Periodic Solutions of Cohen-grossberg-type Bam Neural Networks with Time-varying Delays

Sufficient conditions to guarantee the existence and global exponential stability of periodic solutions of a Cohen-Grossberg-type BAM neural network are established by suitable mathematical transformation.


Introduction
Many important results on the existence and global exponential stability of equilibria of neural networks with time delays have been widely investigated and successfully applied to signal processing system.However, the research of neural networks involves not only the dynamic analysis of equilibrium point but also that of periodic oscillatory solution.In practice, the dynamic behavior of periodic oscillatory solution is very important in learning theory [1,2], which is motivated by the fact that learning usually requires repetition, some important results for periodic solutions of Hopfield neural networks or Cohen-Grossberg neural networks with delays have been obtained in Refs.[3][4][5][6][7][8][9][10][11][12][13][14][15].
The objective of this paper is to study the existence and global exponential stability of periodic solutios of a class of Cohen-Grossberg-type BAM neural networks (CGBAMNNs) with time-varying delays by suitable mathematical transformation.
The rest of this paper is organized as follows: preliminaries are given in Section 2. Sufficient conditions which guarantee the existence and global exponential stability of periodic solutions for the CGBAMNNs are established Section 3.An example is given in Section 4 to demonstrate the main results.
Throughout this paper, we assume for system (1) that (H 1 ) Amplification functions i are continuous and there exist constants  are T-periodic about the first argument and there exist continuous T-periodic functions For any , and for any The initial conditions of system (1) are given by where denotes any solution of the system (1) with initial value 1) is said to be globally exponentially stable, for any solutions x t   of the system (1), if there exist positive constant 0 Lemma 1.Under assumptions (H 1 )-(H 3 ), system (1) has a T-periodic solution which is globally exponentially stable, if the following conditions hold.
(H 4 ) Assume that there exist constants [14] reduces to the system (1), we know that Lemma 1 holds from Theorem 3.1 with r = 1 in [14].

Periodic Solutions of CGBAMNNs with Time Varying Delays
Consider the following CGBAMNNs with time-varying delays: and with common period .Throughout this paper, we assume for system (4) that (H 6 ) Amplification functions i and j c  are continuous and there exist positive constants , i i a a and ,  ,  are T-periodic about the first argument and there exist continuous T-periodic functions The initial conditions of system (4) are given by Under assumptions (H 6 )-(H 10 ), system (4) has a T-periodic solution which is globally exponentially stable, if the following condition holds.
(H 9 ) Assume that there exist constants    and ji such that and hold for .
The following M is a nonsingular M-matrix, and , It follows that system (4) can be rewrote as Hence system ( 7) is a special case of system (1) in mathematical form in which there are n+m neurons and connection weights for and     .Under conditions (H 6 )-(H 10 ), from Lemma 1, we obtain that system (7) has a T-periodic solution which is globally exponentially stable, if the following matrix  is a M-matrix, and where Then, we know from ( 6) and ( 9) that Theorem 1 holds.

Remark 1
The results in [3,15] have more restrictions than the results in this paper because conditions for the results in [3,15] are relevant to amplification functions.In addition, in view of proof of Theorem 1, since CGBAMNNs with time-varying delays is a special case of CGNNs time-varying delays in form as BAM neural networks is a special case of Hopfield neural networks, many results of CGBAMNNs can be directly obtained from the ones of CGNNs, needing no repetitive discussions, which coincide with the conclusion in [16,17].
the state variables of the ith neuron,  f j  denote the signal functions of the jth neuron at time t;  i  