Global existence of periodic solutions of BAM neural networks with variable coefficients

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1 Physics Letters A ) Global existence of periodic solutions of BAM neural networks with variable coefficients Shangjiang Guo a,, Lihong Huang a, Binxiang Dai a, Zhongzhi Zhang b a College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 4182, PR China b Department of Mathematics, Hunan City University, Yiyang, Hunan 413, PR China Received 2 December 22; received in revised form 16 August 23; accepted 18 August 23 Communicated by A.P. Fordy Abstract In this Letter, we study BAM bidirectional associative memory) networks with variable coefficients. By some spectral theorems and a continuation theorem based on coincidence degree, we not only obtain some new sufficient conditions ensuring the existence, uniqueness, and global exponential stability of the periodic solution but also estimate the exponentially convergent rate. Our results are less restrictive than previously known criteria and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity. Moreover, these conclusions are presented in terms of system parameters and can be easily verified for the globally Lipschitz and the spectral radius being less than 1. Therefore, our results should be useful in the design and applications of periodic oscillatory neural circuits for neural networks with delays. 23 Elsevier B.V. All rights reserved. PACS: Sn; e Keywords: Periodic solution; BAM neural networks; Coincidence degree 1. Introduction Theoretical and experimental studies have indicated that a mammalian brain may be exploiting dynamical attractors for its storage of associative memories rather than static equilibrium-type) attractors. Dynamic attractors in the form of limit cycles, strange attractors and other transient characteristics have been used to represent encoded temporal patterns as associative memories. In addition, the periodic nature of neural impulses is of fundamental significance in the control of regular dynamical functions. However, neural networks are complex and large-scale This work was partially supported by the Science Foundation of Hunan University, by the National Natural Science Foundation of PR China Grant No ), by the Foundation for University Excellent Teacher by the Ministry of Education, and by the Key Project of the Ministry of Education No. [2278). * Corresponding author. address: shangjguo@etang.com S. Guo) /$ see front matter 23 Elsevier B.V. All rights reserved. doi:1.116/j.physleta 转载

2 98 S. Guo et al. / Physics Letters A ) nonlinear dynamical systems. For simplicity, many researchers have studied the existence of bifurcating periodic solutions in a two-neuron system see, for example, [1 6). To the best of our knowledge, few authors have considered periodic oscillatory solutions for large-scale networks with delays for more details about delayed neural networks, we refer to [11). On the other hand, the existing literature on artificial neural networks is predominantly concerned with autonomous systems containing temporal uniform networks parameters and input stimuli. Literature dealing with time dependent stimuli or networks parameters appears to be scarce. Such studies, however, are very important to understanding the dynamics of neural networks in time-varying environments. In this Letter, we study a class of neural networks with periodic variable coefficients of the form: ẋ i t) = a i t)x i t) + ẏ j t) = b j t)y j t) + a ij t)f j t,y j t τij t) )) + I i t), i = 1, 2,...,n, b ji t)g i t,x i t σji t) )) + J j t), j = 1, 2,...,p, 1.1) where ẋ i t) = dx i t)/dt, x i t) and I i t) represent the activation and external input of the ith neuron in the I -layer at time t, respectively; y j t) and J j t) represent the activation and external input of the jth neuron in the J -layer, respectively; the time delays τ ij t) and σ ji t) correspond to the finite speed of the axonal transmission of signal, for example, σ ji t) corresponds to the time the ith neuron in the I -layer emits a signal and the moment this signal become available for the jth neuron in the J -layer of 1.1); a ij t) and b ji t) represent the strengths of synaptical connections. Throughout this Letter, we always assume that a i t), b j t), a ij t), b ji t), τ ij t), σ ji t), I i t) and J j t) are continuously periodic functions defined on t [, ) with common period >. Moreover, a i t), b j t), τ ij t) and σ ji t) are positive everywhere, f j t, x) and g j t, x) are continuous and -periodic with respect to t. Let τ = maxτ ij t); 1 i n, 1 j p, t [, )}, σ = maxσ ji t); 1 i n, 1 j p,t [, )}. Then we let K =[ σ, [ τ, and use CK) =Φ : K R n+p is continuous} with the supernorm as the Banach space for system 1.1) and we will always tacitly use the identification CK) = C [ σ,; R n) C [ τ,; R p). For each given initial value Φ = ϕ, ψ) T CK) with ϕ C[ σ,; R n ) and ψ C[ τ,; R p ), one can solve system 1.1) by method of steps to obtain a unique pair of continuous maps x : [ σ, ) R n and y : [ τ, ) R p such that x, y) T :, ) R n+p is continuously differentiable and satisfies 1.1) for t>, x [ σ, = ϕ and y [ τ, = ψ. Recently, by constructing a suitable Lyapunov functional, Cao and Wang [7 studied the existence and uniqueness of periodic oscillatory solutions of system 1.1) with constant parameters. However, finding Lyapunov functions is still a difficult task. In this Letter, by means of some spectral theorems, inequality analysis and a continuation theorem based on coincidence degree, we not only obtain some new sufficient conditions ensuring the existence, uniqueness, and global exponential stability of the periodic solution but also estimate the exponentially convergent rate. These conclusions are presented in terms of system parameters, and should be useful in the design and applications of periodic oscillatory neural circuits for neural networks with delays. Our results impose weaker constraint on the system 1.1) and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity. We not only unify and improve some previous results, but also give some new criteria expressed in terms of matrix norms.

3 S. Guo et al. / Physics Letters A ) Main results and proofs For convenience, in this Letter we use the following notations: f = 1 ft)dt, [ft) + = max t [, ft) }, [ft)l = min t [, ft) }, where f is a continuous periodic -periodic function. For matrix A = a ij ) n n,letρa) denote the spectral radius of A. A matrix or a vector A means that all the entries of A are greater than or equal to zero, similarly define A>. Our main result on the global existence of positive periodic solution of 1.1) is the following theorem. Theorem 2.1. Assume that a i t) > and b j t) > for t and i = 1, 2,...,nand j = 1, 2,...,p. Moreover, H1) there exist non-negative constants p j, q i, α j and β i such that f j t, u) p j u +α j, g i t, u) q i u +β i for any t,u R, i = 1, 2,...,n, j = 1, 2,...,p; H2) ρm)< 1,whereM = m ij ) n+p) n+p) and, if 1 i, j n or n + 1 i, j n + p, m ij = [ a i,j n t) p j n /a i t) +, if 1 i n and n + 1 j n + p, [ b i n,j t) q j /b i n t) +, if n + 1 i n + p and 1 j n. 2.1) Then 1.1) has at least one w-periodic solution. Proof. In order to use the continuation theorem for 1.1), we denote by Z respectively, X) asthesetof all continuously respectively, differentiably) -periodic functions ut) = x 1 t),..., x n t), y 1 t),..., y p t)) T defined on R and denote u = max [xi t) +, [y j t) +}, u 1 = max } u, u. 1 i n,1 j p Then X and Z are Banach spaces when they are endowed with the norms 1 and, respectively. For u X and z Z, set and Lu)t) = ut), P u = 1 w Nu) i t) = a i t)x i t) + ut) dt, Qz = 1 w zt) dt, a ij t)f j t,y j t τij t) )) + I i t) for i = 1, 2,...,n Nu) n+j t) = b j t)y j t) + b ji t)g i t,x i t σji t) )) + J j t) for j = 1, 2,...,p. It is not difficult to show that Ker L = R n+p,andthatiml =z Z: zt) dt = } is closed in Z and that dim Ker L = n + p = codimim L, andthatp,q are continuous projectors such that Im P = Ker L and Ker Q = Im L = ImI Q).

4 1 S. Guo et al. / Physics Letters A ) It follows that L is a Fredholm mapping of index zero. Furthermore, the generalized inverse of L) K P :ImL Ker P Dom L reads K P u) i t) = t u is) ds 1 t u is) ds dt for u = ut) Z. Thus, it is easy to see that QN and K p I Q)N are continuous. An application of the Arzela Ascoli theorem to K p I Q)N results in the fact that K p I Q)N Ω) is compact for any open bounded Ω X.Moreover,QN Ω) is clearly bounded. Thus, N is L-compact on Ω with any open bounded set Ω X. Now we reach the position to search for an appropriate open bounded subset Ω for the application of the continuation theorem. Corresponding to the operator equation Lu = λnu, λ, 1),wehave ẋ i t) = λa i t)x i t) + λ a ij t)f j t,y j t τij t) )) + λi i t), i = 1, 2,...,n, 2.2) ẏ j t) = λb j t)y j t) + λ b ji t)g i t,x i t σji t) )) + λj j t), j = 1, 2,...,p. Assume that u = ut) X is a solution of 2.2) for a certain λ, 1). Thenx i t) and y j t), as the components of ut), are all continuously differentiable. Thus, there exist t i,t j [, such that x it i ) =[x i t) + and y j t j ) =[y j t) +. Hence, ẋ i t i ) =ẏ j t j ) =. This implies that a i t i )x i t i ) = a ij t i )f j t i,y j ti τ ij t i ) )) + I i t i ), b j t j )y jt j ) = b ji t j )g i t j,x i t j σ ji t j ))) + J j t j ). It follows from the first equation of 2.3) that for i = 1, 2,...,n,wehave a ij t i ) x i t i ) = a i t i ) f j t i,y j ti τ ij t i ) )) + I it i ) a i t i ) m i,n+j y j t j ) +D i, where D i = p [a ij t)α j /a i t) + +[I i t)/a i t) +. Similarly, we have y j t j ) m n+j,i x i t i ) +D j+n, j = 1, 2,...,p, where D n+j = n [b ji t)β i /b j t) + +[J j t)/b j t) +.InviewofρM) < 1, we have E M) T and h = E M) 1 D see [9 for more details), where D = D 1,D 2,...,D n+p ) T. It follows from 2.4) and 2.5) that [x i t) + h i and [y j t) + h n+j, for i = 1, 2,...,n,j = 1, 2,...,p, where h i is the ith component of vector h. Clearly, h i, i = 1, 2,...,n+ p, are independent of λ. Moreover, it follows from the first equation of 2.2) that [ [ẋ i t) + max λ a i t) x i t) +λ a ij t) f j t,y j t τij t) )) + λ Ii t) t [, [ max a it) h i + m i,n+j h n+j + D i = 2h i [a i t) +. t [, 2.3) 2.4) 2.5) 2.6)

5 S. Guo et al. / Physics Letters A ) Namely, we have [ẋ i t) + 2h i [a i t) +, i = 1, 2,...,n. 2.7) Similarly, we have [ẏ j t) + 2h n+j [b j t) +, j = 1, 2,...,p. 2.8) Let A = max 1 i n,1 j p h i 1 + 2[a i t) + ), h n+j 1 + 2[b j t) + )}. Then there exists some d>1 such that dh i >Afor all i = 1, 2,...,n+ p. Wetake Ω =u X; dh< ut) < dh for all t}. If u = x 1,x 2,...,x n,y 1,y 2,...,y p ) T Ω Ker L = Ω R n+p,thenu is a constant vector in R n+p with x i =dh i and y j =dh n+j for i = 1, 2,...,n, j = 1, 2,...,p. It follows that and QNu) i = 1 QNu) n+j = 1 We claim that [ a i t)x i + [ a ij t)f j t, y j ) + I i t) dt b j t)y j + b ji t)g i t, x i ) + J j t) dt for i = 1, 2,...,n, j = 1, 2,...,p. QNu) i > fori = 1, 2,...,n+ p. 2.9) 1 By a way of contrary. Suppose that there exists some i 1, 2,...,n} such that QNu) i =, i.e., [ a it)x i + p a ij t)f j t, y j ) + I i t) dt =. Then there exists some t [,, such that Thus, a i t )x i + dh i = x i a ij t )f j t,y j ) + I i t ) =. a ij t ) a i t ) f j t,y j ) + I it ) a i t ) m i,n+j dh n+j + D i. In view of d>1anddh = dmh + D) > Mdh + D, wehavedh i > n+p m ij dh j + D i for i = 1, 2,...,n. It follows from the above inequality that dh i <dh i, which is a contradiction. Thus, QNu) i > forall i 1, 2,...,n}. Similarly, QNu) i > foralli n + 1,n+ 2,...,n+ p}. Therefore, 2.9) holds, and hence QNu, for u Ω Ker L. Define ψ : Ω Ker L [, 1 X/ ImL = X c by ψu,µ) = µ diag ā 1,..., ā n, b 1,..., b p )u + 1 µ)qnu for all u = x 1,...,x n,y 1,...,y p ) T Ω Ker L = Ω R n+p and µ [, 1.

6 12 S. Guo et al. / Physics Letters A ) When u Ω Ker L and µ [, 1, u = x 1,...,x n,y 1,...,y p ) T is a constant vector in R n+p with x i =dh i i = 1, 2,...,n)and y j =dh n+j j = 1, 2,...,p). Thus, ψu,µ) = We claim that ψu,µ) >. max 1 i n,1 j p [ 1 ā ix i + 1 µ) [ 1 b j y j + 1 µ) By a way of contrary. Suppose that ψu,µ) =, then we have ā i x i + b j y j + 1 µ) 1 µ) a ij t)f j t, y j ) dt + 1 µ)ī i =, b ji t)g i t, x i ) dt + 1 µ) J j = a ij t)f j t, y j ) dt + I i, b ji t)g i t, x i ) dt + J j }. 2.1) 2.11) for all j = 1, 2,...,n,j = 1, 2,...,p. It follows from the first equality that there exists some t [, such that Thus, a i t )x i + 1 µ) dh i = x i 1 µ) a ij t )f j t,y j ) + 1 µ)i i t ) =. a ij t ) a i t ) f j t,y j ) +1 µ) I it ) a i t ) m i,n+j dh n+j + D i, which contradicts that dh i > n+p m ij dh j + D i for i = 1, 2,...,n. Thus, 2.1) holds. Therefore, ψu,µ), for any u Ω Ker L. Using the property of topological degree and taking J to be the identity mapping I :ImQ Ker L,wehave degj QN, Ω Ker L,) = deg ψ, ), Ω Ker L, ) = deg ψ, 1), Ω Ker L, ) = deg diag ā 1,..., ā n, b 1,..., b p ), Ω Ker L, ) = 1. Therefore, according to the continuation theorem of Gaines and Mawhin [8, system 1.1) has at least one - periodic solution. The proof is complete. Remark 2.1. In particular, if the activation functions in system 1.1) are all bounded on R CK),i.e.,thereexist positive constants α j and β i such that f j t, u) <α j and g i t, u) <β i for all t R and u CK), i = 1, 2,...,n, j = 1, 2,...,p, then system must have at least one -periodic solution. For example, in CNN network model, the activation function takes the form fu)=.5 u + 1 u 1 ), which is bounded.

7 S. Guo et al. / Physics Letters A ) Remark 2.2. It should be noted that the assumptions of the activation functions in Theorem 2.1 are very general, assuming neither differentiability nor strict monotonicity. Of course, the computation of ρm) could be expensive for a large network. Recall that for a given matrix M, its spectral radius ρm) is equal to the minimum of its all matrix norms of M, i.e., for any matrix norm, ρm) M. Therefore, we have the following two corollaries. Especially, Corollary 2.1 puts the constraints directly on the elements of the connection matrix and decay rate, and can be used more conveniently. Corollary 2.1. Under the assumption H1), if there exist positive constants d i i = 1, 2,...,n+ p) such that one of the following inequalities holds: i) n [a ij t)p j d n+j /a i t)d i + < 1 and p [b jit)q i d i /b j t)d n+j + < 1 for i = 1, 2,...,n and j = 1, 2,...,p; ii) d i a i t) + p a ij t) p j d n+j < and d n+j b j t) + n b ji t) q i d i < for t [, ), i = 1, 2,...,n, j = 1, 2,...,p; iii) d i [a i t) l + p [b jit) + q i d n+j < and d n+j [b j t) l + n [a ij t) + p j d i < for i = 1, 2,...,nand j = 1, 2,...,p; iv) n+p n+p d 1 i d j m ij ) 2 < 1,whereM = m ij ) n+p) n+p) is defined as in hypothesis H2), then system 1.1) has at least one -periodic solution. Proof. For any matrix norm and any nonsingular matrix S, M S = S 1 MS also defines a matrix norm. Let D be positive diagonal matrix D = diagd 1,d 2,...,d n+p }. Then, conditions i) iv) in Corollary 2.1 correspond to the row norms, column norms, and Frobenius norm or Euclidean norm) of matrix DMD 1, respectively. Corollary 2.2. In addition to the assumption H1), suppose that ρa T A) < 1 and ρb T B) < 1 where matrices A = [a ij t)/a i t) + ) n p and B = [b ij t)/b i t) + ) p n.thensystem1.1) has at least one -periodic solution. If we further assume that f j j = 1, 2,...,p)and g i i = 1, 2,...,n)are globally Lipschitz with respect to the second variables, then we shall obtain the global exponential stability of the -periodic solution. Namely, Theorem 2.2. Assume that f j t, ) and g i t, ) are bounded with respect to t and that there exist nonnegative constants p j and q i such that f j t, x) f j t, y) p j x y and g i t, x) g i t, y) q i x y for all t,x,y R, i = 1, 2,...,n, j = 1, 2,...,p.Furthermore,ρM) < 1, where matrix M is defined as in hypothesis H2).Thensystem1.1) has a unique -periodic solution and all other solutions converge exponentially to it as n. Moreover, the globally exponentially convergent rate λ can be estimated by the following inequality. a e λτ max a λ, b e λσ } ρm)< 1, b λ 2.12) where a = min 1 i n [a i t) l } and b = min 1 j p [b j t) + }. Proof. It follows from the assumption about f j and g i that f j t, x) p j x + f j t, ) and g i t, x) q i x + g i t, ). Hence, hypothesis H1) holds with α j =[f j t, ) + j = 1, 2,...,p)andβ i =[g i t, ) + i = 1, 2,...,n). Thus, from Theorem 2.1, system 1.1) has at least one -periodic solution denoted by x t), y t)) T. Let xt), yt)) T be an arbitrary solution of system 1.1) and define ut) = xt) x t) and vt) = yt) y t),

8 14 S. Guo et al. / Physics Letters A ) then we have u i t) = a i t)u i t) + v j t) = b j t)v j t) + a ij t)f j t,v j t τji t) )), b ji t)g i t,u i t σji t) )) 2.13) for i = 1, 2,...,nand j = 1, 2,...,p,whereF j t, v j t τ ij t))) = f j t, y j t τ ij t))) f j t, y j t τ ij t))) and G i t, u i t σ ji t))) = g i t, x i t σ ji t))) g i t, x i t σ jit))). Thus, it is sufficient to prove that,,...,) T is globally exponentially stable for system 2.13). Since ρm) < 1, E M is a M-matrix see [1), where E denotes an identity matrix of size n + p. Therefore, there exists a diagonal matrix Q = diagd 1,d 2,...,d n+p ) with positive diagonal elements such that the product matrix E M)Q is strictly diagonally dominant with positive diagonal entries, i.e., n+p d 1 i d j m ij < 1, i = 1, 2,...,n+ p. Thus, there must exist a sufficiently small constant λ, mina,b }) such that [a i t) l e λτ n+p max 1 i n,1 j p [a i t) l di 1 d k m ik, [b } jt) l e λσ n+p λ [b j t) l dn+j 1 λ d km n+j.k < 1. k=1 k=1 2.14) Let ut), vt)) T be a solution of system 2.13) with any initial value Φ = ϕ, ψ) T CK). SetU i t) = di 1 u i t) e λt if t andu i t) = di 1 u i t) otherwise, V j t) = dn+j 1 v j t) e λt if t andu i t) = dn+j 1 v j t) otherwise. Then, it follows form 2.13) that for t andi = 1, 2,...,n,j = 1, 2,...,p, U i t) exp + V j t) exp + [ t [ t ai s) λ ) ds t di 1 d n+j e λτ bj s) λ ) ds t dn+j 1 d ie λσ U i ) exp exp [ s ai θ) λ ) dθ a ij s) p j V j s) τ ds, t V i ) [ s t bj θ) λ ) dθ b ji s) q i U i s) σ ds, where U i t) σ = sup s [t σ,t U i s) } and V j t) τ = sup s [t τ,t V j s) }.Set m = max 1 i n,1 j p d 1 i ϕ i σ,d 1 n+j ψ j τ }. We will prove that, for any sufficiently small constant ε>, 2.15) U i t) < m + ε, V j t) < m + ε, 2.16)

9 S. Guo et al. / Physics Letters A ) for all t andi 1, 2,...,n} and j 1, 2,...,p}. By a way of contrary. Without loss of generality, we assume that there must exist some andk 1, 2,...,n} such that U k ) = m + ε, U i t) m + ε, V j t) m + ε, for t [, ), i 1, 2,...,n} and j 1, 2,...,p}. It follows from 2.15) that m + ε = U k ) [ exp ak s) λ ) ds U k ) + m + ε)exp + m + ε) m + ε)exp dk 1 d n+j e λτ [ [ exp [ s ak s) λ ) ds dk 1 d n+j e λτ ak s) λ ) ds n+p + m + ε) dk 1 d j e λτ m kj m + ε)exp [ ak θ) λ ) dθ exp ak s) λ ) ds exp [ s [ s ak θ) λ ) dθ a kj s) p j V j s) τ ds ak θ) λ ) dθ n+p + m + ε) dk 1 [a k t) l e λτ d j m kj [a k t) l 1 exp λ < m + ε)exp + m + ε) = m + ε, [ ak s) λ ) ds 1 exp [ ak s) λ ) ds } [ a kj s) p j ds a k s) ds ak s) λ ) ds which is a contradiction. Therefore, 2.16) holds. Let ε. Then we have } U i t) m, V j t) m,

10 16 S. Guo et al. / Physics Letters A ) for all t andi 1, 2,...,n} and j 1, 2,...,p}. Namely, U i t) d 1 r ϕ r σ, V j t) d 1 r ϕ r σ, for all t andi 1, 2,...,n} and j 1, 2,...,p}. This implies that there exists M >1 such that Ut),Vt) ) T M Φ for t. Thus, there exist some constant m 1 such that ut), vt) ) T m Φ e λt for t, which implies the equilibrium,,...,) T of system 2.13) is globally exponentially stable. This completes the proof of Theorem 2.2. Remark 2.3. Theorem 2.2 implies that there exists a unique pattern or memory associated with each set of the periodic external inputs. Moreover, we have derived the conditions for the global attractivity of the unique periodic solution x t), y t)) T of system 1.1). Recall that for a given matrix A, its spectral radius ρa) is equal to the infimum of its all matrix norms of A, we have the following conclusion. Corollary 2.3. Assume that f j t, ) and g i t, ) are bounded with respect to t and that there exist non-negative constants p j and q i such that f j t, x) f j t, y) p j x y and g i t, x) g i t, y) q i x y for any t,x,y R, i = 1, 2,...,n, j = 1, 2,...,p. Then each one of conditions in Corollaries 2.1 and 2.2 ensures the existence, uniqueness, and global exponential stability of -periodic solution of system 1.1). Remark 2.4. The sufficient conditions that Cao and Wang obtained in paper [7 to ensure the existence of globally exponentially stable -periodic solution is a direct consequence of Corollary 2.3. Therefore, Corollary 2.3 not only unify and improve some previous results, but also give some new criteria expressed in terms of matrix norms. References [1 K. Gopalsamy, I. Leung, Physica D ) 395. [2 L.O. Lien, J. Belair, Physica D ) 344. [3 J. Wei, S. Ruan, Physica D ) 255. [4 S. Guo, L. Huang, J. Wu, J. Math. Anal. Appl ) 633. [5 S. Guo, L. Huang, J. Wu, Electron. J. Differential Equations 23 23) 61. [6 S. Guo, L. Huang, L. Wang, Linear stability and Hopf bifurcation in a two-neuron network with three delays, Int. J. Bifur. Chaos, in press. [7 J. Cao, L. Wang, Phys. Rev. E 61 2) [8 R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, [9 J.P. Lassalle, The Stability of Dynamical System, SIAM, Philadelphia, [1 A. Berman, R.J. Plemmons, Non-negative Matrices in the Mathematical Science, Academic Press, New York, [11 J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, Gruyter, New York, 21.