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 . In 1987, Kosko [42] proposed the following BAM neural network: ⎧ ⎨ ⎩ du i (t) dt = -a i u i (t) + n j=1 a ij f j (v j (tσ j (t))) + I i , dv i (t) dt = -b i v i (t) + n j=1 b ij g j (u j (tτ j (t))) + J i , (1.1) where i = 1, 2, . . . , n, t > 0. Here, a i > 0, b i > 0 denote the time scales of the respective layers of the network; -a i u i (t) and -b i v i (t) 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][44][45][46][47]), Gopalsmay [48] studied the stability of the equilibrium and periodic solutions for the following BAM neural network: where i = 1, 2, . . . , n, t > 0. 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: dt = -a i x i (tδ i (t)) + n j=1 a ij f j (y j (tσ ij (t)) + I i , 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][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][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: (1.4) The main objective of this article is to analyze the exponential stability behavior of antiperiodic 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 antiperiodic 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 V = (v 1 , v 2 , . . . , v n ) T and matrix D = (d ij ) n×n , we define the norm as respectively. Let We assume that system (1.4) always satisfies the following initial conditions: . , x n (t)) T , y(t) = (y 1 (t), y 2 (t), . . . , y n (t)) T be the solution of system (1.4) with initial conditions (2.1). We say the solution for all t ∈ R and i = 1, 2, . . . , n, where T is a positive constant. Throughout this paper, for i, j = 1, 2, . . . , n, it will be assumed that there exist constants such that We also assume that the following conditions hold.

Definition 2.1
The solution (x * (t), y * (t)) T of system (1.4) is said to globally exponentially stable if there exist constants β > 0 and M > 1 such that for each solution (x(t), y(t)) T 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
By the definition of matrix norm, we get

Main results
In this section, we present our main result that there exists an exponentially stable antiperiodic solution of (1.4).
In fact, together with the continuity of the right-hand side of system (1.4), let k → ∞, we can easily get (3.14) Therefore, (x * (t), y * (t)) T is a solution of (1.4). Finally, by applying Theorem 3.1, it is easy to check that (x * (t), y * (t)) T is globally exponentially stable. This completes the proof of Theorem 3.2.
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.

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.