1 Introduction
Since Chua and Yang proposed cellular neural networks (CNNs) in 1988 [
1], various dynamical behaviors of CNNs, such as the existence and stability of the equilibrium, periodic solutions, anti-periodic solutions, almost periodic solutions, and pseudo-almost periodic solutions, have been studied by many scholars [
2‐
15].
On the one hand, quaternion-valued neural networks (QVNNs), as an extension of the complex-valued neural networks (CVNNs), can deal with multi-level information and require only half the connection weight parameters of CVNNs [
16]. Moreover, compared with CVNNs, QVNNs perform more prominently when it comes to geometrical transformations, like 2D affine transformations or 3D affine transformations. 3D geometric affine transformations can be represented efficiently and compactly based on QVNNs, especially spatial rotation [
17]. Since the multiplication of quaternion is not commutative due to Hamilton rules:
\(ij=-ji=k\),
\(jk=-kj=i, ki=-ik=j, i^{2}=j^{2}=k^{2}=ijk=-1\), the analysis for QVCNNs becomes difficult. However, with the continuous development of the theory of quaternion, there are some results about the dynamics of QVNNs. For example, the authors of [
18,
19] studied the existence and global exponential stability of equilibrium point for QVNNs; the authors of [
20] investigated the robust stability of QVNNs with time delays and parameter uncertainties; the authors of [
21] considered the existence and stability of pseudo almost periodic solutions for a class of QVCNNs on time scales by a special decomposition method; the authors of [
22,
23] investigated the existence and global
μ-stability of an equilibrium point for QVNNs; the authors of [
24] dealt with the existence and stability of periodic solutions for QVCNNs by using a continuation theorem of coincidence degree theory; the authors of [
25] studied the almost periodic synchronization for QVCNNs. Although non-autonomous neural networks are more general and practical than the autonomous ones, up to now, there have been only few results about the dynamic behaviors of non-autonomous QVNNs.
On the other hand, it is well known that the periodicity, almost periodicity, pseudo almost periodicity, and so on are the very important dynamics for non-autonomous systems [
10,
12,
26]. Moreover, the almost periodicity is more general than the periodicity. In addition, the pseudo almost periodicity is a natural generalization of almost periodicity. In the past few years, the pseudo almost periodicity of real-valued neural networks (RVNNs) has been studied by many authors [
13‐
15,
27‐
34]. Besides, as we all know, time delay is universal and can change the dynamical behavior of the system under consideration [
3,
5,
29,
30,
35,
36]. Therefore, it is important and necessary to consider the neural network model with time delay. However, to the best of our knowledge, there is no paper published on the existence and stability of pseudo almost periodic solutions for quaternion-valued cellular neural networks (QVCNNs) with discrete and distributed delays.
Motivated by the above, in this paper, we are concerned with the following QVCNN with discrete and distributed delays:
$$\begin{aligned} x_{p}'(t) =&-c_{p}(t)x_{p}(t)+ \sum_{q=1}^{n}a_{pq}(t)f_{q} \bigl(x_{q} \bigl(t-\tau _{pq}(t) \bigr) \bigr) \\ &+\sum_{q=1}^{n}b_{pq}(t) \int _{0}^{\infty }K_{pq}(u)g_{q} \bigl(x_{q}(t-u) \bigr)\,\mathrm{d}u+u_{p}(t), \end{aligned}$$
(1)
where
\(p\in \{1,2,\ldots ,n\}:=\Lambda \),
\(x_{p}(t)\in \mathbb{Q}\) is the state vector of the
pth unit at time
t,
\(c_{p}(t)>0\) represents the rate at which the
pth unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs,
\(a_{pq}(t), b_{pq}(t)\in \mathbb{Q}\) are the synaptic weights of delayed feedback between the
pth neuron and the
qth neuron,
\(f_{q}, g_{q}:\mathbb{Q}\rightarrow \mathbb{Q}\) are the activation functions of signal transmission,
\(\tau _{pq}(t)\geq 0\) denotes the transmission delay,
\(u_{p}(t)\in \mathbb{Q}\) denotes the external input on the
pth neuron at time
t.
Throughout this paper, we denote by \(BC(\mathbb{R},\mathbb{R}^{n})\), the set of all bounded continuous functions from \(\mathbb{R}\) to \(\mathbb{R}^{n}\).
The initial value is given by
$$\begin{aligned} x_{p}(s)=\phi _{p}(s),\quad s\in (-\infty ,0], p\in \Lambda , \end{aligned}$$
where
\(\phi _{p}\in BC((-\infty ,0],\mathbb{Q})\).
Our main aim in this paper is to study the existence and global exponential stability of pseudo almost periodic solutions of (
1). The main contributions of this paper are listed as follows.
(1)
To the best of our knowledge, this is the first time to study the existence and stability of pseudo almost periodic solutions for QVCNNs with discrete and distributed delays.
(2)
The stability of QVNNs with distributed delays has not been reported yet. Therefore, our result about the stability of QVNNs is new, and most of the existing results about the stability of QVNNs are obtained by using the theory of linear matrix inequalities but ours are not.
(3)
The method that we use to transform QVNNs into RVNNs is different from that used in [
18,
20‐
23].
(4)
QVCNN (
1) contains RVCNNs and CVCNNs as its special cases.
Throughout this paper,
\(\mathbb{R}^{n\times n}\),
\(\mathbb{Q}^{n\times n}\) denote the set of all
\(n\times n\) real-valued and quaternion-valued matrices, respectively. The skew field of quaternion is denoted by
$$\begin{aligned} \mathbb{Q}:= \bigl\{ x=x^{R}+ix^{I}+jx^{J}+kx^{K} \bigr\} , \end{aligned}$$
where
\(x^{R}\),
\(x^{I}\),
\(x^{J}\),
\(x^{K}\) are real numbers and the elements
i,
j, and
k obey Hamilton’s multiplication rules.
For the convenience, we will introduce the notations: \(\bar{h}=\sup_{t\in \mathbb{R}}\vert h(t) \vert \), \(\underline{h}=\inf_{t\in \mathbb{R}}\vert h(t) \vert \), where \(h(t)\) is a bounded continuous function.
This paper is organized as follows. In Sect.
2, we introduce some definitions, make some preparations for later sections. In Sect.
3, by utilizing Banach’s fixed point theorem and differential inequality techniques, we establish the existence and global exponential stability of pseudo almost periodic solutions of (
1). In Sect.
4, we give an example to demonstrate the feasibility of our results. This paper ends with a brief conclusion in Sect.
5.
2 Preliminaries
In this section, we shall first recall some fundamental definitions, lemmas which are used in what follows.
Let
$$\begin{aligned} PAP_{0} \bigl(\mathbb{R},\mathbb{R}^{n} \bigr) = \biggl\{ f \in BC \bigl(\mathbb{R},\mathbb{R}^{n} \bigr) \Bigm| \lim _{r\rightarrow +\infty }\frac{1}{2r} \int _{-r}^{r}\bigl\Vert f(t) \bigr\Vert \, \mathrm{d}t=0 \biggr\} . \end{aligned}$$
From the above definitions, it is easy to see that \(AP(\mathbb{R},\mathbb{R}^{n})\subset PAP(\mathbb{R},\mathbb{R}^{n})\).
Consider the following pseudo almost periodic system:
$$\begin{aligned} x'(t)=A(t)x(t)+f(t), \end{aligned}$$
(3)
where
\(A(t)\) is an almost periodic matrix function,
\(f(t)\) is a pseudo almost periodic vector function.
In order to decompose the quaternion-valued system (
1) into a real-valued system, we need the following assumption:
\((S_{1})\)
Let
\(x_{p}=x_{p}^{R}+ix_{p}^{I}+jx_{p}^{J}+kx_{p}^{K}\),
\(x_{p}^{l}\in \mathbb{R}, l\in E\). Then the activation functions
\(f_{q}(x_{q})\) and
\(g_{q}(x_{q})\) of (
1) can be expressed as
$$\begin{aligned}& f_{q}(x_{q}) = f_{q}^{R} \bigl(x_{q}^{R},x_{q}^{I},x_{q}^{J},x_{q}^{K} \bigr) \\& \hphantom{f_{q}(x_{q}) =}{}+if_{q}^{I} \bigl(x_{q}^{R},x_{q}^{I},x_{q}^{J},x_{q}^{K} \bigr)+jf_{q}^{J} \bigl(x_{q}^{R},x_{q}^{I},x_{q}^{J},x_{q}^{K} \bigr) +kf_{q}^{K} \bigl(x_{q}^{R},x_{q}^{I},x_{q}^{J},x_{q}^{K} \bigr), \\& g_{q}(x_{q}) = g_{q}^{R} \bigl(x_{q}^{R},x_{q}^{I},x_{q}^{J},x_{q}^{K} \bigr) +ig_{q}^{I} \bigl(x_{q}^{R},x_{q}^{I},x_{q}^{J},x_{q}^{K} \bigr)+jg_{q}^{J} \bigl(x_{q}^{R},x_{q}^{I},x_{q}^{J},x_{q}^{K} \bigr) \\& \hphantom{f_{q}(x_{q}) =}{} +kg_{q}^{K} \bigl(x_{q}^{R},x_{q}^{I},x_{q}^{J},x_{q}^{K} \bigr), \end{aligned}$$
where
\(f_{q}^{l}, g_{q}^{l}:\mathbb{R}^{4}\rightarrow \mathbb{R}\),
\(p\in \Lambda , l\in E\).
Under assumption
\((S_{1})\), system (
1) can be decomposed into the following four real-valued sub-systems:
$$\begin{aligned}& \bigl(x_{p}^{R}(t) \bigr)' =-c_{p}(t)x_{p}^{R}(t)+ \sum _{q=1}^{n} \bigl(a_{pq}^{R}(t)f_{q}^{R}[t,x] -a_{pq}^{I}(t)f_{q}^{I}[t,x] \\ & \hphantom{ (x_{p}^{R}(t) )' =}{}-a_{pq}^{J}(t)f_{q}^{J}[t,x] -a_{pq}^{K}(t)f_{q}^{K}[t,x] \bigr)+\sum _{q=1}^{n} \biggl(b_{pq}^{R}(t) \int _{0}^{\infty }K_{pq}(u) \\ & \hphantom{ (x_{p}^{R}(t) )' =}{}\times g_{q}^{R}[t,u,x]\,\mathrm{d}u -b_{pq}^{I}(t) \int _{0}^{\infty }K_{pq}(u)g_{q}^{I}[t,u,x] \,\mathrm{d}u \\ & \hphantom{ (x_{p}^{R}(t) )' =}{}-b_{pq}^{J}(t) \int _{0}^{\infty }K_{pq}(u)g_{q}^{J}[t,u,x] \,\mathrm{d}u -b_{pq}^{K}(t) \int _{0}^{\infty }K_{pq}(u) \\ & \hphantom{ (x_{p}^{R}(t) )' =}{}\times g_{q}^{K}[t,u,x]\,\mathrm{d}u \biggr)+u_{p}^{R}(t), \end{aligned}$$
(4)
$$\begin{aligned}& \bigl(x_{p}^{I}(t) \bigr)' = -c_{p}(t)x_{p}^{I}(t)+ \sum _{q=1}^{n} \bigl(a_{pq}^{R}(t)f_{q}^{I}[t,x] +a_{pq}^{I}(t)f_{q}^{R}[t,x] \\ & \hphantom{ (x_{p}^{I}(t) )'=}{} +a_{pq}^{J}(t)f_{q}^{K}[t,x] -a_{pq}^{K}(t)f_{q}^{J}[t,x] \bigr)+\sum _{q=1}^{n} \biggl(b_{pq}^{R}(t) \int _{0}^{\infty }K_{pq}(u) \\ & \hphantom{ (x_{p}^{I}(t) )'=}{} \times g_{q}^{I}[t,u,x]\,\mathrm{d}u +b_{pq}^{I}(t) \int _{0}^{\infty }K_{pq}(u)g_{q}^{R}[t,u,x] \,\mathrm{d}u \\ & \hphantom{ (x_{p}^{I}(t) )'=}{} +b_{pq}^{J}(t) \int _{0}^{\infty }K_{pq}(u)g_{q}^{K}[t,u,x] \,\mathrm{d}u -b_{pq}^{K}(t) \int _{0}^{\infty }K_{pq}(u) \\ & \hphantom{ (x_{p}^{I}(t) )'=}{} \times g_{q}^{J}[t,u,x]\,\mathrm{d}u \biggr)+u_{p}^{I}(t), \end{aligned}$$
(5)
$$\begin{aligned}& \bigl(x_{p}^{J}(t) \bigr)' = -c_{p}(t)x_{p}^{J}(t)+ \sum _{q=1}^{n} \bigl(a_{pq}^{R}(t)f_{q}^{J}[t,x] +a_{pq}^{J}(t)f_{q}^{R}[t,x] \\ & \hphantom{ (x_{p}^{I}(t) )'=}{} -a_{pq}^{I}(t)f_{q}^{K}[t,x] +a_{pq}^{K}(t)f_{q}^{I}[t,x] \bigr)+\sum _{q=1}^{n} \biggl(b_{pq}^{R}(t) \int _{0}^{\infty }K_{pq}(u) \\ & \hphantom{ (x_{p}^{I}(t) )'=}{} \times g_{q}^{J}[t,u,x]\,\mathrm{d}u +b_{pq}^{J}(t) \int _{0}^{\infty }K_{pq}(u)g_{q}^{R}[t,u,x] \,\mathrm{d}u \\ & \hphantom{ (x_{p}^{I}(t) )'=}{} -b_{pq}^{I}(t) \int _{0}^{\infty }K_{pq}(u)g_{q}^{K}[t,u,x] \,\mathrm{d}u +b_{pq}^{K}(t) \int _{0}^{\infty }K_{pq}(u) \\ & \hphantom{ (x_{p}^{I}(t) )'=}{} \times g_{q}^{I}[t,u,x]\,\mathrm{d}u \biggr)+u_{p}^{J}(t), \end{aligned}$$
(6)
$$\begin{aligned}& \bigl(x_{p}^{K}(t) \bigr)' = -c_{p}(t)x_{p}^{K}(t)+ \sum _{q=1}^{n} \bigl(a_{pq}^{R}(t)f_{q}^{K}[t,x] +a_{pq}^{K}(t)f_{q}^{R}[t,x] \\ & \hphantom{ (x_{p}^{I}(t) )'=}{} +a_{pq}^{I}(t)f_{q}^{J}[t,x] -a_{pq}^{J}(t)f_{q}^{I}[t,x] \bigr)+\sum _{q=1}^{n} \biggl(b_{pq}^{R}(t) \int _{0}^{\infty }K_{pq}(u) \\ & \hphantom{ (x_{p}^{I}(t) )'=}{} \times g_{q}^{K}[t,u,x]\,\mathrm{d}u +b_{pq}^{K}(t) \int _{0}^{\infty }K_{pq}(u)g_{q}^{R}[t,u,x] \,\mathrm{d}u \\ & \hphantom{ (x_{p}^{I}(t) )'=}{} +b_{pq}^{I}(t) \int _{0}^{\infty }K_{pq}(u)g_{q}^{J}[t,u,x] \,\mathrm{d}u -b_{pq}^{J}(t) \int _{0}^{\infty }K_{pq}(u) \\& \hphantom{ (x_{p}^{I}(t) )'=}{} \times g_{q}^{I}[t,u,x]\,\mathrm{d}u \biggr)+u_{p}^{K}(t), \end{aligned}$$
(7)
where
\(f_{q}^{l}[t,x]\triangleq f_{q}^{l} (x_{q}^{R}(t-\tau _{pq}(t)),x_{q}^{I}(t-\tau _{pq}(t)),x_{q}^{J}(t-\tau _{pq}(t)), x_{q}^{K}(t-\tau _{pq}(t)) )\),
\(g_{q}^{l}[t,u,x]\triangleq g_{q}^{l} (x_{q}^{R}(t-u),(x_{q}^{I}(t-u)),(x_{q}^{J}(t-u)),(x_{q}^{K}(t-u)) )\), and
$$\begin{aligned}& a_{pq}(t) = a_{pq}^{R}(t)+ia_{pq}^{I}(t)+ja_{pq}^{J}(t)+ka_{pq}^{K}(t), \\& b_{pq}(t) = b_{pq}^{R}(t)+ib_{pq}^{I}(t)+jb_{pq}^{J}(t)+kb_{pq}^{K}(t), \\& u_{p}(t) = u_{p}^{R}(t)+iu_{p}^{I}(t)+ju_{p}^{J}(t)+ku_{p}^{K}(t). \end{aligned}$$
According to (
4)–(
7), one can obtain that
$$\begin{aligned} X_{p}'(t) =&-c_{p}(t)X_{p}(t)+ \sum_{q=1}^{n}A_{pq}(t)F_{q}[t,x] \\ &+\sum_{q=1}^{n}B_{pq}(t) \int _{0}^{\infty }K_{pq}(u)G_{q}[t,u,x] \,\mathrm{d}u+U_{p}(t),\quad p\in \Lambda , \end{aligned}$$
(8)
where
$$\begin{aligned} A_{pq}(t)&= \left ( \textstyle\begin{array}{cccc} a_{pq}^{R}(t)&-a_{pq}^{I}(t)&-a_{pq}^{J}(t)&-a_{pq}^{K}(t) \\ a_{pq}^{I}(t)&a_{pq}^{R}(t)&-a_{pq}^{K}(t)&a_{pq}^{J}(t) \\ a_{pq}^{J}(t)&a_{pq}^{K}(t)&a_{pq}^{R}(t)&-a_{pq}^{I}(t) \\ a_{pq}^{K}(t)&-a_{pq}^{J}(t)&a_{pq}^{I}(t)&a_{pq}^{R}(t) \end{array}\displaystyle \right ) , \\ B_{pq}(t)&= \left ( \textstyle\begin{array}{cccc} b_{pq}^{R}(t)&-b_{pq}^{I}(t)&-b_{pq}^{J}(t)&-b_{pq}^{K}(t) \\ b_{pq}^{I}(t)&b_{pq}^{R}(t)&-b_{pq}^{K}(t)&b_{pq}^{J}(t) \\ b_{pq}^{J}(t)&b_{pq}^{K}(t)&b_{pq}^{R}(t)&-b_{pq}^{I}(t) \\ b_{pq}^{K}(t)&-b_{pq}^{J}(t)&b_{pq}^{I}(t)&b_{pq}^{R}(t) \end{array}\displaystyle \right ) ,\qquad X_{p}(t)= \left ( \textstyle\begin{array}{cccc} x_{p}^{R}(t)\\ x_{p}^{I}(t)\\ x_{p}^{J}(t) \\ x_{p}^{K}(t) \end{array}\displaystyle \right ) , \\ U_{p}(t) &= \left ( \textstyle\begin{array}{cccc} u_{p}^{R}(t)\\ u_{p}^{I}(t)\\ u_{p}^{J}(t) \\ u_{p}^{K}(t) \end{array}\displaystyle \right ) ,\qquad F_{q}[t,x]= \left ( \textstyle\begin{array}{cccc} f_{q}^{R}[t,x]\\ f_{q}^{I}[t,x]\\ f_{q}^{J}[t,x] \\ f_{q}^{K}[t,x] \end{array}\displaystyle \right ) ,\qquad G_{q}[t,u,x]= \left ( \textstyle\begin{array}{cccc} g_{q}^{R}[t,u,x]\\ g_{q}^{I}[t,u,x]\\ g_{q}^{J} [t,u,x]\\ g_{q}^{K}[t,u,x] \end{array}\displaystyle \right ) . \end{aligned}$$
The initial condition associated with (
8) is of the form
$$\begin{aligned} X_{p}(s)=\Phi _{p}(s),\quad p\in \Lambda , s\in (-\infty ,0], \end{aligned}$$
where
$$\begin{aligned} \Phi _{p}(s)= \bigl( \phi _{p}^{R}(s), \phi _{p}^{I}(s), \phi _{p}^{J}(s), \phi _{p}^{K}(s) \bigr) ^{T} \end{aligned}$$
and
\(\phi _{p}^{l}(s)\in BC((-\infty ,0],\mathbb{R}), p\in \Lambda , l\in E\).
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.