Anti-periodic oscillations of bidirectional associative memory (BAM) neural networks with leakage delays.

In this article, we discuss anti-periodic oscillations of BAM neural networks with leakage delays. A sufficient criterion guaranteeing the existence and exponential stability of the involved model is presented by utilizing mathematic analysis methods and Lyapunov ideas. The theoretical results of this article are novel and are a key supplement to some earlier studies.

Introduction

In the past several decades, the dynamics of BAM neural networks has been widely investigated for their essential applications in classification, pattern recognition, optimization, signal and image processing, and so on [1-41]. In 1987, Kosko [42] proposed the following BAM neural network: where . Here, denote the time scales of the respective layers of the network; and stand for the stabilizing negative feedback of the model. Noticing that the leakage delay often appears in the negative feedback term of neural networks (see [43-47]), Gopalsmay [48] studied the stability of the equilibrium and periodic solutions for the following BAM neural network: where . Since the delays in neural networks are usually time-varying in the real world, Liu [49] discussed the global exponential stability for the following general BAM neural network with time-varying leakage delays: However, so far, there have been rare reports on the existence and exponential stability of anti-periodic solutions of neural networks, especially for neural networks with leakage delays. Furthermore, the existence of anti-periodic solutions can be applied to help better describe the dynamical properties of nonlinear systems [49-65]. So we think that the investigation on the existence and stability of anti-periodic solutions for neural networks with leakage delays has significant value. Inspired by the ideas and considering the change of system parameters in time, we can modify neural network model (1.3) as follows: The main objective of this article is to analyze the exponential stability behavior of anti-periodic oscillations of (1.4). Based on the fundamental solution matrix, Lyapunov function, and fundamental function sequences, we establish a sufficient condition ensuring the existence and global exponential stability of anti-periodic solutions of (1.4). The derived findings can be used directly to numerous specific networks. Besides, computer simulations are performed to support the obtained predictions. Our findings are a good complement to the work of Gopalsmay [48] and Liu [49].

The paper is planned as follows. In Sect. 2, several notations and preliminary results are prepared. In Sect. 3, we give a sufficient condition for the existence and global exponential stability of anti-periodic solution of (1.4). In Sect. 4, we present an example to show the correctness of the obtained analytic findings.

Remark 1.1

A time delay that exists in the negative feedback term (or called leakage term or forgetting term) of neural networks is called leakage delay. If there exists an anti-periodic solution in a dynamical system, then we can say that the system has anti-periodic oscillations.

Preliminary results

In this segment, several notations and lemmas will be given.

For any vector and matrix , we define the norm as respectively. Let where , we define We assume that system (1.4) always satisfies the following initial conditions: Let be the solution of system (1.4) with initial conditions (2.1). We say the solution is T-anti-periodic on if for all and , where T is a positive constant.

Throughout this paper, for , it will be assumed that there exist constants such that and .

We also assume that the following conditions hold.

(H1) For , there exist constants , and such that for all .

(H2) For all and , where T is a positive constant.

It is clear that the conditions can be fulfilled; for example, let , then we have .

(H3) The following inequality holds: where .

Definition 2.1

The solution of system (1.4) is said to globally exponentially stable if there exist constants and such that for each solution of system (1.4).

Next, we present three important lemmas which are necessary for proving our main results in Sect. 3.

Lemma 2.1

Let then we have for all .

Proof

Since it follows that By the definition of matrix norm, we get  □

Lemma 2.2

Assume that where are any constants. Then there exists such that

Let Obviously, is a continuously differential function. We can easily check that By using the intermediate value theorem, we have that there exist constants such that Let , then it follows that and This completes the proof of Lemma 2.2. □

Lemma 2.3

Assume that (H1), (H3), and (H4) are satisfied. Then, for any solution of system (1.4), there exists a constant such that for all .

From (1.4), we have Then we have Thus Let where then system (1.4) can be written in the following equivalent form: Solving inequality (2.5), we have It follows from Lemma 2.1 that Then Let Then it follows that , for all . This completes the proof of Lemma 2.3. □

Main results

In this section, we present our main result that there exists an exponentially stable anti-periodic solution of (1.4).

Theorem 3.1

Assume that (H1)–(H4) hold true. Then any solution of system (1.4) is globally exponentially stable.

Let . It follows from system (2.4) that which leads to In view of condition (H1), we get Then where .

Now we consider the following Lyapunov function: where β is given by Lemma 2.2. Differentiating along solutions to system (1.4), together with (3.3), we have It follows from Lemma 2.2 that , which implies that for all . Thus Let and choose Then Eq. (3.7) can be rewritten as for all . Then for all . Thus the solution of system (1.4) is globally exponentially stable. □

Theorem 3.2

Assume that (H1)–(H4) are satisfied. Then system (1.4) has exactly one T-anti-periodic solution which is globally stable.

It follows from system (1.4) and (H2) that for each , we have Let Obviously, for any is also a solution of system (1.4). If the initial function is bounded, it follows from Theorem 3.1 that there exists a constant such that where . Since for any we have then By Lemma 2.3, we know that the solutions of system (1.4) are bounded. In view of (3.10) and (3.12), we can easily know that uniformly converges to a continuous function on any compact set of . In a similar way, we can easily prove that uniformly converges to a continuous function on any compact set of .

Now we show that is a T-anti-periodic solution of (1.4). Firstly, is T-anti-periodic since Then we can conclude that is T-anti-periodic on R. Similarly, is also T-anti-periodic on R. Thus we can conclude that is the solution of system (1.4).

In fact, together with the continuity of the right-hand side of system (1.4), let , we can easily get Therefore, is a solution of (1.4). Finally, by applying Theorem 3.1, it is easy to check that is globally exponentially stable. This completes the proof of Theorem 3.2. □

An example

In this section, we give an example to illustrate our main results derived in the previous sections. Consider the following BAM neural network with time-varying delays in the leakage terms: where Set . Then . It is easy to verify that and Then all the conditions (H1)–(H4) hold. Thus system (4.1) has exactly one π-anti-periodic solution which is globally exponentially stable. The results are illustrated in Fig. 1.

Figure 1

Transient response of state variables , and

Conclusions

In this paper, we have investigated the asymptotic behavior of BAM neural networks with time-varying delays in the leakage terms. Applying the fundamental solution matrix of coefficient matrix, we obtained a series of new sufficient conditions to guarantee the existence and global exponential stability of an anti-periodic solution for the BAM neural networks with time-varying delays in the leakage terms. The obtained conditions are easy to check in practice. Finally, an example is included to illustrate the feasibility and effectiveness.