Dynamics in BAM Fuzzy Neural Networks with Delays and Reaction-Diffusion Terms 1

International Mathematical Forum, 3, 2008, no. 20, 979-1000 Dynamics in BAM Fuzzy Neural Networks with Delays and Reaction-Diffusion Terms 1 Zuoan Li Department of Mathematics Sichuan University of Science & Engineering Sichuan 643000, China lizuoan@suse.edu.cn Abstract In this paper, employing some analysis techniques and Lyapunov functional, the existence, uniqueness and global exponential stabilities of both the equilibrium point and the periodic solution are investigated for a class of bi-directional associative memory BAM fuzzy neural networks with delays and reaction-diffusion terms. We obtain two concise sufficient conditions ensuring the existence, uniqueness and global exponential stability of both the equilibrium point and the periodic solution. Moreover, an illustrate example is given to show the effectiveness of obtained results. Mathematics Subject Classification: 92B20, 34K20, 34K13 Keywords: Bi-directional associative memory; fuzzy neural networks; reaction-diffusion; delays; global exponential stability; periodic solution 1 Introduction The bi-directional associative memory BAM neural networks was first introduced by Kosto 1. It is important model with the ability of information memory and information association, which is crucial for application in pattern recognition, solving optimization problems and automatic control engineering 2-4. In such applications, the stability of networks plays an important role, it is of significance and necessary to investigate the stability. In both biological 1 This work was jointly supported by grant 2006A109 and 07ZA047 from the Scientific Research Fund of Sichuan Provincial Education Department.

980 Zuoan Li and man-made neural networks, the delays arise because of the processing of information 5. Time delays may lead to oscillation, divergence, or instability which may be harmful to a system 5, 6. Therefore, study of neural dynamics with consideration of the delayed problem becomes extremely important to manufacture high quality neural networks. Recently, BAM neural networks have been extensively studied both in theory and applications, for example, see 3-17 and references therein. On the other hand, the fuzzy cellular neural networks FCNN was introduced by Yang in 1996 18, it combines fuzzy logic with traditional cellular neural networks CNN. Studies have shown the potential of FCNN in image producing and pattern recognition. Many stability conditions have been given for FCNN 19-21. In 19, the authors have obtained some conditions for the existence and the global stability of the equilibrium point of FCNN without delay. In 20, Liu and Tang have considered FCNN with either constant delays or time-varying delays, several sufficient conditions have been obtained to ensure the existence and uniqueness of the equilibrium point and its global exponential stability. In 21, Yuan, Cao and Deng have given several criteria of dynamics for FCNN with time-varying delays. However, as pointed out in 16, 22, 23, diffusion effect cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic fields, so we must consider the activations vary in space as well as in time. Recently, Huang has considered the stability of FCNN with diffusion terms and time-varying delay, the model is expressed by partial differential equations 23. To the best of our knowledge, few authors have considered BAM fuzzy neural networks model with delays and reaction-diffusion terms. Moreover, studies on neural dynamical systems not only involve the discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillatory behavior, bifurcation, and chaos. In many applications, the properties of periodic oscillatory solutions and global exponential stability are of great interest. For example, the human brain has been in periodic oscillatory or chaos state, hence it is of prime importance to study periodic oscillation, global exponential stability and chaos phenomenon of neural networks. Motivated by the above discussions, the objective of this paper is to study the existence, uniqueness and global exponential stability of both the equilibrium point and the periodic solution for BAM fuzzy neural networks with delays and reaction-diffusion terms. The model is described by the following

Dynamics in BAM fuzzy neural networks 981 functional differential equation u i t,x = l u D i t,x ik a i u i t, x m a ji g j v j t, x m ã ji w j ti i t m α ji g j v j t τ ji,xds m α ji g j v j t τ ji,x m T ji w j t m H ji w j t, v j t,x = l D v j t,x jk b j v j t, x n b ij f i u i t, x n bij w i tj j t n β ij f i u i t σ ij,xds n β ij f i u i t σ ij,x n T ij w i t n H ij w i t, for i I:={1, 2,,n}, j J:={1, 2,,m}, t>0, where x =x 1,x 2,,x l, T R l, is a compact set with smooth boundary and mes > 0 in space R l ; u =u 1,u 2,,u n T R n,v =v 1,v 2,,v m T R m. u i t, x and v j t, x are the state of the ith neuron and the jth neuron at time t and in space x, respectively; f i and g j denote the signal functions of the ith neuron and the jth neuron at time t and in space x, respectively; w i t and w j t denote inputs of the ith neuron and the jth neuron at the time t, respectively; and I i t and J j t denote bias of the ith neuron and the jth neuron at the time t, respectively; a i > 0,b j > 0,a ji, ã ji,α ji, α ji,b ij, b ij,β ij, β ij are constants, a i and b j represent the rate with which the ith neuron and the jth neuron will reset their potential to the resting state in isolation when disconnected from the networks and external inputs, respectively; a ji,b ij and ã ji, b ij denote connection weights of feedback template and feedforward template, respectively; α ji,β ij and α ji, β ij denote connection weights of the delays fuzzy feedback MIN template and the delays fuzzy feedback MAX template, respectively; T ji, T ij and H ji, H ij are elements of fuzzy feedforward MIN template and fuzzy feedforward MAX template, respectively; and denote the fuzzy AND and fuzzy OR operation, respectively; smooth functions D ik = D ik t, x, u 0 and D jk = D jk t, x, v 0 correspond to the transmission diffusion operators along the ith neuron and the jth neuron, respectively. The boundary conditions and initial conditions are given by { ui ñ v j ñ := u i x 1, u i x 2,, u i x l T =0, i I, := v j x 1, v j x 2,, v j x l T =0, j J, 1 2 and { ui t, x =φ uit, x, τ t o, τ = max 1 i n,1 j m{τ ji}, v j t, x =φ vj t, x, σ t 0, σ = max 1 i n,1 j m {σ ij }, 3

982 Zuoan Li where φ ui t, x,φ vj t, x i I,j J are bounded and continuous on τ,0, σ, 0, respectively. Throughout this paper, we make the following assumptions: H1 The neurons activation functions f i and g j i I,j J are Lipschitzcontinuous, that is, there exist constants F i > 0 and G j > 0 such that f i ξ 1 f i ξ 2 F i ξ 1 ξ 2, g j ξ 1 g j ξ 2 G j ξ 1 ξ 2 for all ξ 1,ξ 2 R. H2 There exist constants λ i > 0,λ nj > 0i I,j J such that 2λ i a i λ i a ji α ji α ji Fi 2 λ nj b ij β ij β ij < 0, 2λ nj b j λ nj b ij β ij β ij G 2 j λ i a ji α ji α ji < 0 for i I,j J. To prove our main results, we need the following lemma: Lemma 1 19 Suppose u and u are two state of model 1, then we have n n α ij f j u j α ij f j u j fj α ij u j f j u, j n β ij f j u j n β ij f j u j fj β ij u j f j u. j 2 Global exponential stability of the equilibrium point In this section, we will discuss the existence, uniqueness and global exponential stability of the equilibrium point of BAM fuzzy neural networks with delays and diffusion terms, and give their proofs. Consider the case of model 1 as w i t = w i, w j t =w j, I i t =I i and J j t =J j i I,j J, i.e., u i t,x = l u D i t,x ik a i u i t, x m a ji g j v j t, x m ã ji w j I i m α ji g j v j t τ ji,x m α ji g j v j t τ ji,x m T ji w j m H ji w j, v j t,x = l D v j t,x jk b j v j t, x n b ij f i u i t, x n bij w i J j n β ij f i u i t σ ij,x n β ij f i u i t σ ij,x n T ij w i n H ij w i 1

Dynamics in BAM fuzzy neural networks 983 for i I,j J. We introduce a notation before giving the following definition. For ut, x = u 1 t, x,u 2 t, x,,u k t, x T R k, denote u i t, x 2 = u i t, x 2 dx 1 2, i =1, 2,,k. Definition 1 The equilibrium point u,v T of model 1 is said to be globally exponentially stable, if there exist constants ε>0 and M 1 such that u i t, x u i 2 2 v j t, x vj 2 2 Me ɛt φ u u φ v v for all t 0, where u 1 t, x,,u n t, x,v 1 t, x,,v m t, x T is any solution of model 1, u =u 1,,u n T, v =v1,,v m T, φ u =φ u1,φ u2,,φ un T, φ v = φ v1,φ v2,,φ vm T, and φ u u = sup φ ui t, x u i 2 2, φ v v = sup σ t 0 φ vj t, x vj 2 2. τ t 0 Theorem 1 Under assumptions H1 and H2, then there is exactly one equilibrium point of model 1. Proof.For the sake of simplification, let Ĩ i = m ã ji w j I i m T ji w j m H ji w j, i I, J j = n bij w i J j n T ij w j n H ij w i, j J, then model 1 is reduced to u i t,x = l u D i t,x ik m α ji g j v j t τ ji,x m a i u i t, x m a ji g j v j t, x α ji g j v j t τ ji,x Ĩi, v j t,x = l D v j t,x jk b j v j t, x n b ij f i u i t, x n β ij f i u i t σ ij,x n β ij f i u i t σ ij,x J j 4 for i I,j J. It is evident that the dynamical characteristics of model 1 is the same as of model 4. We denote

984 Zuoan Li hu 1,,u n,v 1,,v m =h 1,,h n, h 1,, h m T, where h i = a i u i m a ji g j v j m α ji g j v j m α ji g j v j Ĩi, i I, h j = b j v j n b ij f i u i n β ij f i u i n β ij f i u i J j, j J. Obviously, the equilibrium points of model 4 are the solutions of system of equations { hi =0, i I, 5 h j =0, j J. Define the following homotopic mapping: Hu 1,,u n,v 1,,v m = λhu 1,,u n,v 1,,v m 1 λu 1,,u n,v 1,,v m T, where λ 0, 1. Let H k k =1, 2,,nm denote the kth component of Hu 1,, u n,v 1,,v m, then from Lemma 1, we can get the following inequalities H i λa i u i λ m a ji α ji α ji g j v j g j 0 λ m a ji α ji α ji g j 0 λ Ĩi, H nj λb j v j λ n b ij β ij β 6 ij f i u i f i 0 λ n b ij β ij β ij f i 0 λ J j for i I,j J. In fact, for i I, we have H i 1 λ u i λa i u i λ λa i u i λ λa i u i λ λ a ji α ji α ji g j v j λ Ĩi a ji α ji α ji g j v j g j 0 g j 0 λ Ĩi a ji α ji α ji g j v j g j 0 a ji α ji α ji g j 0 λ Ĩi, also, for j J, we have H nj λb j v j λ λ b ij β ij β ij f i u i f i 0 b ij β ij β ij f i 0 λ J j.

Dynamics in BAM fuzzy neural networks 985 It follows from the inequality 2ab a 2 b 2 and H1 that λ i u i H i λ λ i a i u i 2 1 2 λ m i u i 2 a ji α ji α ji 1 2 λ i λ a ji α ji α ji G 2 j v j 2 λ i u i Ĩi a ji α ji α ji g j 0 7 and λ nj v j H nj λ λ nj b j v j 2 1 2 λ nj v j 2 b ij β ij β ij 1 2 λ nj λ b ij β ij β ij F 2 i u i 2 λ nj v j J i We obtain from 7 and 8 that where ρ = λ i u i H i min {ρ k}, 1 k nm b ij β ij β ij f i 0. 8 λ nj v j H nj λρ w 2 λμ n m w, 9 ρ i = λ i a i 1 2 λ i m a ji α ji α ji 1 2 F 2 i λ nj b ij β ij β ij, ρ nj = λ nj b j 1 2 λ nj n b ij β ij β ij 1 2 G2 j μ = max 1 k n {μ k}, μ i = λ i Ĩi m a ji α ji α ji g j 0, μ nj = λ nj J j n b ij β ij β ij f i 0, w = w 2 1 w2 n v 2 1 v2 m. λ i a ji α ji α ji,

986 Zuoan Li Define Γ= { w =u 1,,u n,v 1,,v m T w n mμ 1 then, it follows that for any w =u 1,,u n,v 1,,v m T Γ, we have w = nmμ1, hence ρ λ i u i H i ρ }, λ nj v j H nj λρ w 2 λμ n m w n mμ 12 = λρ ρ 2 λμ n mμ 1 n m ρ > 0, λ 0, 1. this means Hu 1,,u n,v 1,,v m 0, for any u 1,,u n,v 1,,v m T Γ, λ 0, 1. Also, as λ =0,Hu 1,,u n,v 1,,v m =u 1,,u n,v 1,,v m 0, for any u 1,,u n,v 1,,v m T Γ. Hence, we have Hu 1,,u n,v 1,,v m 0, for any u 1,,u n,v 1,,v m T Γ, λ 0, 1. From homotopy invariance theorem 24, we get degh, Γ, 0 = degh, Γ, 0 = 1, by topological degree theory, we know that 4 has at least one solution in Γ. That is, model 1 has at least an equilibrium point. If u 1,,u n,v 1,,v m T and u 1,,u n,v 1,,v m T are two equilibrium points of the model 4, by using of Lemma1, H1 and the inequality 2ab a 2 b 2,we easily obtain the following inequalities λ i a i u i u i 2 1 2 λ m i u i u i 2 a ji α ji α ji and 1 2 λ i a ji α ji α ji G 2 j v j vj 2, 10 λ nj b j v j vj 2 1 2 λ nj v j vj 2 b ij β ij β ij 1 2 λ nj b ij β ij β ij Fi 2 u i u i 2. 11 It follows from 10 and 11 that λ i a i 1 2 λ i a ji α ji α ji

Dynamics in BAM fuzzy neural networks 987 1 2 F 2 i 1 2 G2 j λ nj b ij β ij β ij u i u i 2 λ nj b j 1 2 λ nj b ij β ij β ij λ i a ji α ji α ji v j vj 2 0. This implies from H2 that u i = u i,v j = vj,i I,j J. Therefore, the system 4 has one unique equilibrium point. The proof is completed. Theorem 2 Under assumptions H1 and H2, the equilibrium point of model 1 is globally exponentially stable. Proof. Since H2 holds, we can choose a small ε>0 such that 2λ i ε a i λ i m a ji α ji α ji F 2 λ nj b ij β ij β ij e 2εσ < 0, 2λ nj ε b j λ nj b ij β ij β ij G 2 j n λ i a ji α ji α ji e 2ετ < 0 i 12 for i I,j J. Let u,v be the unique equilibrium point of the system 1, rewrite model 1 as u i u i = l u D i u i ik m m v j v j = l α ji g j v j t τ ji,x m a i u i u i m a ji g j v j g j v j α ji g j v j t τ ji,x m D v j vj jk α ji g j v j α ji g j v j, b j v j vj n b ij f i u i f i u i n β ij f i u i t σ ij,x n β ij f i u i n β ij f i u i t σ ij,x n β ij f i u i 13 for i I,j J. By using of Lemma 1 and hypotheses H1, we can obtain as

988 Zuoan Li following inequalities u i u i u i u i u i u i l u D i u i ik a i u i u i 2 u i u i m a ji G j v j vj u i u i m α ji α ji G j v j t τ ji,x vj, v j v j v j v j v j v j l D v j vj jk b j v j vj 2 v j vj b ij F i u i u i v j vj β ij β ij F i u i t σ ij,x u i. Now we construct the Lyapunov functional as follows V t = λ i u i u i 2 e 2εt α ji α ji G 2 j t v j s, x vj 2 e 2εsτji ds dx t τ ji λ nj v j vj 2 e 2ɛt t u i s, x u i 2 e 2εsσij ds dx. t σ ij β ij β ij F 2 i 14 Calculating the upper right Dini derivative D V t ofv t along the solution of model 1, and by using of some analysis techniques, we have D V t = λ i 2u i u i u i u i e 2εt 2ɛe 2ɛt u i u i 2 α ji α ji G 2 j v j t, x vj 2 e 2εtτ ji α ji α ji G 2 j v jt τ ji,x vj 2 e 2εt dx λ nj 2v j vj v j vj e 2ɛt 2εe 2εt v j vj 2 β ij β ij F i u i t, x u i 2 e 2εtσ ij β ij β ij Fi 2 u it σ ij,x u i 2 e 2εt dx.

Dynamics in BAM fuzzy neural networks 989 λ i 2e 2ɛt u i u i l u i u i D ik m 2e 2ɛt ɛ a i u i u i 2 2e 2ɛt u i u i a ji G j v j vj 2e 2ɛt u i u i α ji α ji G j v j t τ ji,x vj α ji α ji G 2 j v jt, x vj 2 e 2εtτ ji e 2εt m α ji α ji G 2 j v j t τ ji,x v j 2 dx λ nj 2e 2ɛt v j vj l 2e 2εt ε b j v j v j 2 2e 2εt v j v j v j vj Djk b ij F i u i u i 2e 2εt v j vj β ij β ij F i u i t σ ij,x u i e 2εt β ij β ij Fi 2 u i u i 2 e 2εσ ij e 2εt e 2ɛt β ij β ij Fi 2 u i t σ ij,x u i 2 dx λ i 2u i u i 2ɛ a i u i u i 2 a ji G 2 j v j vj 2 l u i u i D ik a ji u i u i 2 α ji α ji u i u i 2 α ji α ji G 2 j v j t τ ji,x vj 2 α ji α ji G 2 j v j vj 2 e 2ετ α ji α ji G 2 j v jt τ ji,x vj 2 dx

990 Zuoan Li e 2ɛt λ nj 2v j v j l 2ε b j v j v j 2 b ij F 2 i u i u i 2 v j vj Djk b ij v j vj 2 β ij β ij v j vj 2 β ij β ij Fi 2 u it σ ij,x u i 2 β ij β ij Fi 2 u i u i 2 e 2εσ β ij β ij Fi 2 u i t σ ij,x u i 2 dx = 2e 2ɛt 2e 2ɛt e 2ɛt F 2 i λ i u i u i l λ nj v j vj u i u i D ik dx l v j vj Djk dx 2λ m i ɛ a i λ i a ji α ji α ji λ nj b ij β ij β ij e 2εσ u i u i 2 dx e 2ɛt G 2 j 2λ nj ɛ b j λ nj b ij β ij β ij λ i a ji α ji α ji e 2ετ v j vj 2 dx. 15 From the boundary condition 2 and the proof in 16, we can get l λ i u i u u i u i i D ik dx and = l λ i λ nj v j vj D ik ui u i 2dx 16 l Djk v j v j dx

Dynamics in BAM fuzzy neural networks 991 = l λ nj D ik vj v j 2dx. 17 Since D ik 0, D jk 0i I,j J,k =1, 2,,l, from 12, 15, 16 and 17, we obtain that D V t 0 for t>0. So V t V 0 for t 0. Moreover, we have V 0 = 0 λ i u i u i 2 α ji α ji G 2 j v j s, x vj 2 e 2εsτji ds dx τ ji λ nj v j vj 2 0 β ij β ij F 2 i u i s, x u i 2 e 2εsσij ds dx. σ ij max {λ i} 1 i n λ nj β ij β ij φ ui s, x u i 2 e 2εσ max {F i 2 } 1 i n 0 φ ui s, x u i 2 e 2εs ds dx σ ij max {λ nj} φ vj s, x vj 2 e 2ετ max 1 j m 1 j m {G2 j } λ m i α ji α ji 0 φ vj s, x vj 2 e 2εs ds dx τ ji max {λ i} φ ui s, x u i 1 i n 2 e 2εσ max {F i 2 } 1 i n λ nj max 1 i n { β ij β ij } 0 σ φ ui s, x u i 2 e 2εs ds dx max {λ nj} φ vj s, x vj 1 j m 2 e 2ετ max 1 j m {G2 j } λ i max { α ji α ji } φ vj s, x vj 2 e 2εs ds dx 1 j m τ max {λ i} σe 2εσ max {F i 2 } 1 i n 1 i n λ nj max { β ij β ij } φ u u i 1 i n max {λ nj} τe 2ετ max 1 j m 1 j m {G2 j } 0

992 Zuoan Li and V t λ i e 2εt max { α ji α ji } φ v vj, 1 j m e 2εt min 1 i nm {λ i} λ i u i u i 2 dx e 2ɛt m λ nj v j v j 2 dx. m u i t, x u i 2 2 v j t, x vj 2 2 for t>0. So m e 2εt min {λ i} u i t, x u i 1 i nm 2 2 v j t, x vj 2 2 Let max {λ i} σe 2εσ max {F 2 1 i n 1 i n max {λ nj} τe 2ετ 1 j m i } max 1 j m {G2 j } M 1 = max {λ i} σe 2εσ max {F i 2 } 1 i n 1 i n λ nj max { β ij β ij } φ u u i 1 i n λ i max { α ji α ji } φ v vj. 1 j m λ nj max { β ij β ij }, 1 i n M 2 = max {λ nj} τe 2ετ max 1 j m 1 j m {G2 j } λ i max { α ji α ji }, 1 j m M = max{m 1,M 2 } min 1 i nm {λ i }, then M 1, and we easily get m u i t, x u i 2 2 v j t, x vj 2 2 Me 2εt φ u u i φ u u i for all t 0, it implies that the equilibrium point u,v T is globally exponentially stable. The proof is completed. 3 Periodic oscillatory solution In this section, we will discuss the periodic oscillatory solutions of model 1. Let w i : R R, w j : R R, I i : R R and J j : R R be continuously periodic functions with period ω, i.e., w i t ω = w i t, w j t ω =w j t, I i t ω =I i t, J j t ω =J j t for i I,j J.

Dynamics in BAM fuzzy neural networks 993 Theorem 3 Under assumptions H1 and H2, then there exists exactly one ω- periodic solution of model 1, and all other solutions of model 1 converge exponentially to it as t. Proof. Let φ = φu φ v =φ u1,,φ un,φ v1,,φ vm T, denote C = { τ,0 R l φ φ : σ, 0 R l R nm }. For φ C, we define φ = φu φ v = φ u φ v, τ,0 R l then C is a Banach space of continuous functions which maps σ, 0 R l into R nm with the topology of uniform convergence. φu ψu For any, C, we denote the solutions of model 1 through φ v ψ v 0 φu 0 ψu, and, as 0 0 φ v ψ v ut, φ u,x=u 1 t, φ u,x,u 2 t, φ u,x,,u n t, φ u,x T, vt, φ v,x=v 1 t, φ v,x,v 2 t, φ v,x,,v m t, φ v,x T and ut, ψ u,x=u 1 t, ψ u,x,u 2 t, ψ u,x,,u n t, ψ u,x T, vt, ψ v,x=v 1 t, ψ v,x,v 2 t, ψ v,x,,v m t, ψ v,x T, respectively. Define u t φ u,x=ut θ, φ u,x, θ τ,0,t 0, v t φ v,x=vt θ, φ v,x, θ σ, 0,t 0,

994 Zuoan Li ut φ then u,x v t φ v,x C for t 0. Also, from model 1, we have u i t,φ u,x u i t,ψ u,x = l u D i t,φ u,x u i t,ψ u,x ik a i u i t, φ u,x u i t, ψ u,x m a ji g j v j t, φ v,x g j v j t, ψ v,x m α ji g j v j t τ ji,φ v,x m α ji g j v j t τ ji,ψ v,x m α ji g j v j t τ ji,φ v,x m α ji g j v j t τ ji,ψ v,x, v j t,φ v,x v j t,ψ v,x = l D v j t,φ v,x v j t,ψ v,x jk b j v j t, φ v,x v j t, ψ v,x n b ij f i u i t, φ u,x f i u i t, ψ u,x n β ij f i u i t σ ij,φ u,x n β ij f i u i t σ ij,ψ u,x n β ij f i u i t σ ij,φ u,x n β ij f i u i t σ ij,ψ u,x 18 for i I,j J. By using of Lemma 1 and assumption H1, we can obtain as following inequalities u i t, φ u,x u i t, ψ u,x u it,φ u,x u i t,ψ u,x u i t, φ u,x u i t, ψ u,x l u D i t,φ u,x u i t,ψ u,x ik a i u i t, φ u,x u i t, ψ u,x 2 u i t, φ u,x u i t, ψ u,x m a ji G j v j t, φ v,x v j t, ψ v,x u i t, φ u,x u i t, ψ u,x m α ji α ji G j v j t τ ji,φ v,x v j t τ ji,ψ v,x, v j t, φ v,x v j t, ψ v,x v j t,φ v,x v j t,ψ v,x v j t, φ v,x v j t, ψ v,x l D v j t,φ v,x v j t,ψ v,x jk b j v j t, φ v,x v j t, ψ v,x 2 v j t, φ v,x v j t, ψ v,x b ij F i u i t, φ u,x u i t, ψ u,x v j t, φ v,x v j t, ψ v,x β ij β ij F i u i t σ ij,φ u,x u i t σ ij,ψ u,x 19

Dynamics in BAM fuzzy neural networks 995 for i I,j J and t 0. Since H2 holds, we can choose a small positive number ε>0 such that 2λ i ε a i λ i a ji α ji α ji Fi 2 λ nj b ij β ij β ij e 2εσ < 0, 2λ nj ε b j λ nj b ij β ij β 20 ij λ i a ji α ji α ji e 2ετ < 0 G 2 j for i I,j J. We consider the another Lyapunov functional V t = λ i u i t, φ u,x u i t, ψ u,x 2 e 2εt dx G 2 j λ nj v j t, φ v,x v j t, ψ v,x 2 e 2ɛt dx t m λ i α ji α ji v j s, φ v,x v j s, ψ v,x 2 e 2εsτji ds dx t τ ji λ nj β ij β ij t Fi 2 u i s, φ u,x u i s, ψ u,x 2 e 2εsσij ds dx. t σ ij By a minor modification of the proof of Theorem 2, we can easily derive u i t, φ u,x u i t, ψ u,x 2 2 v j t, φ v,x v j t, ψ v,x 2 2 Me 2εt φ u ψ u φ v ψ v for all t 0, where M 1 is a constant. Hence we have u i t, φ u,x u i t, ψ u,x 2 2 Me 2εt φ u ψ u φ v ψ v and v j t, φ v,x v j t, ψ v,x 2 2 Me 2εt φ u ψ u φ v ψ v for all t 0. We can choose a positive integer N such that Me 2εNω τ 1 6, Me 2εNω σ 1 6.

996 Zuoan Li Now, we define a Poincare mapping C C by φu uω φ P = u,x v ω φ v,x φ v, then Let t = Nω, then P N φu φ v P N φu φ v = P N ψu ψ v unω φ u,x v Nω φ v,x 1 3 φu φ v. ψu ψ v. This implies that P N is a contraction mapping, hence there exist a unique fixed φ point u φ C such that v P N φ u φ v = φ u φ v, since then P φ u φ v P N P φ u φ v = P P N φ u φ v C is a fixed point of P N, and so = P φ u φ v φ P u φ φ = u uω φ v φ u, i.e., φ v v ω φ = u v φ. v ut, φ Let u,x 0 φ vt, φ v,x be the solution of model 1 through u 0 φ v ut ω, φ u,x vt ω, φ v,x is also a solution of model 1. Obviously, we have utω φ u,x v tω φ v,x = ut u ω φ u,x v t v ω φ v,x =, ut φ u,x v t φ v,x, then for all t 0. Hence ut ω, φ u,x vt ω, φ v,x = ut, φ u,x vt, φ v,x ut, φ This shows that u,x vt, φ v,x is exactly one ω-periodic solution of model 1 and all other solutions of model 1 converge exponentially to it as t. The proof is completed..

Dynamics in BAM fuzzy neural networks 997 4 An illustrate example Consider the following BAM fuzzy neural networks with delays and diffusion terms: u i t,x v j t,x = x D i u it,x x 2 2 = a i u i t, x 2 a ji g j v j t, x 2 ã ji w j ti i t α ji g j v j t τ ji,xds 2 T ji w j t 2 H ji w j t, α ji g j v j t τ ji,x x D v j t,x j x b j v j t, x 2 b ij f i u i t, x 2 bij w i tj j t 2 β ij f i u i t σ ij,xds 2 β ij f i u i t σ ij,x 2 T ij w i t 2 H ij w i t, for i =1, 2, j =1, 2, t>0, where 21 D 1 = t 2 u 2 1, D 2 = t 4 u 6 2, D1 = t 4 v 2 1, D2 = t 2 v 8 2, a 1 = a 2 =1, a 11 =0.25, a 21 = 0.25, a 12 =0.25, a 22 =0.25, α 11 =0.125, α 21 = 0.125, α 12 =0.125, α 22 =0.125, α 11 =0.1, α 21 =0.1, α 12 = 0.1, α 22 =0.1, b 1 = b 2 =2, b 11 =0.2, b 21 =0.2, b 12 = 0.2, b 22 =0.2, β 11 =0.1, β 21 = 0.1, β 12 = 0.1, β 22 =0.1, β11 =0.125, β21 = 0.125, β12 =0.125, β22 =0.125. Let f i y = 1 1e,i =1, 2; g y j y = y1 y 1 2,j =1, 2, then we have F 1 = F 2 = 1,G 1 = G 2 = 1, and let I 1 = I 2 = J 1 = J 2 =2, w 1 = w 2 = w 1 = w 2 =1, 0 τ ji π 2,i =1, 2,j =1, 2, 0 σ ij π 2,i =1, 2,j =1, 2, the matrices T = T ji, T = T ij,h =H ji and H = H ij are the identity matrix.

998 Zuoan Li Take constants λ 1 = λ 2 = λ 3 = λ 4 = 1, we have 2λ 1 a 1 λ 1 F1 2 2 a j1 α j1 α j1 2 λ 2j b 1j β 1j β 1j 0.2 < 0, 2λ 2 a 2 λ 2 F2 2 2 a j2 α j2 α j2 2 λ 2j b 2j β 2j β 2j 0.2 < 0, 2λ 3 b 1 λ 3 G 2 1 2 b i1 β i1 β i1 2 λ i a 1i α 1i α 1i 0.2 < 0, 2λ 4 b 2 λ 4 G 2 2 2 b i2 β i2 β i2 2 λ i a 2i α 2i α 2i 0.2 < 0. It follows that the assumption H2 is satisfied. From Theorem 1 and Theorem 2, the system 21 has exactly one equilibrium point, and the equilibrium point is globally exponentially stable. Furthermore, as the inputs w i t, w j t,i i t and J j t are periodic functions with common period, for example, w i t = cos t, i =1, 2, w j t = sin t, I i t = 3 sin t and J j t = 2 cos t, here, tkae λ 1 = λ 2 = λ 3 = λ 4 = 1, by Theorem 3, we conclude that there exists exactly one 2π-periodic solution of the system 21, and all other solutions of 21 converge exponentially to it as t. 5 Conclusions We have dealt with the problem of global exponential stability analysis and the existence of both the equilibrium point and the periodic solution for model 1. The general sufficient conditions have been obtained to ensure the existence, uniqueness and global exponential stability of both the equilibrium point and the periodic solution for BAM fuzzy neural networks with delays and reaction-diffusion terms. In particular, an illustrate example is given to show the effectiveness of obtained results. In addition, the sufficient conditions what we obtained are delay-independent, and are easily verified. This has practical benefits, since easily verifiable conditions for the global exponential stability are important in the design and applications of neural networks.

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