Periodic solutions for discrete-time Cohen–Grossberg neural networks with delays

Highlights

• We study a model of discrete-time Cohen–Crossberg neural networks with delays.

• We present a sufficient condition for the existence of periodic solutions for DCGNND.

• We mainly apply graph theory, coincidence degree theory, and Lyapunov method.

• An example and its numerical simulation are given.

Abstract

In this paper, the existence of periodic solutions for general discrete-time Cohen–Grossberg neural networks with delays (DCGNND) is investigated. Based on graph theory, coincidence degree theory, and Lyapunov method, a sufficient criterion ensuring the existence of periodic solutions for DCGNND is established. In the end, an example and its numerical simulation are given to demonstrate the effectiveness of the theoretical result.

Introduction

In the past few decades, discrete-time Cohen–Grossberg neural networks (DCGNN) have been an active research area for their widespread applications such as image and signal processing, pattern recognition, and combinatorial optimization (see [1], [2], [3], [4], [5], [6] and the references therein). In addition, time delays are inevitably encountered in DCGNN due to the finite switching speed of amplifiers. In this paper, we investigate the model of discrete-time Cohen–Grossberg neural networks with delays (DCGNND) in the following

$$x_i(k+1)=x_i(k)-a_i(x_i(k),k)\times (b_i(x_i(k))-\sum _{j=1}^l{c}_{ij}{f}_j(x_j(k))-\sum _{j=1}^l{d}_{ij}{f}_j(x_j(k-v_{ij}))+J_i(k)),i\in \mathbb{L}.$$

Here $l\ge 2$ is the number of neurons in the networks. Denote $x_i\in \mathbb{R}$ as the state associated with the i-th neuron. Let $a_i(\cdot ,k)$, an ω-periodic function in k, represent the amplification function. In addition, $b_i$ is an appropriately behaved function and define matrices ${(c_{ij})}_{l\times l},{(d_{ij})}_{l\times l}$ for the synaptic connection weights from unit j to unit i. Function $f_i$ is a measure of response or activation to its incoming potential and $v_{ij}$ represents the transmission delay along the axon from unit j to unit i. ω-periodic function $J_i(k)$ is the external input of the networks at time k.

As is known to all, most successful applications on DCGNND are mainly based on their dynamic properties, especially periodicity. In fact, the existence of periodic solutions for DCGNND can represent various storage patterns or memory patterns in some applications. With high dimensions, DCGNND are very complicated, especially when considering time delays into the models. The existence of periodic solutions for DCGNND is not only based on the properties of plenty of subsystems in DCGNND, but also related to the topological structure of DCGNND. Recently, coincidence degree theory is a powerful tool to study the existence of periodic solutions for many systems, such as continuous-time systems [7], [8], [9], discrete-time systems [10], [11], and delay systems [9], [12], [13]. It is a vital step to estimate the priori bounds of unknown solutions for the equation $Lx=\lambda Nx$. Some scholars try to use inequality $|x(t)|\le |x(t_0)|+{\int }_0^{\omega }|\dot{x}(t)|\mathrm{d}t$ and matrix's spectral theory to obtain the priori bounds in previous literatures [7], [8], [11], [12]. However, when we focus on the existence of their periodic solutions, it is not an easy task due to the intricacy of DCGNND and some new skills should be introduced. Based on the discussions above, the existence of periodic solutions for DCGNND is studied in this paper.

In reference to the existing results, our contributions are in the following.

• DCGNND are discussed and a novel easily checked sufficient criterion in practice is proposed, which is verified by an example and its numerical simulation.

• Combining graph theory, coincidence degree theory, and Lyapunov method, a systematic method is provided to investigate the existence of periodic solutions for DCGNND.

• A new skill is given to estimate the priori bounds of unknown solutions for the equation $Lx=\lambda Nx$.

The rest of this paper is organized as follows. In Section 2, some preliminaries about graph theory and coincidence degree theory are introduced. The existence of periodic solutions for DCGNND (1) is discussed in Section 3. Besides, an example and its numerical simulation are given to verify the effectiveness of the theoretical result in Section 4. Finally, conclusions are presented in Section 5.

Notations: Let $\mathbb{R}$ and ${\mathbb{R}}^n$ represent the set of real numbers and n-dimensional Euclidean space, respectively. Set $\mathbb{L}=\{1,2,\dots ,l\}$, $\mathbb{N}=\{0,1,2,\dots \}$, and ${\mathbb{N}}^{+}=\{1,2,\dots \}$. Write $k\in \mathbb{N}$ and $\omega \in {\mathbb{N}}^{+}$. Let $|\cdot |$ and the superscript “T” denote the Euclidean norm and the transpose for vectors or matrices, respectively.