Global Exponential Stability of Periodic Oscillation for Nonautonomous BAM Neural Networks with Distributed Delay

Abstract

We derive a new criterion for checking the global stability of periodic oscillation of bidirectional associative memory (BAM) neural networks with periodic coefficients and distributed delay, and find that the criterion relies on the Lipschitz constants of the signal transmission functions, weights of the neural network, and delay kernels. The proposed model transforms the original interacting network into matrix analysis problem which is easy to check, thereby significantly reducing the computational complexity and making analysis of periodic oscillation for even large-scale networks.

1. Introduction

The bidirectional associative memory (BAM) neural network which was first introduced by Kosko in 1987 [1, 2] is formed by neurons arranged in two layers. The neurons in one layer are fully interconnected to the neurons in the other layer, while there are no interconnections among neurons in the same layer. Through iterations of forward and backward information flows between the two layers, it performs a two-way associative search for stored bipolar vector pairs and generalizes the single-layer autoassociative hebbian correlation to a two-layer pattern matched heteroassociative.

As it is well known, research on neural dynamical systems not only involves a discussion of stability properties, but also involves many dynamic behavior such as periodic oscillatory behavior, bifurcation, and chaos [3–19]. In the application of neural networks to some practical problems, the properties of equilibrium points play important roles. An equilibrium point can be looked as a special periodic solution of neural networks with arbitrary period. In this sense, the analysis of periodic solutions of neural networks could be more general than that of equilibrium points. There are some results on the existence and stability of periodic solution of BAM neural networks. Liu et al. [20, 21] obtained several sufficient conditions which ensure existence and stability of periodic solution for BAM neural networks with periodic coefficients and time-varying delays. Subsequently, Guo et al. [22] obtained some sufficient conditions ensuring the existence, uniqueness, and stability of the periodic solution for BAM neural networks with periodic variable coefficients and variable delays, and they also estimated the exponentially convergent rate. Song et al. obtained several sufficient conditions which ensure existence and stability of periodic solution for BAM neural networks with periodic coefficients and periodic time-varying delays [23]. Moreover, neural networks usually has a spatial extent due to the presence of an amount of parallel pathways with a variety of axon sizes and lengths. Thus, the delays in neural networks are usually continuously distributed. Recently, there are some authors studied the BAM neural networks with distributed delays and constants coefficients [24–26].

Until recently, few studies have considered periodic solution for the BAM neural networks with periodic coefficients and distributed delays. Zhou et al. considered the periodic solution for the BAM neural networks with period coefficients and continuously distributed delays [27]. However, the result in Zhou et al. contains two limitations, one made for periodic and the other is , which also in the Wang et al. [28]. This limitation is being removed by this paper. Based on the continuation theorem of Mawhin's coincidence degree theory, the nonsingular M-matrix and Lyapunov functionals, we derive a new global exponential stability criterion in matrix form for periodic oscillation of BAM neural networks with period coefficients and distributed delay. Moreover, our criterion is easy to check out.

The paper is organized as follows. Our model and some preliminaries are given in Section 2. The existence of periodic solution is proved in Section 3. The exponential stability of periodic oscillator is considered in Section 4. An example is shown in Section 5. Several summary remarks are finally given in Section 6.

2. Preliminaries

In this paper, we study the BAM neural networks with periodic coefficients and continuously distributed delays modeled by the following system:

(2.1)

where ; ; and denote the rate with which the cells and reset their potential to the resting state when isolated from the other cells and inputs; and are connection weights of the neural network; , denote the th and the th component of an external input source introduced from outside the network to the th cell and th cell at time , respectively. Moreover, the th cell has an impact on the th cell in the time of and the th cell has an impact on the th cell in the time of .

If and satisfy system (2.1) and , , then they are -periodic solutions of system (2.1). The initial conditions associated with system (2.1) are given as follows:

(2.2)

where are continuous function ().

Throughout this paper, we make the following assumptions.

Assumption 2.1.

, , , , and are continuous -periodic functions on . In addition, , , , , , , , and .

Assumption 2.2.

Signal transmission functions , are bounded on , and there exist number and such that

(2.3)

for each , and .

Assumption 2.3.

The delay kernels are continuous and integrable and satisfy

(2.4)

Assumption 2.4.

The delay kernels satisfy

(2.5)

where is a bounded positive real number.

Now, we give some useful notations, definitions, and lemmas as follows: and , where is -periodic function.

Assume that , then we have the following.

Lemma 2.5 (see [17, 29, 30]).

Let . Then, each of the following conditions is equivalent to the statement ` is a nonsingular M -matrix':

(a)all of the principal minors of are positive;

(b)the real parts of all the eigenvalue of are positive;

(c) is inverse-positive; that is, exists and ;

(d)there is a vector (or ), whose elements are all positive, such that the elements of (or ) are all positive;

(e) has all positive diagonal elements and there exists a positive diagonal matrix such that is strictly diagonally dominant; that is,

(2.6)

In the following, we introduce some concepts and results from the book by Gaines and Mawhin [31].

Let and be two Banach spaces, a linear mapping, and a continuous mapping. The mapping will be called a Fredholm mapping of index zero if = and is closed in . If is a Fredholm mapping of index zero and there exist continuous projectors and such that = , = = ), it follows that mapping is invertible. We denote the inverse of that mapping by . If is an open bounded subset of , the mapping will be called -compact on if is bounded and is compact. Since is isomorphic to , there exists an isomorphism .

Lemma 2.6 (Mawhin's continuation theorem).

Let and be two Banach spaces and be a Fredholm mapping of index zero. Assume that is an open bounded set and is a continuous operator which is -compact on . Then has at least one solution in , if the following conditions are satisfied:

(a)for each ;

(b)for each , ;

(c),

where is an isomorphism.

3. Existence of Periodic Solutions

Now we give the following sufficient conditions on the existence of periodic solutions.

Theorem 3.1.

Assume that Assumptions 2.1–2.3 hold. Then, system (2.1) has at least one T-periodic solution, if

(3.1)

is a nonsingular , where is unite matrix and ; .

Proof.

Let , then is a Banach space with the norm

Let , , , and be given by the following:

(3.2)

It is easy to see that is a linear operator with = . is closed in , and . Therefore, is a Fredholm mapping of index zero. It is easy to prove that and are two projectors, and = , = = . By using the Arzelá-Ascoli theorem, it is easy to prove that for every bounded subset are relatively compact on in , that is, is -compact on .

Consider the operator equation

(3.3)

that is

(3.4)

where and Denote . By using (3.4), we obtain

(3.5)

Multiplying both sides of (3.5) by , we have

(3.6)

Integrating the inequality above from 0 to [27], we obtain

(3.7)

Hence,

(3.8)

for . So we have

(3.9)

that is,

(3.10)

which implies that is bounded and similarly . By the Assumption 2.3, we know that

(3.11)

is uniformly convergent. Therefore, the following iterated integral:

(3.12)

can be changed integrating order.

Suppose that is any periodic solution of system (2.1) for a certain . Multiplying both sides of

(3.13)

by and integrating from 0 to , we obtain

(3.14)

From Assumptions 2.1, 2.2, and 2.3 and noting that , we have

(3.15)

that is,

(3.16)

By similar argument, we have

(3.17)

It follows (3.16) and (3.17) that

(3.18)

where is a unite matrix and ; , , , , , .

Application of Lemma 2.5 yields

(3.19)

which implies that

(3.20)

It is not difficult to check that there exist such that

(3.21)

for and . Multiplying both sides of (3.13) by and integrating from 0 to , we obtain

(3.22)

Therefore,

It is easy check that

(3.23)

Let and (. We take

(3.24)

Obviously, condition (a) of Lemma 2.6 is satisfied. When = , is a constant vector in with . Then, we have

(3.25)

We claim that there exists some or (, ) such that

(3.26)

If and , then, we obtain

(3.27)

Which implies

(3.28)

By a similar argument, we obtain

(3.29)

On the other hand,

(3.30)

where , and . It follows that there exists some or such that

(3.31)

or

(3.32)

which is contradiction with (3.28) and (3.29). Then, there exists some or (, ) such that or . Therefore,

(3.33)

This indicates that condition (b) of Lemma 2.6 is satisfied.

Define , , where . When , we have

(3.34)

According to the invariant of homology, we have

(3.35)

where is an isomorphism. Therefore, according to the continuation theorem of Gaines and Mawhin, system (2.1) has at least one -periodic solution. The proof is completed.

Remark 3.2.

In [27], the period was assumed to be . In [28], the inequations must hold and also in [27]. Here, the limitation is being removed.

4. Global Exponential Stability of Periodic Solution

In this section, we discuss the global exponential stability of the periodic solution of system (2.1). Under the assumptions of Theorem 3.1, system (2.1) has at least one -periodic solution

(4.1)

Now, we give the following definition about global exponential of periodic solution:

Definition 4.1.

The periodic solution of model (2.1) is said to be globally exponentially stable, if there exist positive constants such that

(4.2)

for all , where , represent the history of and on respectively, and

Definition 4.2.

A matrix is said to be diagonally dominant if for all , for at least one , where denotes the entry in the th row and th column.

Theorem 4.3.

Assume that Assumptions 2.1, 2.2, 2.4 hold and is a nonsingular , where is the same as in Theorem 3.1. System (2.1) has a unique -periodic solution, which is globally exponentially stable, if and is a weakly diagonally dominant matrix as

(4.3)

where ; , ; , ;

Proof.

Let . Then,

(4.4)

for and

We define the following Lyapunov functionals:

(4.5)

Calculating the Dini upper right derivative of and along the solution of (2.1), and estimating it via the assumptions [32], we have

(4.6)

Let

(4.7)

where ; , ; , ;

Consider the Lyapunov functional

(4.8)

When and is a weakly diagonally dominant matrix, calculating the Dini upper right derivative of along the solution of (2.1), we have

(4.9)

Therefore, for ,

(4.10)

Thus,

(4.11)

that is,

(4.12)

This means that periodic solution of system (2.1) is globally exponentially stable. The proof is completed.

Remark 4.4.

For system (2.1), when delay kernels , are -functions, that is, we take and , then system (2.1) can be reduced to a fixed time delay system and all above theorems are hold.

5. Example

In this section, we present an example to show the effectiveness and correctness of our theoretical results.

Consider the following BAM neural networks:

(5.1)

We select a set of parameters as , , , , and , , , ,

(5.2)

Remark 5.1.

A frequently used model for distributed time delays in biological, neural networks applications is to choose for the Gamma kernel due to mathematical difficulties. In this paper, we set a weak delay kernel, that is, an exponential kernel.

It is easy to check that and . Let and . By some computations, we obtain

(5.3)

It is easy to check that is a nonsingular -matrix and is a weakly diagonally dominant matrix. From Theorem 4.3 system (5.1) has -periodic oscillation, which is globally exponentially stable. In Figures 1, 2, 3, 4, 5, and 6, we plot the trajectories of and , respectively.

Figure 1

Time evolution of of system (5. 1).

Figure 2

Time evolution of of system (5. 1).

Figure 3

Time evolution of of system (5. 1).

Figure 4

Time evolution of of system (5. 1).

Figure 5

Time evolution of of system (5. 1).

Figure 6

Time evolution of of system (5. 1).

6. Conclusion

In this paper, we derive a new criterion for checking the global stability of periodic oscillation of BAM neural networks with distributed delay and periodic external input sources and find that the criterion rely on the Lipschitz constants of the signal transmission functions, weights of the neural network and delay kernels by using the continuation theorem of Mawhin's coincidence degree theory, the nonsingular M-matrix and Lyapunov function. The proposed model transforms the original interacting network into matrix analysis, thereby significantly reducing the computational complexity and making analysis of periodic oscillation for even large-scale networks. Most importantly, our result is very practical in the design of BAM neural networks.