Global exponential periodicity and stability of recurrent neural networks with multi-proportional delays

ISA Transactions 60 (2016) 89–95

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Global exponential periodicity and stability of recurrent neural networks with multi-proportional delays Liqun Zhou n, Yanyan Zhang School of Mathematics Science, Tianjin Normal University, Tianjin 300387, China

art ic l e i nf o

a b s t r a c t

Article history: Received 1 July 2015 Received in revised form 1 October 2015 Accepted 6 November 2015 Available online 27 November 2015 This paper was recommended for publication by Dr. Oscar Camacho

In this paper, a class of recurrent neural networks with multi-proportional delays is studied. The nonlinear transformation transforms a class of recurrent neural networks with multi-proportional delays into a class of recurrent neural networks with constant delays and time-varying coefficients. By constructing Lyapunov functional and establishing the delay differential inequality, several delay-dependent and delay-independent sufficient conditions are derived to ensure global exponential periodicity and stability of the system. And several examples and their simulations are given to illustrate the effectiveness of obtained results. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Recurrent neural networks (RNNs) Proportional delay factor Global exponential periodicity Global exponential stability Nonlinear transformation

1. Introduction In recent years, delayed recurrent neural networks (DRNNs), such as delayed cellular neural networks, delayed Hopfield neural networks and delayed Cohen–Grossberg neural networks, have received great attention due to their wide applications in classification of image processing, pattern recognition, signal processing, associative memories, optimization problem, and so forth. And such applications depend on the stability of the equilibrium of the designed networks. Up to now, many results on stability of DRNNs have been developed [1–7]. At the same time, other dynamical properties of DRNNs have been widely studied, such as periodicity, synchronization and oscillation. It is well known that an equilibrium point can be viewed as a special periodic solution of neural networks with arbitrary period. In this sense, the stability analysis of periodic solutions of DRNNs may be considered to be more general than that of equilibrium points. At present, a lot of periodicity results for DRNNs are obtained in [8–26]. Time delays considered for RNNs can be classified as constant delays [9–11,18,22,23,25,26], time-varying delays [8,12,14,21,24], distributed delays [13,15–17,19,20]. Proportional delay is a kind of time-varying unbounded delay which is different from the aboven

Corresponding author. E-mail addresses: [email protected], [email protected] (L. Zhou). http://dx.doi.org/10.1016/j.isatra.2015.11.008 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

mentioned delays. Proportional delay is a kind of objective existence, for example, in Web quality of service (QoS) routing decision, the proportional delay is usually required [27,28]. Since a neural network usually has a spatial nature due to the presence of an amount of parallel pathways of a variety of axon sizes and lengths, it is desired to model by introducing continuously proportional delay. As an important mathematical model, the proportional delay system often rises in some fields such as physics, biology systems and control theory [29–31]. With the increase of time, the proportional delay function ð1  qÞt is monotonically increasing, thus it may be convenient to control the network's running time according to the allowed delays. From the viewpoint of time delay, RNNs with proportional delays are different from the above other DRNNs, so those obtained dynamics properties of DRNNs [8–26] cannot be directly applied to RNNs with proportional delays. It is important to ensure that the designed network is stable in the presence of proportional delay. Therefore, it has important theoretical significance to study dynamic behavior of NNs with proportional delays. To the best of our knowledge, so far a few works have been done on the dynamical behaviors for the neural networks with proportional delays [32–38]. In [32–38], by a nonlinear transformation, a class of neural networks with proportional delays is transformed equivalently into a class of neural networks with constant delays and time-varying coefficients, then using such approaches as Lyapunov functional [19–21,23,29], algebraic inequalities [15,18,22,25,26], matrix theory [24], and so on, the

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L. Zhou, Y. Zhang / ISA Transactions 60 (2016) 89–95

their dynamics behaviors are studied. In [32], dissipativity of a class of cellular neural networks (CNNs) with proportional delays was investigated by using the inner product properties. Zhou in [33–35] had discussed the global exponential stability and asymptotic stability of CNNs with proportional delays by employing matrix theory and constructing Lyapunov functional, respectively. In [36], by constructing appropriate delay differential inequalities, several new delay-independent and delay-dependent global exponential stability sufficient conditions of equilibrium point of the BAM neural networks with proportional delays were obtained. In [37], the stability criteria for high-order networks with proportional delay was studied based on matrix measure and Halanay inequality. In [38], new explicit conditions ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval were obtained by matrix inequalities. In this case, it is interesting and challenging to further study the periodic solutions of neural networks with proportional delays. But the dynamical behavior of periodic solutions of DRNNs with proportional delays has never been tackled. Motivated by the discussion above, the previous criteria for checking the stability and periodicity of the addressed neural networks are somewhat conservative due to the construction of Lyapunov functionals and the technicality of the mathematical method used. This paper attempts to fill the gap by considering RNNs with multi-proportional delays. In this paper, criteria on the global exponential periodicity and stability of RNNs with multi-proportional delays can be derived by constructing a suitable Lyapunov functional and establishing a delay differential inequality. The delay differential inequality is established in this paper, which is similar to the Halanay inequality [18,22,26] or generalized Halanay inequality [15], but it is not Halanay inequality or generalized Halanay inequality. This method might have an important role in studying the dissipativity, instability, and synchronization for complex neural networks with proportional delays. The remainder of this paper is organized as follows. Model description and preliminaries are given in Section 2. In Section 3, several delay-dependent and delay-independent sufficient conditions are given by constructing appropriate Lyapunov functional and delay differential inequality. Numerical examples and their simulations are presented in Section 4 to illustrate the effectiveness of obtained results. Finally, conclusions are given in Section 5. Notations: R denotes real number. Rn denotes n dimensions Euclidean space. Let C ¼ C ð½q; 1; Rn Þ be the Banach space of all continuous function from ½q; 1 to Rn with the topology of uniform P convergence. For any φðsÞ A C , let J φðsÞ J ¼ ni¼ 1 supq r s r 1 j φi ðsÞj . n Let C ¼ Cð½  τ; 0; R Þ be the Banach space of all continuous function from ½  τ; 0 to Rn with the topology of uniform convergence. P For any ψ A C, let J ψ ðtÞ J ¼ ni¼ 1 sup  τ r t r 0 j ψ i ðtÞj . Define xt ¼ xðs þ tÞ, xt ðx0 Þ ¼ xðs þ t; x0 Þ, s A ½q; 1, t Z 1. Denote J xt J ¼ Pn i ¼ 1 supq r s r 1 j xi ðs þ tÞj . DefinePyt ¼ yðt þ θ Þ, yt ðφÞ ¼ yðt þ θ ; φÞ, θ A ½  τ; 0; t Z 0. Denote J yt J ¼ ni¼ 1 sup  τ r θ r 0 j yi ðt þ θÞj .

2. System description and preliminaries Consider a class of recurrent neural networks with multiproportional delays described by 8 n X > > x_ ðtÞ ¼  di xi ðtÞ þ ½aij f j ðxj ðtÞÞ > > < i j¼1 ð2:1Þ > þ bij g j ðxj ðpj tÞÞ þcij hj ðxj ðqj tÞÞ þ I i ðtÞ; > > > : xi ðsÞ ¼ φi ðsÞ; s A ½q; 1; for i ¼ 1; 2; …; n; t Z 1. In (2.1), xi(t) denotes the state variable of the ith neuron at time t; di 4 0 denotes the rate with which the ith neuron resets its potential to the resting state when isolated from

the other neuron and inputs; aij ; bij and cij denote the connection weights between the ith neuron and the jth neuron at time t; pj t and qj t, respectively; pj and qj are proportional delay factors and satisfy 0 o pj ; qj r1, pj t ¼ t  ð1  pj Þt; qj t ¼ t  ð1  qj Þt, in which ð1 pj Þt and ð1  qj Þt denote the transmission delays; q ¼ min1 r j r n fpj ; qj g; Ii(t) is the external input periodic function with period ω, i.e. there exists a constant ω 4 0 such that I i ðt þ ωÞ ¼ I i ðtÞ; i ¼ 1; 2; …; n; φi ðsÞ is an initial function at s A ½q; 1, φðsÞ ¼ ðφ1 ðsÞ; φ2 ðsÞ; …; φn ðsÞÞT A Cð½q; 1; Rn Þ. f i ðÞ; g i ðÞ and hi ðÞ : R-R are the nonlinear activation functions, and assumed to satisfy the following requirements: 8 > < j f i ðuÞ f i ðvÞj rLi j u  vj ; j g i ðuÞ  g i ðvÞj r M i j u  vj ; ð2:2Þ > : j h ðuÞ  h ðvÞj r N j u vj ; i i i for i ¼ 1; 2; …; n, u; v A R, where Li ; M i ; N i are nonnegative constants. As a special case of (2.1), the DRNNs with constant input Ii are described by the following functional differential equations: 8 n X > > ½aij f j ðxj ðtÞÞ > x_ i ðtÞ ¼ di xi ðtÞ þ > < j¼1 ð2:3Þ > þ bij g j ðxj ðpj tÞÞ þ cij hj ðxj ðqj tÞÞ þ I i ; > > > : x ðsÞ ¼ φ ðsÞ; s A ½q; 1; i ¼ 1; 2; …; n: i i Remark 2.1. The transmission delays are ð1  pj Þt- þ 1, ð1  qj Þt- þ 1, as t- þ 1; j ¼ 1; 2; …; n in (2.1) and (2.3), so (2.1) and (2.3) are different from models in [8–26]. Those stability results in [8–26] cannot be directly applied to (2.1). Let us consider the following nonlinear transformation defined by yi ðtÞ ¼ xi ðet Þ;

ð2:4Þ

then it is easy to prove that systems (2.1) and (2.3) are equivalent to the following RNNs with constant delays and time-varying coefficients, respectively (see, [33]) 8 ( n X > > > y_ ðtÞ ¼ et  di y ðtÞ þ ½aij f j ðyj ðtÞÞ > i i > > > j¼1 > < ) ð2:5Þ > þ b g ðy ðt  τ ÞÞ þ c h ðy ðt  ς ÞÞ þ u ðtÞ ; > ij j j j ij j j i j > > > > > > : y ðsÞ ¼ ψ ðsÞ; s A ½  τ; 0; i i and 8 ( n X > > t > _ y ðtÞ ¼ e ½aij f j ðyj ðtÞÞ  di yi ðtÞ þ > i > > > j¼1 > <

) > þ b g ðy ðt  τ ÞÞ þ c h ðy ðt  ς ÞÞ þ I > ij j j j ij j j i ; j > > > > > > : y ðsÞ ¼ ψ ðsÞ; s A ½  τ; 0; i

ð2:6Þ

i

for t Z 0, i ¼ 1; 2; …; n. Where τj ¼  log pj Z 0, ςj ¼  log qj Z 0, τ ¼ max1 r j r n fτj ; ςj g, ui ðtÞ ¼ Ii ðet Þ. ψ i ðsÞ ¼ φi ðes Þ A Cð½  τ; 0; RÞ, i ¼ 1; 2; …; n. ψ ¼ ðψ 1 ; ψ 2 ; …; ψ n ÞT A Cð½  τ; 0; Rn Þ. Remark 2.2. Eqs. (2.5) and (2.6) are different from models in [14,15,21,24]. The coefficients of models in [14,15,21,24] are bounded time variable functions, but the coefficients which contain et in (2.5) and (2.6) are unbounded time-varying functions. Let xn ¼ ðxn1 ; xn2 ; …; xnn ÞT be the equilibrium point of system (2.3), P we denote J φ  xn J ¼ ni¼ 1 supq r s r 1 j φi ðsÞ  xni j . Let x(t) be an arbitrary solution of system (2.3) for t Z 1, where xðtÞ ¼ ðx1 ðt; φÞ; x2 ðt; φÞ; …; xn ðt; φÞÞT . Let yn ¼ ðyn1 ; yn2 ; …; ynn ÞT be the equilibrium point of system (2.6), P we denote J ψ  yn J ¼ ni¼ 1 sup  τ r θ r 0 j ψ i ðθÞ  yni j . Let y(t) be an

L. Zhou, Y. Zhang / ISA Transactions 60 (2016) 89–95

arbitrary solution of system (2.6) for t Z 0, where yðtÞ ¼ ðy1 ðt; ψ Þ; y2 ðt; ψ Þ; …; yn ðt; ψ ÞÞT . Definition 2.3. An equilibrium point xn A Rn of system (2.3) is said to be globally exponentially stable if there exist two positive constants β Z 1 and α 4 0 such that J xðtÞ  x J r β J φ  x J t n

n

α

;

t Z1;

n

for any xðtÞ A R with J xðtÞ  xn J ¼

along (3.4) n  X

D þ VðtÞ r

þ

n X

Pn

n i ¼ 1 j xi ðtÞ  xi j .

þ

r

Given any φðsÞ; φðsÞ A C , let xðt; φÞ ¼ ðx1 ðt; φÞ; x2 ðt; φÞ; …; xn ðt; φÞÞT and xðt; φ s Þ ¼ ðx1 ðt; φ Þ; x2 ðt; φ Þ; …; xn ðt; φ ÞÞT be the solutions of (2.1) starting from φ and φ , respectively. Given any ψ ; ψ A C, let yðt; ψ Þ ¼ ðy1 ðt; ψ Þ; y2 ðt; ψ Þ; …; yn ðt; ψ ÞÞT and yðt; ψ Þ ¼ ðy1 ðt; ψ Þ; y2 ðt; ψ Þ; …; yn ðt; ψ ÞÞT be the solutions of (2.5) starting from ψ and ψ , respectively. Theorem 3.1. Under condition (2.2), if there exists a positive number σ 4 1, such that ðj aji j Li þ j bji j eστi M i þ j cji j eσςi N i Þ Z 0;

for i ¼ 1; 2; …; n, then for every periodic input Ii(t), system (2.1) is globally exponentially periodic, where τi ¼  log pi Z 0, ςi ¼  log qi Z 0.

þ

By applying (3.1) in the above inequality, we deduce that D þ VðtÞ r0 for t Z0 with the implication V ðtÞ r Vð0Þ for t Z0. Using this and (3.5), we have n X

Y i ðtÞ ¼ eσ t j yi ðt; ψ Þ  yi ðt; ψ Þj :

i¼1

Substituting (3.3) into (3.2) yields 8 n
j¼1

þ j cij j N j ςj eσςj r

n X

Y i ð0Þ þ

j cij j N j Y j ðt  ςj

Þeσςj

9 = ;



sup

 τj r s r 0

Y j ðsÞ

Y j ðsÞ

sup

 ςj r s r 0

n X n X

ðj bji j M i τeστ

i¼1j¼1

Y i ðsÞ

sup

τrsr0

1 þ τeστ

n X

ðj bji j M i þ j cji j N i Þ

j¼1

 X n

sup

i ¼ 1 τrsr0

Y i ðsÞ;

n o P where β ¼ max1 r i r n 1 þ τeστ nj¼ 1 ðj bji j M i þ j cji j N i Þ Z 1. Hence, by (3.3) and (3.6), we obtain n X

j yi ðt; ψ Þ  yi ðt; ψ Þj r βe  αt

i¼1

n X

sup

i ¼ 1 τrsr0

j yi ðt; ψ Þ  yi ðt; ψ Þj ; ð3:7Þ

for t Z 0, where α ¼ σ  1 40. Let ψ ðtÞ ¼ φðet Þ; ψ ðtÞ ¼ φ ðet Þ, it follows from (2.4) and (3.7) that n X

j¼1

j¼1

1rirn

ð3:3Þ

j bij j M j Y j ðt  τj

ð3:6Þ

And, it follows from (3.5) that n  n  X X Vð0Þ r Y i ð0Þ þ j bij j M j τj eστj

r max

ð3:2Þ

þ

e  t Y i ðtÞ r VðtÞ rV ð0Þ:

i¼1



for t Z 0; i ¼ 1; 2; …; n, where D þ denotes the upper right derivation. Accordingly, we then consider function Yi(t) defined by

n X

j¼1

i¼1

 j cij j Nj j yj ðt  ςj ; ψ Þ  yj ðt  ςj ; ψ Þj ;

Þeστj þ

n  n X X ðj aji j Li þ j bji j M i eστi ðdi  σ Þ 

þ j cji j N i τeστ Þ

j¼1

n X

j¼1



 þ j cji j Ni eσςi Þ Y i ðtÞ:

j bij j M j j yj ðt  τj ; ψ Þ  yj ðt  τj ; ψ Þj

n X

n  n X X ð1  σ Þe  t þ di  ðj aji j Li þ j bji j M i eστi

i¼1

j aij j Lj j yj ðt; ψ Þ  yj ðt; ψ Þj

j¼1



þ j cji j Ni eσςi Þ Y i ðtÞ

j¼1

þ

j cij j N j et Y j ðtÞeσςj

i¼1

Proof. By yt ðψ Þ ¼ yðt þ θ; ψ Þ, then yt ðψ Þ A C for all t Z 0. By using (2.2), it follows from (2.5) that  D þ j yi ðt; ψ Þ  yi ðt; ψ Þj r et  di j ðyi ðt; ψ Þ  yi ðt; ψ Þj :

n X

j bij j M j et Y j ðtÞeστj

j¼1

ð3:1Þ

j¼1

þ

n X

n X

j¼1

3. Main results

n X

j aij j Lj et Y j ðtÞ þ

j¼1

¼

di  σ 

  e  t Y i ðtÞ þ e  t σ Y i ðtÞ  di et Y i ðtÞ::

i¼1

Definition 2.4 (Zhou and Hu [17]). System (2.1) is said to be globally exponentially periodic if there exists one ω-periodic solution of system (2.1) and all other solutions of the system converge exponentially to it as t-1.

n X

91

j xi ðet ; φðet ÞÞ  xi ðet ; φ ðet ÞÞj

i¼1

r β e  αt

;

n X

sup

i ¼ 1 τ rsr0

j xi ðes ; φðes ÞÞ  xi ðes ; φ ðes ÞÞj :

ð3:8Þ

ð3:4Þ for i ¼ 1; 2; …; n. We then construct a Lyapunov functional of the form V ðtÞ ¼

8 n < n X X e  t Y i ðtÞ þ :

i¼1

j¼1

Z j bij j M j

t t  τj

eστj Y j ðsÞ ds þ j cij j Nj

Z

t t  ςj

!) eσςj Y j ðsÞ ds

;

ð3:5Þ for t Z 0, σ 4 1, and calculating the rate of change D þ VðtÞ of V(t)

Let et ¼ η, then η Z1 and t ¼ log η Z 0; Let es ¼ ξ, then s ¼ log ξ A ½  τ; 0 and ξ A ½q; 1. Thus, by (3.8), we get n X

j xi ðη; φðηÞÞ  xi ðη; φ ðηÞÞj

i¼1

r βe  α log η

n X

sup j φi ðξÞ  φi ðξÞj ;

i ¼ 1 qrξr1

η Z1:

ð3:9Þ

92

L. Zhou, Y. Zhang / ISA Transactions 60 (2016) 89–95

Taking η ¼ t, we obtain that

0.6

J xðt; φÞ  xðt; φ Þ J r β J φ  φ J e  α log t ¼ β J φ  ψ J t  α ;

0.4

ð3:10Þ

Pn

where J φ  φ J ¼ that

t Z 1;

i¼1

supq r ξ r 1 j φi  φ i j . It follows from (3.10) t Z 1:

ð3:11Þ

0 X

J xt ðφÞ  xt ðφ Þ J r β J φ  φ J ðqtÞ  α ;

0.2

We can choose a positive integer m such that 1 4

βðqmωÞ  α r :

ð3:12Þ

Now, define a Poincaré mapping H : C -C by H φ ¼ xω ðφÞ. Then we can derive from (3.11) and (3.12) that J H m φ H m φ J r 14 J φ  φ J ;

ð3:13Þ

0

where H φ ¼ xmω ðφÞ. This implies that H is a contraction mapping. Therefore, there exists a unique fixed point φn A C such that H m φn ¼ φn . Note that m

m

0.2

0.4

0.6

0.8

X

Fig. 1. Periodicity of RNNs (4.1) with proportional delays.

H m ðH φn Þ ¼ HðH m φn Þ ¼ H φn :

2

This shows that H φn A C is also a fixed point of H , hence, H φn ¼ φn , that is, xω ðφn Þ ¼ φn . Let xðt; φn Þ be the solution of (2.1) through ð0; φn Þ. By using Iðt þ ωÞ ¼ IðtÞ for t Z 1, xðt þ ω; φn Þ is also a solution of (2.1), and note that m

1.5 1

xt þ ω ðφn Þ ¼ xt ðxω ðφn ÞÞ ¼ xt ðφn Þ;

0.5

for t Z 1, then xðt þ ω; φ Þ ¼ xðt; φ Þ for t Z 1. This shows that xðt; φn Þ is a periodic solution of (2.1) with period ω. From (3.11), it is easy to see that all other solution of (2.1) converge to this periodic solution exponentially as t-1.□ n

X

n

0

A sufficient condition of delay-dependent global exponentially periodic for (2.1) is obtained in Theorem 3.1. In the following, we give a sufficient condition of delay-independent global exponentially periodic for (2.1). Theorem 3.2. Under condition (2.2), if di 

n X

0

0.5

1

1.5

X

ðj aij j Lj þ j bij j M j þ j cij j N j Þ 4 0;

i ¼ 1; 2; …; n

ð3:14Þ

Fig. 2. Periodicity of RNNs (4.2) with proportional delays.

j¼1

hold, then for every periodic input I(t), the system (2.1) is globally exponentially periodic.

for i ¼ 1; 2; …; n. Accordingly, we define the function ZðÞ as follows:

Proof. Let us consider function P i ðvi Þ defined by

Z i ðtÞ ¼ eηt j yi ðt; ψ Þ  yi ðt; ψ Þj ;

P i ðvi Þ ¼ di vi 

n X

ðj aij j Lj þ j bij j M j evi τj þ j cij j Nj evi ςj Þ;

ð3:15Þ

j¼1

for vi A ½0; 1Þ, i ¼ 1; 2; …; n. We notice from (3.14) that di 

n X

ðj aij j Lj þ j bij j M j þ j cij j N j Þ Z δ;

i ¼ 1; 2; …; n;

D þ Z i ðtÞ r ηZ i ðtÞ  di et Z i ðtÞ þet ð3:16Þ

j aij j Lj Z j ðtÞ þ j bij j M j Z j ðt  τj Þeητj

o þ j cij j N j Z j ðt  ςj Þeηςj r  ðdi  ηÞet Z i ðtÞ

follows from (3.15) that P i ð0Þ Z δ for i ¼ 1; 2; …; n. We observe that P i ðvi Þ is continuous for vi A ½0; 1Þ, and P i ðvi Þ-  1 as vi -1. Thus there exist constants v~ i A ð0; 1Þ, i ¼ 1; 2; …; n, such that ðj aij j Lj þ j bij j M j ev~ i τj þ j cij j N j ev~ i ςj Þ ¼ 0:

n  X j¼1

n o P where δ ¼ min1 r i r n di  nj¼ 1 ðj aij j Lj þ j bij j M j þ j cij j N j Þ 4 0. It

n X

ð3:19Þ

We then use (3.2) and (3.19) to derive the following inequality for t 40 given by

j¼1

P i ðv~ i Þ ¼ di  v~ i 

t A ½  τ; þ 1Þ:

þ et

n X

ðj aij j Lj þ j bij j M j eητj þ j cij j N j eηςj Þ sup Z j ðsÞ: τrsrt

j¼1

ð3:20Þ

ð3:17Þ

j¼1

Therefore, there must exist a positive constant η A ð0; minfv~ i gÞ, such that P i ðηÞ ¼ di  η 

n X

ðj aij j Lj þ j bij j M j eητj þ j cij j Nj eηςj Þ 40;

j¼1

ð3:18Þ

Let  T ¼ max

1rirn

sup

τrsr0

where T 4 0.

 j yi ðs; ψ Þ  yi ðs; ψ Þj ;

ð3:21Þ

L. Zhou, Y. Zhang / ISA Transactions 60 (2016) 89–95

We notice from (3.21) and (3.19) that Z i ðtÞ rT for i ¼ 1; 2; …; n; t A ½  τ; 0, where T 4 0 is given by (3.21). We claim Z i ðtÞ r T;

i ¼ 1; 2; …; n; t A ½0; þ 1Þ:

ð3:22Þ

93

Corollary 3.4. Under condition (2.2), if there exists a positive number σ 4 1, for i ¼ 1; 2; …; n such that di  σ 

n X

ðj aji j Li þ j bji j M i e  σ log

pi

þ j cji j Ni e  σ log

qi

Þ 40

j¼1

Here we first prove that for any d 4 1, there are Z i ðtÞ o dT;

i ¼ 1; 2; …; n; t A ½0; þ 1Þ:

ð3:23Þ

hold, then the equilibrium point of system (2.3) is globally exponentially stable. Corollary 3.5. Under condition (2.2), if

Suppose that this claim (3.23) does not hold in this sense that there is one component among Z i ðÞ (say Z k ðÞ) and a first time t 1 4 0 such that

di 

Z k ðtÞ o dT; t A ½  τ; t 1 Þ;

ð3:24Þ

hold, then the equilibrium point of system (2.3) is globally exponentially stable.

ð3:25Þ

Corollary 3.6. Under condition (2.2), if there exists a positive number η 4 0, for i ¼ 1; 2; …; n such that

Z k ðt 1 Þ ¼ dT;

D þ Z k ðt 1 Þ Z 0

and Z i ðtÞ r dT; i ak; t A ½  τ; t 1 :

On the other hand, substituting (3.24) and (3.25) into (3.20) yields   n X 0 r D þ Z k ðt 1 Þ r  dk  η  ðj akj j Lj þ j bkj j Mj eητj þ j ckj j N j eηςj et1 dT; j¼1

ð3:26Þ and by applying (3.18) to the above inequality, we lead to 0 r D þ Z k ðt 1 Þ o 0 which means a contradiction. Thus, for t Z 0, Z i ðtÞ o dT. As d-1, we obtain that the claim (3.22) must hold. It follows (3.19) and (3.22) that j yi ðt; ψ Þ  yi ðt; ψ Þj r Te  ηt ;

i ¼ 1; 2; …; n; t Z0:

ðj aij j Lj þ j bij j M j þ j cij j Nj Þ 4 0;

i ¼ 1; 2; …; n

j¼1

di  η 

n X

ðj aij j Lj þ j bij j e  η log

pj

M j þ j cij j e  η log

qj

N j Þ Z 0;

i ¼ 1; 2; …; n

j¼1

hold, then the equilibrium point of system (2.3) is globally exponentially stable. Remark 3.7. Corollary 3.4 in this paper is the same as Corollary 3.4 in [33]. Remark 3.8. In (2.1), if pj ¼ qj ¼ 1; j ¼ 1; 2; …; n, then system (2.1) becomes standard RNNs without delays. Thus, results in the paper are also applicable to standard RNNs without delays.

ð3:27Þ 4. Illustrative examples

Let

λ¼

n X

max1 r i r n fsup  τ r s r 0 j yi ðs; ψ Þ  yi ðs; ψ Þj g ; sup  τ r s r 0 j yi ðs; ψ Þ  yi ðs; ψ Þj

ð3:28Þ

then λ Z1. And according to (3.21) and (3.28) yields T ¼λ

sup

τ rsr0

j yi ðs; ψ Þ  yi ðs; ψ Þj :

In the following, we will give several examples to illustrate our results. Example 4.1. Consider the following system

ð3:29Þ

x_ i ðtÞ ¼  di xi ðtÞ þ

t Z 0:

ð4:1Þ ð3:30Þ

It follows from (3.30) that J yt ðψ Þ  yt ðψ Þ J r λ J ψ  ψ J e  ηðt  τÞ :

ð3:31Þ

Then, by yi ðtÞ ¼ xi ðe Þ, we have t

J xt ðφÞ xt ðφ Þ J r λ J φ  φ J ðqtÞ  α ;

t Z 1:

ð3:32Þ

By the same arguments as in Theorem 3.1, we obtain that xðt; φn Þ is a periodic solution of (2.1) with period ω. From (3.32), it is easy to see that all other solutions of (2.1) converge to this periodic solution exponentially as t- þ 1.□ In addition, in view of the proof of Theorem 3.2, the following result is obtained. Theorem 3.3. Under condition (2.2), if there exists a positive number

η 4 0, such that di  η 

n X

½aij f j ðxj ðtÞÞ þbij f j ðxj ðpj tÞÞ þcij f j ðxj ðqj tÞÞ þ I i ðtÞ;

j¼1

Substituting (3.29) and (3.19) into (3.22) yields J yðt; ψ Þ  yðt; ψ Þ J r λ J ψ  ψ J e  ηt ;

2 X

ðj aij j Lj þ j bij j eητj M j þ j cij j eηςj N j Þ 4 0;

j¼1

for i ¼ 1; 2; …; n, then for every periodic input I(t), system (2.1) is globally exponentially periodic, where τi ¼  log pi Z 0, ςi ¼ log qi Z 0. Further, by Theorems 3.1–3.3, we obtain the following corollaries.

for i ¼ 1; 2; t Z 1, where d1 ¼ d2 ¼ 8, a11 ¼ 0:5, a12 ¼ 1, a21 ¼  1, a22 ¼ 1, b11 ¼ 1, b12 ¼ 1, b21 ¼  1, b22 ¼ 1:5, c11 ¼ 0, c12 ¼ 1, c21 ¼  1, c22 ¼ 2. I 1 ðtÞ ¼ sin ðtÞ; I 2 ðtÞ ¼ cos ðtÞ. The activation func

tions are f i ðxi Þ ¼ sin 13xi þ 13xi ; i ¼ 1; 2. And pj ¼ 0:4; qj ¼ 0:8; j ¼ 1; 2. Obviously f i ðxi Þ; i ¼ 1; 2 are Lipschitz continuous with Lipschitz constant Li ¼ 23; i ¼ 1; 2. τj ¼  log 0:4 ¼ 0:9163, ςj ¼  log 0:8 ¼ 0:2231; j ¼ 1; 2. Taking σ ¼ 1:2, we have 2 d1  σ  ½ðj a11 j þ j a21 j Þ þ ðj b11 j þ j b21 j Þeστ1 þ ðj c11 j þ j c21 j Þeσς1  3 ¼ 0:9249 4 0; 2 d2  σ  ½ðj a12 j þ j a22 j Þ þ ðj b12 j þ j b22 j Þeστ2 þ ðj c12 j þ j c22 j Þeσς2  3 ¼ 0:8499 4 0: Theorem 3.1 is satisfied. Thus, by Theorem 3.1, there exists one periodic solution of (4.1) with 2π as its period and all other solutions of (4.1) converge exponentially to it as t-1. The existence of exponentially periodic solutions can be clearly observed in Fig. 1. Example 4.2. Consider the following system x_ i ðtÞ ¼  di xi ðtÞ þ

2 X

½aij f j ðxj ðtÞÞ þbij g j ðxj ðpj tÞÞ þ cij hj ðxj ðqj tÞÞ þ I i ðtÞ;

j¼1

ð4:2Þ

94

L. Zhou, Y. Zhang / ISA Transactions 60 (2016) 89–95 1.5

Example 4.3. Consider the following system x_ i ðtÞ ¼  di xi ðtÞ þ

1

2 X

½aij f j ðxj ðtÞÞ þ bij g j ðxj ðpj tÞÞ þ cij hj ðxj ðqj tÞÞ þ I i ðtÞ;

j¼1

ð4:4Þ

X

0.5

for i ¼ 1; 2; t Z1, where d1 ¼ 3, d2 ¼ 4, a11 ¼ 0:5, a12 ¼ 1, a21 ¼ 1, a22 ¼  1, b11 ¼ 0:5, b12 ¼  0:5, b21 ¼ 0:6, b22 ¼ 1, c11 ¼ 1, c12 ¼ 1, c21 ¼ 0:5, c22 ¼ 1. I 1 ðtÞ ¼ cos ð2tÞ; I 2 ðtÞ ¼ sin ð2tÞ. The activation



functions are f i ðxi Þ ¼ tanh 12xi , g i ðxi Þ ¼ cos 14xi þ 14xi and hi ðxi Þ ¼ 1 4ðj xi þ 1j  j xi  1j Þ, i ¼ 1; 2; and pj ¼ 0:3, qj ¼ 0:5, j ¼ 1; 2. Obviously f i ðxi Þ, g i ðxi Þ and hi ðxi Þ, i ¼ 1; 2, are Lipschitz continuous with Lipschitz constants Li ¼ 12; M i ¼ 12 and N i ¼ 14, i ¼ 1; 2. τj ¼  log 0:3 ¼ 1:2040, ςj ¼  log 0:5 ¼ 0:6931, j ¼ 1; 2. Taking η ¼ 0:5, we have

0

0

1

2

3

4

X

d1  η  ½ðj a11 j L1 þ j a12 j L2 Þ þ ðj b11 j M 1 eητ1 þ j b12 j M 2 eητ2 Þ þ ðj c11 j eης1 þ j c12 j eης2 Þ ¼ 0:1301 4 0; d2  η  ½ðj a21 j L1 þ j a22 j L2 Þ þ ðj b21 j M 1 eητ1 þ j b22 j M 2 eητ2 Þ

Fig. 3. Periodicity of RNNs (4.4) with proportional delays.

þ ðj c21 j eης1 þ j c22 j eης2 Þ ¼ 0:5091 4 0: 2

Theorem 3.3 is satisfied. Thus, by Theorem 3.3, there exists one periodic solution of (4.4) with π as its period and all other solutions of (4.4) converge exponentially to it as t-1. The existence of exponentially periodic solutions can be clearly observed in Fig. 3.

1.5 1

Example 4.4. Consider the following system

X

0.5

x_ i ðtÞ ¼  di xi ðtÞ þ

0

2 X

½aij f j ðxj ðtÞÞ þ bij g j ðxj ðpj tÞÞ þ cij hj ðxj ðqj tÞÞ þ I i ;

j¼1

ð4:5Þ for i ¼ 1; 2, t Z 1, where I 1 ¼ 1, I 2 ¼ 2. The rest of this example are exactly the same as Example 4.2. So, (4.3) is also established. By Corollary 3.5, system (4.5) is globally exponentially stable, whose equilibrium point is ð  0:3318; 1:0510ÞT by using Matlab. The stability of system (4.5) can be clearly observed in Fig. 4. 0

0.5

1

1.5

X

Fig. 4. Stability of RNNs (4.5) with proportional delays.

5. Conclusions

for i ¼ 1; 2; t Z 1, where d1 ¼ 4, d2 ¼ 3:5, a11 ¼ 1, a12 ¼ 0, a21 ¼  2, a22 ¼ 1, b11 ¼  1, b12 ¼  2, b21 ¼ 0, b22 ¼ 1, c11 ¼ 0, c12 ¼ 1, c21 ¼ 1, c22 ¼ 0. I 1 ðtÞ ¼ sin ðtÞ; I 2 ðtÞ ¼ cos ðtÞ. The activation functions

are f i ðxi Þ ¼ 14ðj xi þ1j  j xi  1j Þ, g i ðxi Þ ¼ cos 12xi þ 14xi and hi ðxi Þ ¼ tanh ðxi Þ with Lipschitz constants Li ¼ 12; M i ¼ 34 and N i ¼ 1; i ¼ 1; 2. And 0 o pj ; qj r 1, j ¼ 1; 2. We have (

 d1  12ð j a11 j þ j a12 j Þ þ 34ð j b11 j þ j b12 j Þ þ ð j c11 j þ j c12 j Þ ¼ 0:25 4 0; 1 d2  2ð j a21 j þ j a22 j Þ þ 34ð j b21 j þ j b22 j Þ þ ð j c21 j þ j c22 j Þ ¼ 0:25 4 0: ð4:3Þ

The sufficient conditions in Theorem 3.2 are satisfied. Therefore, by Theorem 3.2, there exists one periodic solution of (4.2) with 2π as its period and all other solutions of (4.2) converge exponentially to it as t-1. The existence of exponentially periodic solutions can be clearly observed in Fig. 2 (taking pj ¼ qj ¼ 0:5; j ¼ 1; 2).

In this paper, the exponential periodic and stability of recurrent neural networks with proportional delays have been studied. The activation functions only need to satisfy the Lipschitz conditions, which need not be differentiable, bounded and monotone increasing. The exponential stability criteria obtained are very simple and easily checked by simple algebraic methods. The nonlinear transform yi ðtÞ ¼ xi ðet Þ can transform RNNs with proportional delays into RNNs with constant delays and time-varying coefficients. Therefore, the stability problem of neural networks with unbounded delay is changed into the stability problem of neural networks with constant time delays, so as to simplify the problem, which might have an impact role in studying the dissipativity, instability, and synchronization.

Acknowledgments The author would like to thank the editor and the anonymous reviewers for their valuable comments and constructive suggestions. The project is supported by the National Science Foundation of China (No. 61374009).

L. Zhou, Y. Zhang / ISA Transactions 60 (2016) 89–95

References [1] Cao J, Wang J. Global asymptotic and robust stability of recurrent neural networks with time delays. IEEE Trans Circuits Syst-I 2005;52(2):417–26. [2] Hu L, Gao H, Zheng W. Novel stability of cellular neural networks with interval-time varying delay. Neural Netw 2008;21(10):1458–63. [3] Huang C, Cao J. Almost sure exponential stability of stochastic cellular neural networks with unbounded distributed delays. Neurocomputing 2009;72(13–15):3352–6. [4] Feng Z, Lam J. Stability and dissipativity analysis of distributed delay cellular neural networks. IEEE Trans Neural Netw 2011;22(6):981–97. [5] Hu J, Wang J. Global stability of complex-valued recurrent neural networks with time-delays. IEEE Trans Neural Netw Learn Syst 2012;23(6):853–65. [6] Sun J, Chen J. Stability analysis of static recurrent neural networks with interval time-varying delay. Appl Math Comput 2013;221(15):111–20. [7] Liu PL. Delay-dependent global exponential robust stability for delayed cellular neural networks with time-varying delay. ISA Trans 2013;52:711–6. [8] Chen B, Wang J. Global exponential periodicity of a class of recurrent neural networks with oscillating parameters and time-varying delays. IEEE Trans Neural Netw 2005;16(6):1440–8. [9] Li C, Liao X. Robust stability and robust periodicity of delayed recurrent neural networks with noise disturbance. IEEE Trans Circuits Syst-I 2006;53(10):2265–73. [10] Li Y, Zhu L, Liu P. Existence and stability of periodic solutions of delayed cellular neural networks. Nonlinear Anal-Real World Appl 2006;7(2):225–34. [11] Yang Y, Cao J. Stability and periodicity in delayed cellular neural networks with impulsive effects. Nonlinear Anal-Real World Appl 2007;8(1):362–74. [12] Lou X, Cui B. Delay-dependent criteria for global robust periodicity of uncertain switched recurrent neural networks with time-varying delay. IEEE Trans Neural Netw 2008;19(4):549–57. [13] Huang Z, Xia Y. Exponential periodic attractor of impulsive BAM networks with finite distributed delays. Chaos Solitons Fractals 2009;39(1):373–84. [14] Tan M, Tan Y. Global exponential stability of periodic solution of neural network with variable coefficients and time-varying delays. Appl Math Modell 2009;33(1):373–85. [15] Huang X, Cao J, Ho DWC. Existence and attractivity of almost periodic solution for recurrent neural networks with unbounded delays and variable coefficients. Nonlinear Dyn 2006;45(3–4):337–51. [16] Liu Y, Wang Z, Liu X. Stability criteria for periodic neural networks with discrete and distributed delays. Nonlinear Dyn 2007;49(1–2):93–103. [17] Zhou L, Hu G. Global exponential periodicity and stability of cellular neural networks with variable and distributed delays. Appl Math Comput 2008;195(2):402–11. [18] Wang X, Li S, Xu D. Globally exponential stability of periodic solutions for impulsive neutral-type neural networks with delays. Nonlinear Dyn 2011;64 (1–2):65–75. [19] Liu Y, Huang Z, Chen L. Almost periodic solution of impulsive Hopfield neural networks with finite distributed delays. Neural Comput Appl 2012;21(5):821–31.

95

[20] Yang G, Kao Y, Li W, Sun X. Exponential stability of impulsive stochastic fuzzy cellular neural networks with mixed delays and reaction–diffusion terms. Neural Comput Appl 2013;23(3–4):1109–21. [21] Xu H, Wu R. Periodicity and exponential stability of discrete-time neural networks with variable coefficients and delays. Adv Differ Equ 2013. http://dx. doi.org/10.1186/1687-1847-2013-226. [22] Xu C, Zhang Q. Existence and exponential stability of anti-periodic solutions for a high-order delayed Cohen–Grossberg neural networks with impulsive effects. Neural Process Lett 2014;40(3):227–42. [23] Shao Y, Zhang Q. Stability and periodicity for impulsive neural networks with delays. Adv Differ Equ 2013. http://dx.doi.org/10.1186/1687-1847-2013-3529. [24] Yang W. Periodic solution for fuzzy Cohen–Grossberg BAM neural networks with both time-varying and distributed delays and variable coefficients. Neural Process Lett 2014;40(1):51–73. [25] Chiu K, Pinto M, Jeng JC. Existence and global convergence of periodic solutions in recurrent neural network models with a general piecewise alternately advanced and retarded argument. Acta Appl Math 2014;133(1):133–52. [26] Duan L, Huang L, Guo Z. Stability and almost periodicity for delayed highorder Hopfield neural networks with discontinuous activations. Nonlinear Dyn 2014;77(4):1469–84. [27] Chen Y, Qiao C. Proportional differentiation: a scalable QoS approach. IEEE Commun Mag 2003;41(6):52–8. [28] Wei J, Xu C, Zhou X, Li Q. A robust packet scheduling algorithm for proportional delay differentiation services. Comput Commun 2006;29(18):3679–90. [29] Liu YK. Asymptotic behavior of functional differential equations with proportional time delays. Eur J Appl Math 1996;7(1):11–30. [30] Iserles A. The asymptotic behaviour of certain difference equations with proportional delay. Appl Math Comput 1994;28(1–3):141–52. [31] Iserles A, Liu Y. On neutral functional–differential equations with proportional delays. J Math Anal Appl 1997;207(1):73–95. [32] Zhou L. Dissipativity of a class of cellular neural networks with proportional delays. Nonlinear Dyn 2013;73(3):1895–903. [33] Zhou L. Delay-dependent exponential stability of cellular neural networks with multi-proportional delays. Neural Process Lett 2013;38(3):321–46. [34] Zhou L, Chen X, Yang Y. Asymptotic stability of cellular neural networks with multi-proportional delays. Appl Math Comput 2014;229(1):457–66. [35] Zhou L. Global asymptotic stability of cellular neural networks with proportional delays. Nonlinear Dyn 2014;77(1):41–7. [36] Zhou L. Novel global exponential stability criteria for hybrid BAM neural networks with proportional delays. Neurocomputing 2015;161:99–106. [37] Zheng C, Li N, Cao J. Matrix measure based stability criteria for high-order networks with proportional delay. Neurocomputing 2015;149:1149–54. [38] Hiena LV, Son DT. Finite-time stability of a class of non-autonomous neural networks with heterogeneous proportional delays. Appl Math Comput 2015;14:14–23.