Periodic Solutions of Cohen-Grossberg-Type BAM Neural Networks with Time-Varying Delays ()
1. 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. [3-15].
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.
2. Preliminaries
Consider the following periodic CGNNs with timevarying delays (see Equation (1)):
For
, 
and
. 
denote the state variables of the ith neuron, 
denote the signal functions of the jth neuron at time t; 
denote inputs of the ith neuron at time t; 
represent amplification functions; 
are appropriately behaved functions; 
and 
and are connection weights of the neural networks, respectively; 
are positive constants which correspond to the neuronal gains associated with the neuronal activations; 
correspond to the finite speed of the axonal signal transmission at time t and there exist constants 
such that
, 
and 
are all continuously periodic functions on [0, +∞) with common period T > 0.
Throughout this paper, we assume for system (1) that
(H1) Amplification functions 
are continuous and there exist constants 
such that 
for
.
(H2) 
are T-periodic about the first argument and there exist continuous T-periodic functions 
such that
.
(H3) For activation functions
, there exist positive constants 
such that
(1)
For any continuous function 
on
, 
and 
denote 
and
respectively.
For any
define
and for any
,
define
in which
.
Denote
is continuous on
.
Then 
is a Banach space with respect to
.
The initial conditions of system (1) are given by
(2)
where
.
Le 
denotes any solution of the system (1) with initial value
.
Definition 1. An solution 
of system (1) is said to be globally exponentially stable, for any solutions 
of the system (1), if there exist positive constant 
and 
such that
(3)
Lemma 1. Under assumptions (H1)-(H3), system (1) has a T-periodic solution which is globally exponentially stable, if the following conditions hold.
(H4) Assume that there exist constants 
such that
,
.
(H5) 
is a nonsingular M-matrix, where
Proof. If
, the model (2.1) in [14] reduces to the system (1), we know that Lemma 1 holds from Theorem 3.1 with r = 1 in [14].
3. Periodic Solutions of CGBAMNNs with Time Varying Delays
Consider the following CGBAMNNs with time-varying delays:
(4)
for
, 
and 
and 
denote the state variables, 
and 
denote the signal functions, 
and 
denote inputs; 
and 
represent amplification functions; 
and 
are appropriately behaved functions;
, 
, 
and 
are the connection weights and
, 
are positive constants, which correspond to the neuronal gains associated with the neuronal activations; Time delays 
and 
correspond to the finite speed of the axonal signal transmission at time t and there exist constants 
and 
such that
,
;
, 
, 
, 
, 
, 
, 
and 
are all continuously periodic functions on 
with common period
.
Throughout this paper, we assume for system (4) that
(H6) Amplification functions 
and 
are continuous and there exist positive constants 
and 
such that
, 
,
.
(H7)
, 
are T-periodic about the first argument and there exist continuous T-periodic functions 
and 
such that
(H8) For activation functions
and
, there exist constants 
and 
such that
The initial conditions of system (4) are given by
where
,
Theorem 1. Under assumptions (H6)-(H10), system (4) has a T-periodic solution which is globally exponentially stable, if the following condition holds.
(H9) Assume that there exist constants 
and 
such that 
and 
hold for
.
(H10) The following
is a nonsingular M-matrix, and
(5)
in which
Proof. Let
(6)
It follows that system (4) can be rewrote as
(7)
for
.
Initial conditions are given by
(8)
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 (H6)-(H10), 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
(9)
where
in which 
Then, we know from (6) and (9) that Theorem 1 holds.
4. An Example
Consider the following CGBAMNNs with time delays:
(10)
Figure 1. Time response of state variables x1, y1 and phase plot in space (t, x1, y1) for system (10).
It is easy to verify system (10) satisfies (H6)-(H9). In addition, system (10) satisfies (H10) because
is a nonsingular M-matrix. According to Theorem 1, system (10) has a 2-periodic solution which is globally exponentially stable. Figure 1 shows the dynamic behaveiors of system (10) with initial conditions (0.8, 0.9).
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].