)||fj (x) − fj (y)|.
j=1
The remainder of this paper is organized as follows. In Section 2, we will give the sufficient conditions to ensure the existence of periodic oscillatory solution for fuzzy cellular neural networks with time-varying delays, and show that all other solutions converge exponentially to it as n → ∞. In Section 3 an example will be given to illustrate effectiveness of our results obtained. We will give a general conclusion in Section 4. II. P ERIODIC OSCILLATORY SOLUTIONS In this section, we will consider the periodic oscillatory solutions of system (1) with τij (t), ci (t), aij (t), αij (t), βij (t), Tij (t), Hij (t), uj (t) and Ii (t) satisfying the following assumptions: (A3) τij ∈ C(R, [0, ∞)) are periodic solutions with period ω for i, j = 1, 2, · · · , n. (A4) ci ∈ C(R, (0, ∞)), aij , αij , βij , Tij , Hij , uj , Ii ∈ C(R, R) are periodic solutions with common periodic ω, and fi ∈ C(R, R), i, j = 1, 2, · · · , n. © 2012 ACADEMY PUBLISHER
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We will use the coincidence degree theory to obtain the existence of an ω-periodic solution to system (1). For the sake of convenience, we briefly summarize the theory as below. Let X and Z be normed spaces, L : DomL ⊂ X 7→ Z be a linear mapping and N : X 7→ Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimKerL = codim ImL < ∞ and ImL is closed in Z. If L is a Fredholm mapping of index zero, then there exist continuous projectors P : X 7→ X and Q : Z 7→ Z such that ImP = KerL and ImL = KerQ = Im(I − Q). It follows that L|DomL ∩ KerP : (I − P )X 7→ ImL is invertible. We denote the inverse of this map by Kp . If Ω is a bounded open subset of X, the mapping N is called L-compact on Ω. if QN (Ω) is bounded and Kp (I − Q)N : Ω 7→ X is compact. Because ImQ is isomorphic to KerL, there exists an isomorphism J : ImQ 7→ KerL. Let Ω T⊂ Rn be open and bounded, f ∈ S C 1 (Ω, Rn ) C(Ω, Rn ) and y ∈ Rn \f (∂Ω Sf ), i.e., y is a regular value of f . Here, Sf = {x ∈ Ω : Jf (x) = 0}, the critical set of f , and Jf is the Jacobian of f at x. Then the degree degf, Ω, y is defined by X deg{f, Ω, y} = sgnJf (x) x∈f −1 (y)
with the agreement that the above sum is zero if f −1 (y) = Φ. Lemma 3: Let L be a Fredholm mapping of index zero and let N be L-compact on Ω. Suppose that (a) for each λ ∈ (0, 1), every solution x of Lx = λN x is such that x ∈ / ∂Ω. T (b) QN x 6= 0 for each x ∈ ∂Ω KerL and \ deg{JQN, Ω KerL, 0} = 6 0. Then the equation T Lx = N x has at least one solution lying in DomL Ω. To be convenience, in the rest of paper, for a continuous function g : [0, ω] 7→ R, we denote Z 1 ω + g = max g(t), g− = min g(t), g¯ = g(t)dt. ω 0 t∈[0,ω] t∈[0,ω] Theorem 1: Under assumptions (A1), (A3) and (A4), kij = c¯1i + ω (|aij | + |αij | + |βij |)pj K = (kij )n×n . Suppose that ρ(K) < 1, then system (1) has at least an ω-periodic solution. Proof: Take X = Z = {x(t) = (x1 (t), x2 (t), · · · , xn (t))T ∈ C(R, Rn ) : x(t + ω) = x(t), t ∈ R} and denote kxk = max1≤i≤n maxt∈[0,ω] |xi (t)|. Equipped with the norm k · k, both X and Z are Banach space. For any x(t) ∈ X, it is easy to check that Θ(xi , t)
:= −ci (t)xi (t) + +
n X
aij (t)fj (xj (t))
j=1
+
n ^ j=1
αij (t)fj (xj (t − τij (t)))
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JOURNAL OF COMPUTERS, VOL. 7, NO. 4, APRIL 2012
+ +
n _ j=1 m ^
Noting assumption (A1), we get
βij (t)fj (xj (t − τij (t))) + Ii (t) Tij (t)uj (t) +
j=1
n _
|xi |− ci
≤
Hij (t)uj (t) ∈ Z
j=1 n X + (|aij | + αij | + |βij |)qj + |Ii |
j=1
Let L : Dom L = {x ∈ X : x ∈ C(R, Rn )} 3 x 7→ x(·) ˙ ∈ Z. Z ω 1 x(t)dt ∈ X. P : X 3 x 7→ ω 0 Z 1 ω Q : Z 3 z 7→ z(t)dt ∈ Z, ω 0 N : X 3 x 7→ Θ(x, ·) ∈ Z. For any V = (v1 , v2 , · · · , vn ) ∈ Rn , we identify it as the constant function in X or Z with the value vector V = (v1 , v2 , · · · , vn ). Then system (1.1) can be reduced to operator equation Lx = N x. It is Reasy to see that ω KerL = Rn , ImL = {z ∈ Z : ω1 0 z(t)dt = 0}. Which is closed in Z, dimKerL = codimImL = n < ∞, and P, Q are continuous projectors such that ImP = KerL, KerQ = ImL = Im(I − Q). It follows that L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) Kp : ImL 7→ KerP ∩ DomL given by Z t Z Z 1 ω s (Kp (z))i (t) = zi (s)ds − zi (v)dvds. ω 0 0 0 Therefore, applying the Arzela-Ascoli theorem, one can ¯ with any bounded easily show that N is L-compact on Ω open subset Ω ⊂ X. Since ImQ = KerL, we take the isomorphism J of ImQ onto KerL to be the identity mapping. Now we need only to show that, for an appropriate open bounded subset Ω, application of the continuation theorem corresponding to the operator equation Lx = λN x, λ ∈ (0, 1), Let x˙ i (t) = λΘ(xi , t) i = 1, 2, · · · , n.
0
0
=
+
0
j=1 n ^
+ +
j=1 n _
|xi |−
≤
n 1 X (|aij | + |αij | + |βij |)pj |xj |+ ci j=1 n X 1 + (|aij | + αij | + |βij |)qj + |Ii | ci j=1 n n _ ^ 1 + |Hij ||uj |+ (7) |Tij ||uj |+ + ci
Note that each xi (t) is continuously differentiable for i = 1, 2, · · · , n, it is certain that there exists ti ∈ [0, ω] such that |xi (ti )| = |xi (t)|− . Set F = (F1 , F2 , · · · , Fn )T , where n X 1 Fi = ( + ω) (|aij | + αij | + |βij |)qj ci j=1 n n ^ _ + |Tij ||uj |+ + |Hij ||uj |+ + |Ii | (8) j=1
j=1
In view of ρ(K) < 1 and Lemma 1, we have (E − K)−1 F = h = (h1 , h2 , · · · , hn )T ≥ 0. where hi is given by n X hi = kij hj + Fi , i = 1, 2, · · · , n. (9) j=1
Set Ω = {(x1 , x2 , · · · , xn )T ∈ Rn : |xi | < hi , i = 1, 2, · · · , n} (10) Then, for t ∈ [ti , ti + ω], we have t
t+ω
|xi (t)|− +
D+ |xi (t)|dt
ti
j=1
n _ + Hij (t)uj (t) + Ii (t) dt
D+ |xi (t)|dt
|xi (ti )| + Z
≤ ≤
βij (t)fj (xj (t − τij (t)))
© 2012 ACADEMY PUBLISHER
j=1
ti
n 1 X (|aij | + |αij | + |βij |)pj |xj |+ ci j=1 n 1 X + (|aij | + αij | + |βij |)qj ci j=1
j=1
(5)
(6)
j=1
j=1
≤
αij (t)fj (xj (t − τij (t)))
j=1
|Hij ||uj |+
| xi (t) |
ci (t)xi (t)dt Z ω X n m ^ aij (t)fj (xj (t)) + Tij (t)uj (t) 0 j=1
n _
It follows that
Z
n ^
|Tij ||uj |+ +
j=1
(3)
Assume that x = x(t) ∈ X is a solution of system (3) for some λ ∈ (0, 1). Integrating (3) over the interval [0, ω], we obtain that Z ω Z ω 0= x˙ i (t)dt = λ Θ(xi , t)dt (4) Hence Rω
n X (|aij | + |αij | + |βij |)pj |xj |+
+
n ^ j=1
|Tij ||uj |+ +
n _ j=1
|Hij ||uj |+ + |Ii |
JOURNAL OF COMPUTERS, VOL. 7, NO. 4, APRIL 2012
Z
937
t+ω
D+ |xi (t)|dt ti n X 1 ( + ω) (|aij | + |αij | + |βij |)pj |xj |+ ci j=1 n X 1 +( + ω) (|aij | + αij | + |βij |)qj ci j=1 n n ^ _ + |Tij ||uj |+ + |Hij ||uj |+ + |Ii |
n X 1 +( + ω) (|aij | + αij | + |βij |)qj ci j=1
+
≤
j=1
n X
j=1
n ^
+
|Tij ||uj |+ +
j=1
=
n X
n _ j=1
|Hij ||uj |+ + |Ii |
kij hj + Fj
(15)
j=1
Which is a contradiction. Therefore (13) holds and QN u 6= 0, for u ∈ ∂Ω ∩ kerL = ∂Ω ∩ Rn .
(16)
(11)
Consider the homotopy Φ : (Ω∩KerL)×[0, 1] 7→ Ω∩KerL defined by
Clearly, hi , i = 1, 2, · · · , n, are independent of λ. Then for ∀λ ∈ (0, 1), x ∈ ∂Ω such that Lx 6= λN x. When u = (x1 , x2 , · · · , xn )T ∈ ∂Ω ∩ KerL = ∂Ω ∩ Rn , u is a constant vector with |xi | = hi , i = 1, 2, · · · , n. Note that QN u = JQN u, when u ∈ KerL, it must be
Φ(u, µ) = µdiag(−c1 , −c2 , · · · , −cn )u + (1 − µ)QN u (17) Note that Φ(·, 0) = JQN . if Φ(u, µ) = 0, then we have n 1 − µ X (aij + αij + βij )f (xj ) |xi | = ci j=1 n n _ ^ Tij uj + Hij uj + Ii + j=1 j=1 n 1 X (|aij | + |αij | + |βij |)pj hj ≤ ci
≤
kij hj + Fi = hi
j=1
= −ci xi +
(QN )ui
n X (aij + αij + βij )f (xj ) j=1
+
n ^
Tij uj +
j=1
n _
Hij uj + Ii
(12)
j=1
We claim that
j=1
k(QN u)i k > 0, i = 1, 2, · · · , n.
(13)
+
ci xi
j=1 n ^
+
Tij uj +
j=1
n _
Hij uj + Ii
+
(14)
j=1
n 1 X = |xi | = (aij + αij + βij )f (xj ) ci j=1 n n ^ _ + Tij uj + Hij uj + Ii j=1 j=1 n 1 X ≤ (|aij | + |αij | + |βij |)pj hj ci j=1
n X + (|aij | + |αij | + |βij |)qj j=1
+
n ^
|Tij ||uj |+ +
j=1
n _ j=1
|Hij ||uj |+ + |Ii |
n X 1 < ( + ω) (|aij | + |αij | + |βij |)pj |hj |+ ci j=1 © 2012 ACADEMY PUBLISHER
n ^
|Tij ||uj |+ +
j=1
Then we have hi
(|aij | + |αij | + |βij |)qj
j=1
On the contrary, suppose that there are some i such that k(QN u)i k = 0, namely n X = (aij + αij + βij )f (xj )
n X
n _ j=1
|Hij ||uj |+ + |Ii |
n X 1 < ( + ω) (|aij | + |αij | + |βij |)pj |hj |+ ci j=1 n X 1 +( + ω) (|aij | + αij | + |βij |)qj ci j=1 n n _ ^ + |Tij ||uj |+ + |Hij ||uj |+ + |Ii | j=1
=
n X
j=1
kij hj + Fj
(18)
j=1
Therefore Φ(u, µ) 6= 0 for any (u, µ) ∈ (Ω ∩ KerL). It follows from the property of invariance under homotopy that deg = = =
{JQN, Ω ∩ KerL, 0} deg{Φ(·, 0), Ω ∩ KerL, 0} deg{Φ(·, 1), Ω ∩ KerL, 0} deg{diag(−c1 , −c2 , · · · , −cn )} = 6 0
Thus, we have shown that Ω satisfies all the assumptions of Lemma 3. Hence, Lu = N u has at least one ω-periodic solution on DomL ∩ Ω.
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Theorem 2: Let τ = max1≤i,j≤n,t∈[0,ω] {τij (t)}. Suppose that (A2), (A3) and (A4) hold, ρ(K) < 1, and that ci −
n X + (|aij | + |αij | + |βij |)pj eci τ > 0
|xi (t) − xi (t)| ≤ −ci (t)|xi (t) − xi (t)| n X + |aij (t)[fj (xj (t)) − fj (xj (t))]| j=1
n ^ αij (t)fj (xj (t − τij (t))) + j=1 n ^ αij (t)fj (xj (t − τij (t))) − j=1 n _ βij (t)fj (xj (t − τij (t))) + j=1 n _ − βij (t)fj (xj (t − τij (t))) j=1
j=1
+ +
j=1 n X
|αij (t)|pj |xj (t − τij (t)) − xj (t − τij (t))|
(22)
It follows that Rt ci (s)ds zi (t) e t0 ≤ |zi (t0 )| Z t X n + (|aij (u)| + |αij (u)| + |βij (u)|) t0 j=1 Ru ci (s)ds t0 ×pj kzu ke du (23) Thus, for any θ ∈ [−τ, 0], we have R t+θ R t R t+θ Rt ci (s)ds ( + )ci (s)ds ci (s)ds−c+ τ i e t0 = e t0 t ≥ e t0 (24) Therefore Rt R t+θ ci (s)ds−c+ τ ci (s)ds i t0 e z (t + θ) ≤ e t0 zi (t + θ) i Z t X n (|aij (u)| + |αij (u)| + |βij (u)|) ≤ kzt0 k + t0 j=1 Ru ci (s)ds t0 × pj kzu ke du
By Gronwall’s inequality, we obtain, for t ≥ t0 , R t c+ τ Pn Rt e γij (u)pj du −ci (s)ds c+ τ j=1 t0 kzt k ≤ e kzt0 ke e t0 (26) where γij (u) = |aij (u)| + |αij (u)| + |βij (u)|.
|βij (t)|pj |xj (t − τij (t)) − xj (t − τij (t))|
j=1
≤
ci (s)ds
t0
It follows that Rt + ci (s)ds e t0 kzt k ≤ eci τ kzt0 k Z t n X + + eci τ (|aij (u)| + |αij (u)| + |βij (u)|) t0 j=1 Ru ci (s)ds t0 ×pj kzu ke du (25)
−ci (t)|xi (t) − x˜i (t)| n X |aij (t)|pj |xj (t) − xj (t)| + n X
Rt +|βij (t)|)pj kzt ke
then system (1) has exactly one ω−periodic solution x ˜(t). Moreover it is globally exponentially stable. Proof: Let C = C([−τ, 0], Rn ) with the norm kϕk = sups∈[−τ,0],1≤i≤n |ϕi (s)|. From (A2), we can get |fj (u)| ≤ pj |u| + |fj (0)| = pj |u|, j = 1, 2, · · · , n. Hence all hypotheses in Theorem 1 with qj = fj (0) = 0 hold. Thus system (1) has at least one ω-periodic solution, say x(t) = (x1 (t), x2 (t), · · · , xn (t))T . Let x(t) = (x1 (t), x2 (t), · · · , xn (t))T be an arbitrary solution of system (1.1). Calculating the right derivative D+ |xi (t) − xi (t)| of |xi (t) − xi (t)| along the solutions of system (1).
≤
j=1
(19)
j=1
D+
Thus, for t > t0 , we have Rt n X ci (s)ds + D (zi (t)e t0 )≤ (|aij (t)| + |αij (t)|
−ci (t)|xi (t) − xi (t)| +
n X (|aij (t)| + |αij (t)| j=1
+|βij (t)|)pj
sup
|xj (s) − xj (s)|
(20)
t−τ ≤s≤t
Let zi (t) = |xi (t) − xi (t)|. Then (20) can be transformed into D+ zi (t) ≤
n X −ci (t)zi (t) + (|aij (t)| + |αij (t)|
sup t−τ ≤s≤t
© 2012 ACADEMY PUBLISHER
= ec
+
×e
zj (s)
(21)
τ
kz0 k R ω[ t ] R t
(
ω
0
+
)e ω[ t ] ω
Pn
c+ τ +(−ci +
≤ e
j=1
c+ τ
×kz0 ke
Pn j=1
γij pj ec
Rt +
j=1
+|βij (t)|)pj
Let t0 = 0, for t ≥ 0, [s] denotes the largest integer less than or equal to s. Noting that [ ωt ] ≥ ωt − 1 and (19), we get R t c+ τ Pn Rt + e γij (u)pj du −c (s)ds j=1 kzt k ≤ ec τ kz0 ke 0 e 0 i
+τ
t )ω[ ω ]
Pn
(−ci (s)+ ω[ t ] ω
Pn
−(ci −
≤ ec τ kz0 ke = M kz0 ke−λt
j=1
γij (s)pj −ci (s)ds
j=1
γij (s)pj e
|γij pj ec
+τ
c
+ τ i )ds
)t
(27)
JOURNAL OF COMPUTERS, VOL. 7, NO. 4, APRIL 2012
+
where M = max1≤i≤n {eci τ }, λ = min1≤i≤n {ci − Pn c+ τ }. From (27), it is clear j=1 (|aij | + |αij | + |βij |)|pj e that periodic solution xi (t) is global exponentially stable.
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u1 (t) = u2 (t) = 10+cos t, τ11 (t) = τ21 (t) = 0.005(1+sin t), τ12 (t) = τ22 (t) = 0.003(1 + cos t). Furthermore, we have
Remark 1. To the best of our knowledge, few authors have considered the existence of periodic solution and global exponential stability for model (1) with coefficients and delays all periodically varying in time. We only find [35] in existing work, however, it is assumed in [35] that coefficients are constants and only delays τij (t) vary in time. Especially, the authors of [35] suppose that fi , i = 1, 2, · · · , n are strictly nondecreasing. In this work, fi , i = 1, 2, · · · , n are only assumed to satisfying (A1) and (A2). It is clear that fi , i = 1, 2, · · · , n can not be strictly monotone. Obviously our model is more general. Therefore, our results are more convenient when design a fuzzy cellular neural networks.
c1 = 4, c2 = 3, |a11 | + |α11 | + |β 11 | =
3 , 40
3 3 , |a21 | + |α21 | + |β 21 | = , 20 40 3 + 1 1 , c = 5, c+ |a22 |+|α22 |+|β 22 | = 2 = 4, p1 = , p2 = . 20 1 3 4 By some simple calculations, we obtain 8π+1 24π+3 160 320 K = 6π+1 , ρ(K) ≈ 0.4936 < 1 6π+1 |a12 | + |α12 | + |β 12 | =
60
80
It is easy to verify that all the conditions of Theorem 2 are satisfied. Therefore, system (28) has an exponentially stable 2π-periodic solution.
III. A N ILLUSTRATIVE EXAMPLE In this section, we give an example to illustrate our results. Example 1. Consider the following fuzzy cellular neural networks with delays 0 P2 x1 (t) = −c1 (t)x1 (t) + j=1 a1j (t)fj (xj (t)) V2 + j=1 α1j (t)fj (xj (t − τ1j (t))) + I1 (t) W2 + j=1 β1j (t)fj (xj (t − τ1j (t))) V2 W2 + j=1 T1j (t)uj (t) + j=1 H1j (t)uj (t) x02 (t)
= −c2 (t)x2 (t) +
P2
j=1
a2j (t)fj (xj (t))
+
V2
α2j (t)fj (xj (t − τ2j (t))) + I2 (t)
+
W2
β2j (t)fj (xj (t − τ2j (t)))
+
V2
T2j (t)uj (t) +
j=1
j=1
j=1
W2
j=1
H2j (t)uj (t) (28)
where 1 c1 (t) = 4 + sin t, c2 (t) = 3 + cos t, f1 (x) = sin( x), 3 1 1 f2 (x) = sin( x), a11 (t) = a21 (t) = + sin t, 4 40 1 1 a12 (t) = a22 (t) = +cos t, α11 (t) = α21 (t) = +sin t, 20 60 1 1 α12 (t) = α22 (t) = +cos t, β11 (t) = β21 (t) = +sin t, 30 30 1 1 β12 (t) = β22 (t) = +cos t, T11 (t) = T21 (t) = +cos t, 15 10 1 1 T12 (t) = T22 (t) = +sin t, H11 (t) = H21 (t) = +cos t, 20 20 1 H12 (t) = H22 (t) = +sin t, I1 (t) = I2 (t) = 20+sin t, 10 © 2012 ACADEMY PUBLISHER
IV. C ONCLUSION In this paper, we have studied the existence, and exponential stability of the periodic solution for fuzzy cellular neural networks with time-varying delays. Some sufficient conditions set up here are easily verified and these conditions are correlated with parameters and time delays of the system (1). The obtained criteria can be applied to design globally exponentially periodic oscillatory fuzzy cellular neural networks. ACKNOWLEDGMENT We would like to extend our thanks to the referees for their suggestions which certainly improved the exposition of this paper. This work is partially supported by the Doctoral Foundation of Guizhou College of Finance and Economics, and supported by the Scientific Research Foundation of Guizhou Science and Technology Department (Dynamics of Impulsive Fuzzy Cellular Neural Networks with Delays) R EFERENCES [1] B. Kosto, “Adaptive bi-directional associative memories,” Appl. Opt., vol. 26, no. 23, pp. 4947–4960, 1987. [2] ——, “Bi-directional associative memories,” IEEE Trans. Syst. Man Cybernet, vol. 18, no. 1, pp. 49–60, 1988. [3] ——, Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence. Prentice-Hall, Englewood Cliffs, NJ, 1992. [4] K. Gopalsmy and X. Z. He, “Delay-independent stability in bi-directional associative memory networks,” IEEE Trans. Neural Networks, vol. 5, no. 6, pp. 998–1002, 1994. [5] J. Cao and L. Wang, “Periodic oscillatory solution of bidirectional associative memory networks with delays,” Phys. Rev. E, vol. 61, no. 2, pp. 1825–1828, 2000. [6] B. Liu and L. Huang, “Global exponential stability of BAM neural networks with recent-history distributed delays and impulse,” Neurocomputing, vol. 69, no. 16, pp. 2090–2096, 2006. [7] J. Cao and L. Wang, “Exponential stability and periodic oscilatory solution in BAM networks with delays,” IEEE Trans. Neural Networks, vol. 13, no. 2, pp. 457–463, 2002.
940
[8] H. Zhao, “Global exponential stability of bidirectional associative memory neural networks with distributed delays,” Phys. Lett. A, vol. 297, no. 3, pp. 182–190, 2002. [9] J. Zhang and Y. Yang, “Global stability analysis of bidirectional associative memory neural networks with time delays,” Int. J. Ciruit Theor. Appl., vol. 29, no. 2, pp. 185– 196, 2001. [10] X. F. Liao, K. W. Wong, and S. Z. Yang, “Convergence dynamics of hybrid bidirectional associative memory neural networks with distributed delays,” Phys. Lett. A, vol. 316, no. 1, pp. 55–64, 2003. [11] X. F. Liao and J. B. Yu, “Qualitative analysis of bidirectional associative memory with time delay,” Int. J. Circ. Theory Appl., vol. 26, no. 3, pp. 219–229, 1998. [12] J. Cao and Q. Jiang, “An analysis of periodic solutions of bi-directional associative memory networks with timevarying delays,” Phys. Lett. A, vol. 330, no. 3, pp. 203–213, 2004. [13] A. Chen, J. Cao, and L. Huang, “Exponential stability of BAM neural networks with transmission delays,” Neurocomputing, vol. 57, pp. 435–454, 2004. [14] A. Chen, L. Huang, and J. Cao, “Existence and stability of almost periodic solution for BAM neural networks with delays,” Appl. Math. Comput., vol. 137, no. 1, pp. 177– 193, 2003. [15] A. Chen, L. Huang, Z. Liu, and J. Cao, “Periodic bidirectional associative memory neural networks with distributed delays,” Journal of Math. Analys. and Appl., vol. 317, no. 1, pp. 80–102, 2006. [16] Z. Liu, A. Chen, J. Cao, and L. Huang, “Existence and global exponential stability of almost periodic solutions of BAM neural networks with continuously distributed delays,” Phys. Lett. A, vol. 319, no. 3, pp. 305–316, 2003. [17] S. Guo, L. Huang, B. Dai, and Z. Zhang, “Global existence of periodic solutions of BAM neural networks with variable coefficients,” Phys. Lett. A, vol. 317, no. 1, pp. 97–106, 2003. [18] J. Cao, “A set of stability criteria for delayed cellular neural networks,” IEEE Trans. Circuits Systems I, vol. 48, no. 4, pp. 494–498, 2001. [19] ——, “Global stability conditions for delayed CNNs,” IEEE Trans. Circuits Systems I, vol. 48, no. 11, pp. 1330– 1333, 2001. [20] J. Cao and J. Wang, “Global asymptotic stability of general class of recurrent neural networks with time-varying delays,” IEEE Trans. Circuits Systems I, vol. 50, no. 1, pp. 34–44, 2003. [21] J. Cao, “New results concerning exponential stability and periodic solutions of delayed cellular neural networks,” Phys. Lett. A, vol. 307, no. 2, pp. 136–147, 2003. [22] J. Cao and J. Wang, “Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays,” Neural Networks, vol. 17, no. 13, pp. 379–390, 2004. [23] ——, “Global exponential stability and periodicity of recurrent neural networks with time delays,” IEEE Trans. Circuits Systems I, vol. 52, no. 5, pp. 920–931, 2005. [24] J. Cao and D. W. C. Ho, “A general framework for global asymptotic stability analysis of delayed neural networks based on LMI approach,” Chaos Solitons Fractals, vol. 24, no. 5, pp. 1317–1329, 2005. [25] J. Cao, D. Huang, and Y. Qu, “Global robust stability of delayed recurrent neural networks,” Chaos Solitons Fractals, vol. 23, no. 1, pp. 221–229, 2005. [26] Q. Song, Z. Zhao, and Y. Li, “Global exponential stability of BAM neural networks with distributed delays and reaction-diffusion terms,” Phys. Lett. A, vol. 335, no. 2, pp. 213–225, 2005. [27] T. Yang and L. B. Yang, “The global stability of fuzzy cel© 2012 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 7, NO. 4, APRIL 2012
[28] [29] [30] [31] [32] [33]
[34]
[35]
lular neural networks,” IEEE Trans. Circ. Syst. I, vol. 43, no. 10, pp. 880–883, 1996. T. Yang, L. B. Yang, C. W. Wu, and L. O. Chua, “Fuzzy cellular neural networks: theory,” in Proc IEEE Int. Workshop Cellular Neural Networks Appl., 1996, pp. 181–186. T. Yang, L. Yang, C. Wu, and L. Chua, “Fuzzy cellular neural networks: applications,” in Proc IEEE Int. Workshop Cellular Neural Networks Appl., 1996, pp. 225–230. T. Huang, “Exponential stability of fuzzy cellular neural networks with distributed delay,” Phys. Lett. A, vol. 351, no. 1, pp. 48–52, 2006. ——, “Exponential stability of delayed fuzzy cellular neural networks with diffusion,” Chaos Solitons Fractals, vol. 31, pp. 658–664, 2007. Q. Zhang and R. Xiang, “Global asymptotic stability of fuzzy cellular neural networks with time-varying delays,” Phys. Lett. A, vol. 372, no. 22, pp. 3971–3977, 2008. Q. Zhang and W. Luo, “Global exponential stability and periodic solutions of FCNNs with constant delays and time-varying delays,” in Proceeding of 2009 International Joint Conference on Computational Sciences and Optimization, vol. 2, 2009, pp. 659–662. Y. Wu and Q. Zhang, “Global exponential stability of fuzzy cellular neural networks with variable delays and distributed delays,” in Proceeding of the 6th Conference of Biomathematics, 2008, pp. 695–699. K. Yuan, J. Cao, and J. Deng, “Exponential stability and periodic solutions of fuzzy cellular neural networks with time-varying delays,” Neurocomputing, vol. 69, no. 13, pp. 1619–1627, 2006.
Qianhong Zhang received the M. S. degree from Southwest Jiaotong University, Chengdu, China in 2004, and the Ph. D degree from Central South University, Changsha, China in 2009, all in mathematics or applied mathematics. From April 2004 to December 2009, he was a Lecture of Hunan Institute of Technology, Hengyang, Hunan, China. In January 2010, he joined Guizhou Key Laboratory of Economics System Simulation, Guizhou College of Finance and Economics, Guiyang China. He is currently an Associate Professor of Guizhou College of Finance and Economics. He also is the author or coauthor of more than 30 journal papers. His research interests include nonlinear systems, neural networks, fuzzy differential and fuzzy difference equation and stability theory.
Lihui Yang received the M. S. degree in department of applied mathematics from Southwest Jiaotong University, Chengdu, China in 2005. From July 2005 to June 2008, he was with Huaiying Institute of Technology, Huai’an, Jiangsu, China. In July 2008, he joined Department of Mathematics, Hunan City University, Yiyang, Hunan,China. He is currently a Lecture of Hunan City University. He also is the author or coauthor of more than 20 journal papers. His research interests include neural networks and fuzzy systems.
Daixi Liao received the B. S. degree in 2003 and the M. S. in 2009 from Xiangtan University, Xiangtan, Hunan, China, all in mathematics. He is currently a Lecture of Hunan Institute of Technology, Hengyang, Hunan, China. His current research interests include neural networks and nonlinear systems.
JOURNAL OF COMPUTERS, VOL. 7, NO. 4, APRIL 2012
Dynamical analysis of fuzzy cellular neural networks with time-varying delays Qianhong Zhang1,∗ , Lihui Yang2 , Daixi Liao3 1. Guizhou Key Laboratory of Economics System Simulation, Guizhou College of Finance and Economics, Guiyang, Guizhou 550004, P.R.China 2. Department of Mathematics, Hunan City University, Yiyang, Hunan 413000, P. R. China 3. Basic Science Department, Hunan Institute of Technology, Hengyang, Hunan 421002, P. R. China Email: [email protected]; [email protected]; [email protected]
Abstract— In this paper, employing continuation theorem of the coincidence degree, and inequality technique, some sufficient conditions are derived to ensure global exponential stability and the existence of periodic solution for fuzzy cellular neural networks with time-varying delays. These results have important leading significance in the design and applications of globally stable neural networks. Moreover an example is given to illustrate the effectiveness and feasibility of results obtained. Index Terms— fuzzy cellular neural networks, periodic solution, global exponential stability, time-varying delays
fuzzy cellular neural networks: x0i (t)
= −ci (t)xi (t) + + +
n ^ j=1 n _ j=1 n ^ j=1
Cellular neural network is formed by many units named cells, and that cell contains linear and nonlinear circuit elements, which typically are linear capacitor, linear resistor, linear and nonlinear controlled source, and independent sources. Nowadays, cellular neural networks(CNNs) are widely used in signal and image processing, associative memories, pattern classification [1]–[6]. In the last decade, dynamic behaviors of CNNs have been intensively studied because of the successful hardware implementation and their widely application(see,for example, [4]–[26]). In this paper, we would like to integrate fuzzy operations into cellular neural networks. Speaking of fuzzy operations, Yang and Yang [27] first introduced fuzzy cellular neural networks (FCNNs) combining those operations with cellular neural networks. So far researchers have founded that FCNNs are useful in image processing, and some results have been reported on stability and periodicity of FCNNs [27]–[35]. However, to the best of our knowledge, few author investigated the stability of fuzzy cellular neural networks with time-varying delays. In this paper, we investigate the existence, and the global exponential stability of periodic solution for the following Corresponding author. Tel. +86 851 6902456. fax: +86 851 6902456. E-mail address: [email protected] (Q.Zhang). Manuscript received January 1, 2011; revised June 1, 2011; accepted July 1, 2011.
© 2012 ACADEMY PUBLISHER doi:10.4304/jcp.7.4.934-940
aij (t)fj (xj (t))
j=1
+ I. I NTRODUCTION
n X
αij (t)fj (xj (t − τij (t))) βij (t)fj (xj (t − τij (t))) + Ii (t) Tij (t)uj (t) +
n _
Hij (t)uj (t)
(1)
j=1
i = 1, 2, · · · , n. where n corresponds to the number of neurons in neural networks. For xi (t) is the activations of the ith neuron at time t. ci (t) denotes the rate with which the ith neuron will reset its potential to the resting state in isolationVwhen W disconnected from the network and external inputs; and denote the fuzzy AND and fuzzy OR operations. aij (t) denotes the strengths of connectivity between cell i and cell j at time t. αij (t), βij (t), Tij (t) and Hij (t) are elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template between cell i and j at time t. τij (t) corresponds to the time delay required in processing and transmitting a signal from the jth cell to the ith cell at time t. uj (t) and Ii (t) denote the external input, bias of the ith neurons at time t, respectively. fj (·) is signal transmission functions. Throughout the paper, we give the following assumptions (A1) |fj (x)| ≤ pj |x| + qj for all x ∈ R, j = 1, 2, · · · , n. where pj , qj are nonnegative constants. (A2) The signal transmission functions fj (·), (j = 1, 2, · · · , n) are Lipschtiz continuous on R with Lipschtiz constants pj , namely, for any x, y ∈ R |fj (x) − fj (y)| ≤ pj |x − y|, fj (0) = 0. Definition 1: If f (t) : R → R is a continuous function, then the upper right derivative of f (t) is defined as 1 D+ f (t) = lim+ sup (f (t + h) − f (t)). h h→0
JOURNAL OF COMPUTERS, VOL. 7, NO. 4, APRIL 2012
Let τ = max1≤i,j≤n supt≥0 {τij (t)}. For functions ϕi defined on [−τ, 0], i = we set Ψ = (ϕ1 , ϕ2 , · · · , ϕn )T . If (x1 (t), x2 (t), · · · , xn (t))T is an ω−periodic system (1.1), then we denote kΨ − xk =
continuous 1, 2, · · · , n, x(t) = solution of
n X ( sup |ϕi (t) − xi (t)|). i=1 −τ ≤t≤0
Assume that system (1) is supplemented with initial value xi (t) = ϕi (t),
−τ ≤ t ≤ 0.
Definition 2: The periodic solution x(t) = (x1 (t), x2 (t), · · · , xn (t))T is said to be globally exponentially stable. If there exist constants λ > 0 and M ≥ 1 such that for any solution x(t) = (x1 (t), x2 (t), · · · , xn (t))T of system (1) |xi (t) − xi (t)| ≤ M kΨ − xke−λt , t ≥ 0. (2) Lemma 1: (see [26]) If ρ(K) < 1 for matrix K = (kij )n×n ≥ 0, then (E − K)−1 ≥ 0, where E denotes the identity matrix of size n. Lemma 2:(see [27]) Suppose x and y are two states of system (1), then we have ^ n ^ n αij (t)fj (y) αij (t)fj (x) − j=1 j=1 ≤
n X
|αij (t)||fj (x) − fj (y)|,
j=1
and n _ βij (t)fj (x) j=1
−
≤
βij (t)fj (y) j=1 n _
n X
|βij (t)||fj (x) − fj (y)|.
j=1
The remainder of this paper is organized as follows. In Section 2, we will give the sufficient conditions to ensure the existence of periodic oscillatory solution for fuzzy cellular neural networks with time-varying delays, and show that all other solutions converge exponentially to it as n → ∞. In Section 3 an example will be given to illustrate effectiveness of our results obtained. We will give a general conclusion in Section 4. II. P ERIODIC OSCILLATORY SOLUTIONS In this section, we will consider the periodic oscillatory solutions of system (1) with τij (t), ci (t), aij (t), αij (t), βij (t), Tij (t), Hij (t), uj (t) and Ii (t) satisfying the following assumptions: (A3) τij ∈ C(R, [0, ∞)) are periodic solutions with period ω for i, j = 1, 2, · · · , n. (A4) ci ∈ C(R, (0, ∞)), aij , αij , βij , Tij , Hij , uj , Ii ∈ C(R, R) are periodic solutions with common periodic ω, and fi ∈ C(R, R), i, j = 1, 2, · · · , n. © 2012 ACADEMY PUBLISHER
935
We will use the coincidence degree theory to obtain the existence of an ω-periodic solution to system (1). For the sake of convenience, we briefly summarize the theory as below. Let X and Z be normed spaces, L : DomL ⊂ X 7→ Z be a linear mapping and N : X 7→ Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dimKerL = codim ImL < ∞ and ImL is closed in Z. If L is a Fredholm mapping of index zero, then there exist continuous projectors P : X 7→ X and Q : Z 7→ Z such that ImP = KerL and ImL = KerQ = Im(I − Q). It follows that L|DomL ∩ KerP : (I − P )X 7→ ImL is invertible. We denote the inverse of this map by Kp . If Ω is a bounded open subset of X, the mapping N is called L-compact on Ω. if QN (Ω) is bounded and Kp (I − Q)N : Ω 7→ X is compact. Because ImQ is isomorphic to KerL, there exists an isomorphism J : ImQ 7→ KerL. Let Ω T⊂ Rn be open and bounded, f ∈ S C 1 (Ω, Rn ) C(Ω, Rn ) and y ∈ Rn \f (∂Ω Sf ), i.e., y is a regular value of f . Here, Sf = {x ∈ Ω : Jf (x) = 0}, the critical set of f , and Jf is the Jacobian of f at x. Then the degree degf, Ω, y is defined by X deg{f, Ω, y} = sgnJf (x) x∈f −1 (y)
with the agreement that the above sum is zero if f −1 (y) = Φ. Lemma 3: Let L be a Fredholm mapping of index zero and let N be L-compact on Ω. Suppose that (a) for each λ ∈ (0, 1), every solution x of Lx = λN x is such that x ∈ / ∂Ω. T (b) QN x 6= 0 for each x ∈ ∂Ω KerL and \ deg{JQN, Ω KerL, 0} = 6 0. Then the equation T Lx = N x has at least one solution lying in DomL Ω. To be convenience, in the rest of paper, for a continuous function g : [0, ω] 7→ R, we denote Z 1 ω + g = max g(t), g− = min g(t), g¯ = g(t)dt. ω 0 t∈[0,ω] t∈[0,ω] Theorem 1: Under assumptions (A1), (A3) and (A4), kij = c¯1i + ω (|aij | + |αij | + |βij |)pj K = (kij )n×n . Suppose that ρ(K) < 1, then system (1) has at least an ω-periodic solution. Proof: Take X = Z = {x(t) = (x1 (t), x2 (t), · · · , xn (t))T ∈ C(R, Rn ) : x(t + ω) = x(t), t ∈ R} and denote kxk = max1≤i≤n maxt∈[0,ω] |xi (t)|. Equipped with the norm k · k, both X and Z are Banach space. For any x(t) ∈ X, it is easy to check that Θ(xi , t)
:= −ci (t)xi (t) + +
n X
aij (t)fj (xj (t))
j=1
+
n ^ j=1
αij (t)fj (xj (t − τij (t)))
936
JOURNAL OF COMPUTERS, VOL. 7, NO. 4, APRIL 2012
+ +
n _ j=1 m ^
Noting assumption (A1), we get
βij (t)fj (xj (t − τij (t))) + Ii (t) Tij (t)uj (t) +
j=1
n _
|xi |− ci
≤
Hij (t)uj (t) ∈ Z
j=1 n X + (|aij | + αij | + |βij |)qj + |Ii |
j=1
Let L : Dom L = {x ∈ X : x ∈ C(R, Rn )} 3 x 7→ x(·) ˙ ∈ Z. Z ω 1 x(t)dt ∈ X. P : X 3 x 7→ ω 0 Z 1 ω Q : Z 3 z 7→ z(t)dt ∈ Z, ω 0 N : X 3 x 7→ Θ(x, ·) ∈ Z. For any V = (v1 , v2 , · · · , vn ) ∈ Rn , we identify it as the constant function in X or Z with the value vector V = (v1 , v2 , · · · , vn ). Then system (1.1) can be reduced to operator equation Lx = N x. It is Reasy to see that ω KerL = Rn , ImL = {z ∈ Z : ω1 0 z(t)dt = 0}. Which is closed in Z, dimKerL = codimImL = n < ∞, and P, Q are continuous projectors such that ImP = KerL, KerQ = ImL = Im(I − Q). It follows that L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) Kp : ImL 7→ KerP ∩ DomL given by Z t Z Z 1 ω s (Kp (z))i (t) = zi (s)ds − zi (v)dvds. ω 0 0 0 Therefore, applying the Arzela-Ascoli theorem, one can ¯ with any bounded easily show that N is L-compact on Ω open subset Ω ⊂ X. Since ImQ = KerL, we take the isomorphism J of ImQ onto KerL to be the identity mapping. Now we need only to show that, for an appropriate open bounded subset Ω, application of the continuation theorem corresponding to the operator equation Lx = λN x, λ ∈ (0, 1), Let x˙ i (t) = λΘ(xi , t) i = 1, 2, · · · , n.
0
0
=
+
0
j=1 n ^
+ +
j=1 n _
|xi |−
≤
n 1 X (|aij | + |αij | + |βij |)pj |xj |+ ci j=1 n X 1 + (|aij | + αij | + |βij |)qj + |Ii | ci j=1 n n _ ^ 1 + |Hij ||uj |+ (7) |Tij ||uj |+ + ci
Note that each xi (t) is continuously differentiable for i = 1, 2, · · · , n, it is certain that there exists ti ∈ [0, ω] such that |xi (ti )| = |xi (t)|− . Set F = (F1 , F2 , · · · , Fn )T , where n X 1 Fi = ( + ω) (|aij | + αij | + |βij |)qj ci j=1 n n ^ _ + |Tij ||uj |+ + |Hij ||uj |+ + |Ii | (8) j=1
j=1
In view of ρ(K) < 1 and Lemma 1, we have (E − K)−1 F = h = (h1 , h2 , · · · , hn )T ≥ 0. where hi is given by n X hi = kij hj + Fi , i = 1, 2, · · · , n. (9) j=1
Set Ω = {(x1 , x2 , · · · , xn )T ∈ Rn : |xi | < hi , i = 1, 2, · · · , n} (10) Then, for t ∈ [ti , ti + ω], we have t
t+ω
|xi (t)|− +
D+ |xi (t)|dt
ti
j=1
n _ + Hij (t)uj (t) + Ii (t) dt
D+ |xi (t)|dt
|xi (ti )| + Z
≤ ≤
βij (t)fj (xj (t − τij (t)))
© 2012 ACADEMY PUBLISHER
j=1
ti
n 1 X (|aij | + |αij | + |βij |)pj |xj |+ ci j=1 n 1 X + (|aij | + αij | + |βij |)qj ci j=1
j=1
(5)
(6)
j=1
j=1
≤
αij (t)fj (xj (t − τij (t)))
j=1
|Hij ||uj |+
| xi (t) |
ci (t)xi (t)dt Z ω X n m ^ aij (t)fj (xj (t)) + Tij (t)uj (t) 0 j=1
n _
It follows that
Z
n ^
|Tij ||uj |+ +
j=1
(3)
Assume that x = x(t) ∈ X is a solution of system (3) for some λ ∈ (0, 1). Integrating (3) over the interval [0, ω], we obtain that Z ω Z ω 0= x˙ i (t)dt = λ Θ(xi , t)dt (4) Hence Rω
n X (|aij | + |αij | + |βij |)pj |xj |+
+
n ^ j=1
|Tij ||uj |+ +
n _ j=1
|Hij ||uj |+ + |Ii |
JOURNAL OF COMPUTERS, VOL. 7, NO. 4, APRIL 2012
Z
937
t+ω
D+ |xi (t)|dt ti n X 1 ( + ω) (|aij | + |αij | + |βij |)pj |xj |+ ci j=1 n X 1 +( + ω) (|aij | + αij | + |βij |)qj ci j=1 n n ^ _ + |Tij ||uj |+ + |Hij ||uj |+ + |Ii |
n X 1 +( + ω) (|aij | + αij | + |βij |)qj ci j=1
+
≤
j=1
n X
j=1
n ^
+
|Tij ||uj |+ +
j=1
=
n X
n _ j=1
|Hij ||uj |+ + |Ii |
kij hj + Fj
(15)
j=1
Which is a contradiction. Therefore (13) holds and QN u 6= 0, for u ∈ ∂Ω ∩ kerL = ∂Ω ∩ Rn .
(16)
(11)
Consider the homotopy Φ : (Ω∩KerL)×[0, 1] 7→ Ω∩KerL defined by
Clearly, hi , i = 1, 2, · · · , n, are independent of λ. Then for ∀λ ∈ (0, 1), x ∈ ∂Ω such that Lx 6= λN x. When u = (x1 , x2 , · · · , xn )T ∈ ∂Ω ∩ KerL = ∂Ω ∩ Rn , u is a constant vector with |xi | = hi , i = 1, 2, · · · , n. Note that QN u = JQN u, when u ∈ KerL, it must be
Φ(u, µ) = µdiag(−c1 , −c2 , · · · , −cn )u + (1 − µ)QN u (17) Note that Φ(·, 0) = JQN . if Φ(u, µ) = 0, then we have n 1 − µ X (aij + αij + βij )f (xj ) |xi | = ci j=1 n n _ ^ Tij uj + Hij uj + Ii + j=1 j=1 n 1 X (|aij | + |αij | + |βij |)pj hj ≤ ci
≤
kij hj + Fi = hi
j=1
= −ci xi +
(QN )ui
n X (aij + αij + βij )f (xj ) j=1
+
n ^
Tij uj +
j=1
n _
Hij uj + Ii
(12)
j=1
We claim that
j=1
k(QN u)i k > 0, i = 1, 2, · · · , n.
(13)
+
ci xi
j=1 n ^
+
Tij uj +
j=1
n _
Hij uj + Ii
+
(14)
j=1
n 1 X = |xi | = (aij + αij + βij )f (xj ) ci j=1 n n ^ _ + Tij uj + Hij uj + Ii j=1 j=1 n 1 X ≤ (|aij | + |αij | + |βij |)pj hj ci j=1
n X + (|aij | + |αij | + |βij |)qj j=1
+
n ^
|Tij ||uj |+ +
j=1
n _ j=1
|Hij ||uj |+ + |Ii |
n X 1 < ( + ω) (|aij | + |αij | + |βij |)pj |hj |+ ci j=1 © 2012 ACADEMY PUBLISHER
n ^
|Tij ||uj |+ +
j=1
Then we have hi
(|aij | + |αij | + |βij |)qj
j=1
On the contrary, suppose that there are some i such that k(QN u)i k = 0, namely n X = (aij + αij + βij )f (xj )
n X
n _ j=1
|Hij ||uj |+ + |Ii |
n X 1 < ( + ω) (|aij | + |αij | + |βij |)pj |hj |+ ci j=1 n X 1 +( + ω) (|aij | + αij | + |βij |)qj ci j=1 n n _ ^ + |Tij ||uj |+ + |Hij ||uj |+ + |Ii | j=1
=
n X
j=1
kij hj + Fj
(18)
j=1
Therefore Φ(u, µ) 6= 0 for any (u, µ) ∈ (Ω ∩ KerL). It follows from the property of invariance under homotopy that deg = = =
{JQN, Ω ∩ KerL, 0} deg{Φ(·, 0), Ω ∩ KerL, 0} deg{Φ(·, 1), Ω ∩ KerL, 0} deg{diag(−c1 , −c2 , · · · , −cn )} = 6 0
Thus, we have shown that Ω satisfies all the assumptions of Lemma 3. Hence, Lu = N u has at least one ω-periodic solution on DomL ∩ Ω.
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JOURNAL OF COMPUTERS, VOL. 7, NO. 4, APRIL 2012
Theorem 2: Let τ = max1≤i,j≤n,t∈[0,ω] {τij (t)}. Suppose that (A2), (A3) and (A4) hold, ρ(K) < 1, and that ci −
n X + (|aij | + |αij | + |βij |)pj eci τ > 0
|xi (t) − xi (t)| ≤ −ci (t)|xi (t) − xi (t)| n X + |aij (t)[fj (xj (t)) − fj (xj (t))]| j=1
n ^ αij (t)fj (xj (t − τij (t))) + j=1 n ^ αij (t)fj (xj (t − τij (t))) − j=1 n _ βij (t)fj (xj (t − τij (t))) + j=1 n _ − βij (t)fj (xj (t − τij (t))) j=1
j=1
+ +
j=1 n X
|αij (t)|pj |xj (t − τij (t)) − xj (t − τij (t))|
(22)
It follows that Rt ci (s)ds zi (t) e t0 ≤ |zi (t0 )| Z t X n + (|aij (u)| + |αij (u)| + |βij (u)|) t0 j=1 Ru ci (s)ds t0 ×pj kzu ke du (23) Thus, for any θ ∈ [−τ, 0], we have R t+θ R t R t+θ Rt ci (s)ds ( + )ci (s)ds ci (s)ds−c+ τ i e t0 = e t0 t ≥ e t0 (24) Therefore Rt R t+θ ci (s)ds−c+ τ ci (s)ds i t0 e z (t + θ) ≤ e t0 zi (t + θ) i Z t X n (|aij (u)| + |αij (u)| + |βij (u)|) ≤ kzt0 k + t0 j=1 Ru ci (s)ds t0 × pj kzu ke du
By Gronwall’s inequality, we obtain, for t ≥ t0 , R t c+ τ Pn Rt e γij (u)pj du −ci (s)ds c+ τ j=1 t0 kzt k ≤ e kzt0 ke e t0 (26) where γij (u) = |aij (u)| + |αij (u)| + |βij (u)|.
|βij (t)|pj |xj (t − τij (t)) − xj (t − τij (t))|
j=1
≤
ci (s)ds
t0
It follows that Rt + ci (s)ds e t0 kzt k ≤ eci τ kzt0 k Z t n X + + eci τ (|aij (u)| + |αij (u)| + |βij (u)|) t0 j=1 Ru ci (s)ds t0 ×pj kzu ke du (25)
−ci (t)|xi (t) − x˜i (t)| n X |aij (t)|pj |xj (t) − xj (t)| + n X
Rt +|βij (t)|)pj kzt ke
then system (1) has exactly one ω−periodic solution x ˜(t). Moreover it is globally exponentially stable. Proof: Let C = C([−τ, 0], Rn ) with the norm kϕk = sups∈[−τ,0],1≤i≤n |ϕi (s)|. From (A2), we can get |fj (u)| ≤ pj |u| + |fj (0)| = pj |u|, j = 1, 2, · · · , n. Hence all hypotheses in Theorem 1 with qj = fj (0) = 0 hold. Thus system (1) has at least one ω-periodic solution, say x(t) = (x1 (t), x2 (t), · · · , xn (t))T . Let x(t) = (x1 (t), x2 (t), · · · , xn (t))T be an arbitrary solution of system (1.1). Calculating the right derivative D+ |xi (t) − xi (t)| of |xi (t) − xi (t)| along the solutions of system (1).
≤
j=1
(19)
j=1
D+
Thus, for t > t0 , we have Rt n X ci (s)ds + D (zi (t)e t0 )≤ (|aij (t)| + |αij (t)|
−ci (t)|xi (t) − xi (t)| +
n X (|aij (t)| + |αij (t)| j=1
+|βij (t)|)pj
sup
|xj (s) − xj (s)|
(20)
t−τ ≤s≤t
Let zi (t) = |xi (t) − xi (t)|. Then (20) can be transformed into D+ zi (t) ≤
n X −ci (t)zi (t) + (|aij (t)| + |αij (t)|
sup t−τ ≤s≤t
© 2012 ACADEMY PUBLISHER
= ec
+
×e
zj (s)
(21)
τ
kz0 k R ω[ t ] R t
(
ω
0
+
)e ω[ t ] ω
Pn
c+ τ +(−ci +
≤ e
j=1
c+ τ
×kz0 ke
Pn j=1
γij pj ec
Rt +
j=1
+|βij (t)|)pj
Let t0 = 0, for t ≥ 0, [s] denotes the largest integer less than or equal to s. Noting that [ ωt ] ≥ ωt − 1 and (19), we get R t c+ τ Pn Rt + e γij (u)pj du −c (s)ds j=1 kzt k ≤ ec τ kz0 ke 0 e 0 i
+τ
t )ω[ ω ]
Pn
(−ci (s)+ ω[ t ] ω
Pn
−(ci −
≤ ec τ kz0 ke = M kz0 ke−λt
j=1
γij (s)pj −ci (s)ds
j=1
γij (s)pj e
|γij pj ec
+τ
c
+ τ i )ds
)t
(27)
JOURNAL OF COMPUTERS, VOL. 7, NO. 4, APRIL 2012
+
where M = max1≤i≤n {eci τ }, λ = min1≤i≤n {ci − Pn c+ τ }. From (27), it is clear j=1 (|aij | + |αij | + |βij |)|pj e that periodic solution xi (t) is global exponentially stable.
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u1 (t) = u2 (t) = 10+cos t, τ11 (t) = τ21 (t) = 0.005(1+sin t), τ12 (t) = τ22 (t) = 0.003(1 + cos t). Furthermore, we have
Remark 1. To the best of our knowledge, few authors have considered the existence of periodic solution and global exponential stability for model (1) with coefficients and delays all periodically varying in time. We only find [35] in existing work, however, it is assumed in [35] that coefficients are constants and only delays τij (t) vary in time. Especially, the authors of [35] suppose that fi , i = 1, 2, · · · , n are strictly nondecreasing. In this work, fi , i = 1, 2, · · · , n are only assumed to satisfying (A1) and (A2). It is clear that fi , i = 1, 2, · · · , n can not be strictly monotone. Obviously our model is more general. Therefore, our results are more convenient when design a fuzzy cellular neural networks.
c1 = 4, c2 = 3, |a11 | + |α11 | + |β 11 | =
3 , 40
3 3 , |a21 | + |α21 | + |β 21 | = , 20 40 3 + 1 1 , c = 5, c+ |a22 |+|α22 |+|β 22 | = 2 = 4, p1 = , p2 = . 20 1 3 4 By some simple calculations, we obtain 8π+1 24π+3 160 320 K = 6π+1 , ρ(K) ≈ 0.4936 < 1 6π+1 |a12 | + |α12 | + |β 12 | =
60
80
It is easy to verify that all the conditions of Theorem 2 are satisfied. Therefore, system (28) has an exponentially stable 2π-periodic solution.
III. A N ILLUSTRATIVE EXAMPLE In this section, we give an example to illustrate our results. Example 1. Consider the following fuzzy cellular neural networks with delays 0 P2 x1 (t) = −c1 (t)x1 (t) + j=1 a1j (t)fj (xj (t)) V2 + j=1 α1j (t)fj (xj (t − τ1j (t))) + I1 (t) W2 + j=1 β1j (t)fj (xj (t − τ1j (t))) V2 W2 + j=1 T1j (t)uj (t) + j=1 H1j (t)uj (t) x02 (t)
= −c2 (t)x2 (t) +
P2
j=1
a2j (t)fj (xj (t))
+
V2
α2j (t)fj (xj (t − τ2j (t))) + I2 (t)
+
W2
β2j (t)fj (xj (t − τ2j (t)))
+
V2
T2j (t)uj (t) +
j=1
j=1
j=1
W2
j=1
H2j (t)uj (t) (28)
where 1 c1 (t) = 4 + sin t, c2 (t) = 3 + cos t, f1 (x) = sin( x), 3 1 1 f2 (x) = sin( x), a11 (t) = a21 (t) = + sin t, 4 40 1 1 a12 (t) = a22 (t) = +cos t, α11 (t) = α21 (t) = +sin t, 20 60 1 1 α12 (t) = α22 (t) = +cos t, β11 (t) = β21 (t) = +sin t, 30 30 1 1 β12 (t) = β22 (t) = +cos t, T11 (t) = T21 (t) = +cos t, 15 10 1 1 T12 (t) = T22 (t) = +sin t, H11 (t) = H21 (t) = +cos t, 20 20 1 H12 (t) = H22 (t) = +sin t, I1 (t) = I2 (t) = 20+sin t, 10 © 2012 ACADEMY PUBLISHER
IV. C ONCLUSION In this paper, we have studied the existence, and exponential stability of the periodic solution for fuzzy cellular neural networks with time-varying delays. Some sufficient conditions set up here are easily verified and these conditions are correlated with parameters and time delays of the system (1). The obtained criteria can be applied to design globally exponentially periodic oscillatory fuzzy cellular neural networks. ACKNOWLEDGMENT We would like to extend our thanks to the referees for their suggestions which certainly improved the exposition of this paper. This work is partially supported by the Doctoral Foundation of Guizhou College of Finance and Economics, and supported by the Scientific Research Foundation of Guizhou Science and Technology Department (Dynamics of Impulsive Fuzzy Cellular Neural Networks with Delays) R EFERENCES [1] B. Kosto, “Adaptive bi-directional associative memories,” Appl. Opt., vol. 26, no. 23, pp. 4947–4960, 1987. [2] ——, “Bi-directional associative memories,” IEEE Trans. Syst. Man Cybernet, vol. 18, no. 1, pp. 49–60, 1988. [3] ——, Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence. Prentice-Hall, Englewood Cliffs, NJ, 1992. [4] K. Gopalsmy and X. Z. He, “Delay-independent stability in bi-directional associative memory networks,” IEEE Trans. Neural Networks, vol. 5, no. 6, pp. 998–1002, 1994. [5] J. Cao and L. Wang, “Periodic oscillatory solution of bidirectional associative memory networks with delays,” Phys. Rev. E, vol. 61, no. 2, pp. 1825–1828, 2000. [6] B. Liu and L. Huang, “Global exponential stability of BAM neural networks with recent-history distributed delays and impulse,” Neurocomputing, vol. 69, no. 16, pp. 2090–2096, 2006. [7] J. Cao and L. Wang, “Exponential stability and periodic oscilatory solution in BAM networks with delays,” IEEE Trans. Neural Networks, vol. 13, no. 2, pp. 457–463, 2002.
940
[8] H. Zhao, “Global exponential stability of bidirectional associative memory neural networks with distributed delays,” Phys. Lett. A, vol. 297, no. 3, pp. 182–190, 2002. [9] J. Zhang and Y. Yang, “Global stability analysis of bidirectional associative memory neural networks with time delays,” Int. J. Ciruit Theor. Appl., vol. 29, no. 2, pp. 185– 196, 2001. [10] X. F. Liao, K. W. Wong, and S. Z. Yang, “Convergence dynamics of hybrid bidirectional associative memory neural networks with distributed delays,” Phys. Lett. A, vol. 316, no. 1, pp. 55–64, 2003. [11] X. F. Liao and J. B. Yu, “Qualitative analysis of bidirectional associative memory with time delay,” Int. J. Circ. Theory Appl., vol. 26, no. 3, pp. 219–229, 1998. [12] J. Cao and Q. Jiang, “An analysis of periodic solutions of bi-directional associative memory networks with timevarying delays,” Phys. Lett. A, vol. 330, no. 3, pp. 203–213, 2004. [13] A. Chen, J. Cao, and L. Huang, “Exponential stability of BAM neural networks with transmission delays,” Neurocomputing, vol. 57, pp. 435–454, 2004. [14] A. Chen, L. Huang, and J. Cao, “Existence and stability of almost periodic solution for BAM neural networks with delays,” Appl. Math. Comput., vol. 137, no. 1, pp. 177– 193, 2003. [15] A. Chen, L. Huang, Z. Liu, and J. Cao, “Periodic bidirectional associative memory neural networks with distributed delays,” Journal of Math. Analys. and Appl., vol. 317, no. 1, pp. 80–102, 2006. [16] Z. Liu, A. Chen, J. Cao, and L. Huang, “Existence and global exponential stability of almost periodic solutions of BAM neural networks with continuously distributed delays,” Phys. Lett. A, vol. 319, no. 3, pp. 305–316, 2003. [17] S. Guo, L. Huang, B. Dai, and Z. Zhang, “Global existence of periodic solutions of BAM neural networks with variable coefficients,” Phys. Lett. A, vol. 317, no. 1, pp. 97–106, 2003. [18] J. Cao, “A set of stability criteria for delayed cellular neural networks,” IEEE Trans. Circuits Systems I, vol. 48, no. 4, pp. 494–498, 2001. [19] ——, “Global stability conditions for delayed CNNs,” IEEE Trans. Circuits Systems I, vol. 48, no. 11, pp. 1330– 1333, 2001. [20] J. Cao and J. Wang, “Global asymptotic stability of general class of recurrent neural networks with time-varying delays,” IEEE Trans. Circuits Systems I, vol. 50, no. 1, pp. 34–44, 2003. [21] J. Cao, “New results concerning exponential stability and periodic solutions of delayed cellular neural networks,” Phys. Lett. A, vol. 307, no. 2, pp. 136–147, 2003. [22] J. Cao and J. Wang, “Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays,” Neural Networks, vol. 17, no. 13, pp. 379–390, 2004. [23] ——, “Global exponential stability and periodicity of recurrent neural networks with time delays,” IEEE Trans. Circuits Systems I, vol. 52, no. 5, pp. 920–931, 2005. [24] J. Cao and D. W. C. Ho, “A general framework for global asymptotic stability analysis of delayed neural networks based on LMI approach,” Chaos Solitons Fractals, vol. 24, no. 5, pp. 1317–1329, 2005. [25] J. Cao, D. Huang, and Y. Qu, “Global robust stability of delayed recurrent neural networks,” Chaos Solitons Fractals, vol. 23, no. 1, pp. 221–229, 2005. [26] Q. Song, Z. Zhao, and Y. Li, “Global exponential stability of BAM neural networks with distributed delays and reaction-diffusion terms,” Phys. Lett. A, vol. 335, no. 2, pp. 213–225, 2005. [27] T. Yang and L. B. Yang, “The global stability of fuzzy cel© 2012 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 7, NO. 4, APRIL 2012
[28] [29] [30] [31] [32] [33]
[34]
[35]
lular neural networks,” IEEE Trans. Circ. Syst. I, vol. 43, no. 10, pp. 880–883, 1996. T. Yang, L. B. Yang, C. W. Wu, and L. O. Chua, “Fuzzy cellular neural networks: theory,” in Proc IEEE Int. Workshop Cellular Neural Networks Appl., 1996, pp. 181–186. T. Yang, L. Yang, C. Wu, and L. Chua, “Fuzzy cellular neural networks: applications,” in Proc IEEE Int. Workshop Cellular Neural Networks Appl., 1996, pp. 225–230. T. Huang, “Exponential stability of fuzzy cellular neural networks with distributed delay,” Phys. Lett. A, vol. 351, no. 1, pp. 48–52, 2006. ——, “Exponential stability of delayed fuzzy cellular neural networks with diffusion,” Chaos Solitons Fractals, vol. 31, pp. 658–664, 2007. Q. Zhang and R. Xiang, “Global asymptotic stability of fuzzy cellular neural networks with time-varying delays,” Phys. Lett. A, vol. 372, no. 22, pp. 3971–3977, 2008. Q. Zhang and W. Luo, “Global exponential stability and periodic solutions of FCNNs with constant delays and time-varying delays,” in Proceeding of 2009 International Joint Conference on Computational Sciences and Optimization, vol. 2, 2009, pp. 659–662. Y. Wu and Q. Zhang, “Global exponential stability of fuzzy cellular neural networks with variable delays and distributed delays,” in Proceeding of the 6th Conference of Biomathematics, 2008, pp. 695–699. K. Yuan, J. Cao, and J. Deng, “Exponential stability and periodic solutions of fuzzy cellular neural networks with time-varying delays,” Neurocomputing, vol. 69, no. 13, pp. 1619–1627, 2006.
Qianhong Zhang received the M. S. degree from Southwest Jiaotong University, Chengdu, China in 2004, and the Ph. D degree from Central South University, Changsha, China in 2009, all in mathematics or applied mathematics. From April 2004 to December 2009, he was a Lecture of Hunan Institute of Technology, Hengyang, Hunan, China. In January 2010, he joined Guizhou Key Laboratory of Economics System Simulation, Guizhou College of Finance and Economics, Guiyang China. He is currently an Associate Professor of Guizhou College of Finance and Economics. He also is the author or coauthor of more than 30 journal papers. His research interests include nonlinear systems, neural networks, fuzzy differential and fuzzy difference equation and stability theory.
Lihui Yang received the M. S. degree in department of applied mathematics from Southwest Jiaotong University, Chengdu, China in 2005. From July 2005 to June 2008, he was with Huaiying Institute of Technology, Huai’an, Jiangsu, China. In July 2008, he joined Department of Mathematics, Hunan City University, Yiyang, Hunan,China. He is currently a Lecture of Hunan City University. He also is the author or coauthor of more than 20 journal papers. His research interests include neural networks and fuzzy systems.
Daixi Liao received the B. S. degree in 2003 and the M. S. in 2009 from Xiangtan University, Xiangtan, Hunan, China, all in mathematics. He is currently a Lecture of Hunan Institute of Technology, Hengyang, Hunan, China. His current research interests include neural networks and nonlinear systems.
