Almost Anti-Periodic Oscillation Excited by External Inputs and Synchronization of Clifford-Valued Recurrent Neural Networks

Abstract

The main purpose of this paper was to study the almost anti-periodic oscillation caused by external inputs and the global exponential synchronization of Clifford-valued recurrent neural networks with mixed delays. Since the space consists of almost anti-periodic functions has no vector space structure, firstly, we prove that the network under consideration possesses a unique bounded continuous solution by using the contraction fixed point theorem. Then, by using the inequality technique, it was proved that the unique bounded continuous solution is also an almost anti-periodic solution. Secondly, taking the neural network that was considered as the driving system, introducing the corresponding response system and designing the appropriate controller, some sufficient conditions for the global exponential synchronization of the driving-response system were obtained by employing the inequality technique. When the system we consider degenerated into a real-valued system, our results were considered new. Finally, the validity of the results was verified using a numerical example.

1. Introduction

A recurrent neural network is a recurrent neural network that takes sequence data as input and recurses in the evolutionary direction of the sequence, with all nodes connected in a chain. Research on recurrent neural networks started in the 1980s and 1990s and developed into one of the deep learning algorithms in the early 2000s [1]. Recurrent neural network has been applied in natural language processing, such as speech recognition, language modeling, machine translation and so on. It is also used for various time series forecasting. These applications of recurrent neural networks are closely related to their dynamic behavior. Therefore, the dynamics of recurrent neural networks have been extensively studied over the past few decades.

As a high-dimensional neural network, the Clifford-valued neural network not only includes real-valued, complex-valued and quaternary-valued neural networks as its special cases, but also has greater advantages than low-dimensional neural networks in dealing with multidimensional data. In recent years, they have attracted more and more attention [2,3,4,5,6,7,8,9,10,11].

On the one hand, it is well known that periodic and almost periodic oscillations are the focus of qualitative research on differential equations [12,13,14]. Anti-periodic oscillation is a special form of periodic oscillation, but it can reflect a particularly accurate oscillation and has many important applications, such as in interpolation problems [15,16], wavelet theory [17], neural networks [18,19,20,21,22,23,24,25,26,27], etc. In the past decade, anti-periodic oscillation has been widely studied. Recently, the concept of almost anti-periodic functions was proposed in [28], which is a generalization of the concept of anti-periodic functions. Any anti-periodic function is an almost anti-periodic function, but an almost periodic function is not necessarily an almost anti-periodic function. At the same time, it is difficult to investigate the existence of almost anti-periodic solutions to differential equations because the space consists of almost anti-periodic functions has no vector space structure. Consequently, it is significant and challenging to study the almost anti-periodic oscillation of neural networks.

On the other hand, the synchronization of neural networks has a wide range of applications in the fields of secure communication [29], image processing [30], and information science [31]. Therefore, for more than ten years, the synchronization of neural networks has become a hot issue in the research of neural network dynamics.

In the light of the above discussion and considering the ubiquitous time delay effect [3,4,5,6,32,33], in this work, we focused on the following Clifford-valued recurrent neural networks with mixed delays:

$$\begin{array}{cc}\hfill z_p^{\prime }\left(t\right)=& -a_p{z}_p\left(t\right)+\sum _{q=1}^n{b}_{pq}^0{A}_q\left(z_q\left(t\right)\right)+\sum _{q=1}^n{c}_{pq}^0{B}_q\left(z_q\left(t-{\tau }_{pq}\right)\right)\hfill \\ \hfill & +\sum _{q=1}^n{d}_{pq}^0{\int }_{-\infty }^t{O}_{pq}(t-s)g_q\left(z_q\left(s\right)\right)ds+J_p\left(t\right),\hfill \end{array}$$

(1)

where $p=1,2,\dots ,n,$$z_p\left(t\right)\in \mathcal{A}$ corresponds to the state of the pth unit at time t and $\mathcal{A}$ is a real Clifford algebra that will be defined in the next section; $A_q,B_q,g_q:\mathcal{A}\to \mathcal{A}$ are the activation functions, $b_{pq}^0,c_{pq}^0,d_{pq}^0\in \mathcal{A}$ are the connection, the discretely delayed connection and the distributively delayed connection weights between the qth neuron and the pth neuron, respectively; $J_p\left(t\right)\in \mathcal{A}$ represents the external input on the pth neuron at time t; ${\tau }_{pq}\ge 0$ corresponds the transmission delay; $a_p>0$ denotes the rate at which the pth unit resets its potential to the quiescent state when disconnected from the network and external input. The kernel function $O_{pq}:\mathbb{R}\to \mathbb{R}$ is a positive continuous integrable function with ${\int }_0^{+\infty }{O}_{pq}\left(s\right)ds=1$.

The system (1) is supplemented with the initial values

$$\left.\left.x_p\left(s\right)={\phi }_p\left(s\right)\in C(\right(-\infty ,0\right],\mathcal{A}),p=1,2,\dots ,n.$$

The main aim of this work was to investigate the existence and global exponential synchronization of almost anti-periodic solutions to the network (1). The important contribution of this paper is that this is the first paper to study the almost anti-periodic oscillation of neural networks by employing the fixed point theorem and inequality technique. The difficulty of this paper is that the sum and product of two almost anti-periodic functions are not necessarily almost anti-periodic functions, and the sum of an almost anti-periodic function plus a non-zero constant is not necessarily almost anti-periodic. This makes it difficult to study the existence of almost anti-periodic solutions of differential equations directly by using the fixed point theorem. In order to overcome this difficulty, we first prove that the network under consideration has a unique bounded continuous solution by using Banach fixed point theorem. Then it is proved that the bounded continuous solution is also an almost anti-periodic solution according to the definition and inequality technique. Therefore, the results and methods of this paper are new.

Remark 1.

In recently published papers [25,26], the existence and exponential stability of almost anti-periodic solutions of inertial neural networks on real sets and time scales were studied by constructing appropriate Lyapunov functional and inequality techniques, respectively. It is worth pointing out that the results and methods in this paper are different from those in [25,26].

The remainder of this paper is organized as follows: In Section 2, we introduce some symbols, definitions, and a preliminary lemma. In Section 3, we establish the existence and uniqueness of almost anti-periodic solutions of (1). In Section 4, we study the synchronization problem. In Section 5, we provide a numerical example to illustrate the validity of our results. Finally, we draw a conclusion in Section 6.

2. Preliminaries

Let $\mathcal{P}=\{\varnothing ,1,2,\dots ,A,\dots ,12\dots d\}$, then a real Clifford algebra over ${\mathbb{R}}^d$ can be described as

$$\mathcal{A}=\left\{\sum _{A\in \mathcal{P}}{u}^A{e}_A,u^A\in \mathbb{R}\right\},$$

in which $e_A=e_{h_1}\dots e_{h_v}$ satisfying $A=h_1\dots h_v,1\le h_1<$$h_2<\dots <h_v\le d$ and $1\le v\le d$. In addition, $e_{\varnothing }=e_0=1$ and $e_h,$$h=1,2,\dots ,d$ are called the generators of $\mathcal{A}$, which meet the multiplication rules: $e_i^2=-1,i=1,2,\dots ,d$, and $e_i{e}_j+e_j{e}_i=0,$$i\ne j,$ $i,j=1,2,\dots ,d$.

For $x={\sum }_{A\in \mathcal{P}}{x}^A{e}_A\in \mathcal{A}$, we denote ${\parallel x\parallel }_{\mathcal{A}}=\underset{A\in \mathcal{P}}{max}\left|x^A\right|$ and for $y=(y_1,y_2,\dots ,y_n)\in {\mathcal{A}}^n$, we denote ${\parallel y\parallel }_{{\mathcal{A}}^n}=\underset{1\le i\le n}{max}{\parallel y_i\parallel }_{\mathcal{A}}$, then $({\mathcal{A}}^n,\parallel ·{\parallel }_{{\mathcal{A}}^n})$ is a Banach space.

Let $BC(\mathbb{R},{\mathcal{A}}^n)$ be the collection of bounded and continuous functions $f:\mathbb{R}\to {\mathcal{A}}^n$, then $BC(\mathbb{R},{\mathcal{A}}^n)$ with the norm ${\parallel \theta \parallel }_0=\underset{t\in \mathbb{R}}{sup}{\parallel \theta \left(t\right)\parallel }_{{\mathcal{A}}^n}$ where $\theta \in BC(\mathbb{R},{\mathcal{A}}^n)$ is a Banach space.

Definition 1

([34]). A function $f\in BC(\mathbb{R},{\mathcal{A}}^n)$ is said to be almost periodic if for each $ϵ>0$, there exists an $l\left(ϵ\right)>0$ such that in every interval with length l contains a number τ satisfying

$${\parallel f(t+\tau )-f\left(t\right)\parallel }_{{\mathcal{A}}^n}<ϵ,\forall t\in \mathbb{R}.$$

We will denote by $AP(\mathbb{R},{\mathcal{A}}^n)$ the set of all such functions.

Definition 2

([28]). A function $f\in BC(\mathbb{R},{\mathcal{A}}^n)$ is said to be almost anti-periodic if for each $ϵ>0$, there exists an $l\left(ϵ\right)>0$ such that in every interval with length l contains a number τ satisfying

$${\parallel f(t+\tau )+f\left(t\right)\parallel }_{{\mathcal{A}}^n}<ϵ,\forall t\in \mathbb{R}.$$

We will denote by $AAP(\mathbb{R},{\mathcal{A}}^n)$ the collection of all such functions.

Example 1.

$\left(a\right)$

Let $f\left(t\right)=cos\left(\pi t\right)+cos\left(\sqrt{2}\pi t\right)$, then one can easily show that f is not periodic or anti-periodic but almost anti-periodic.

$\left(b\right)$

Let $g\left(t\right)=cos\left(\pi t\right)+cos\left(\sqrt{2}\pi t\right)+3$, then one can easily show that $g\left(t\right)$ is almost periodic, not almost anti-periodic.

From Definitions 1 and 2 and Example 1, one can conclude that

$$AAP(\mathbb{R},{\mathcal{A}}^n)⫋AP(\mathbb{R},{\mathcal{A}}^n).$$

Definition 3.

Function $f=(f_1,f_2,\dots ,f_n):\mathbb{R}\to {\mathcal{A}}^n$ is called almost anti-periodic if for every $i=1,2,\dots ,n$, $f_i$ is almost anti-periodic.

The symbols we use are as follows:

$$b_{pq}=\parallel b_{pq}^0{\parallel }_{\mathcal{A}},c_{pq}=\parallel c_{pq}^0{\parallel }_{\mathcal{A}},d_{pq}={\parallel d_{pq}^0\parallel }_{\mathcal{A}}.$$

We need the following hypothesises to prove our main results:

$\left({\mathrm{H}}_1\right)$

For $p,q=1,2,\dots ,n$, $a_p,{\tau }_{pq}>0$, $J_p\in ANP(\mathbb{R},\mathcal{A})$ and $O_{pq}$ are continuous and integrable such that ${\int }_0^{+\infty }{O}_{pq}\left(s\right)ds=1$.

$\left({\mathrm{H}}_2\right)$

For $p=1,2,\dots ,n$, there exist positive constants $L_p^A$, $L_p^B$ and $L_p^g$ such that for all $x,y\in \mathcal{A}$,

$$\begin{array}{cc}\hfill & ∥A_p\left(x\right)\pm A_p\left(y\right)∥\le L_p^A∥x\pm y∥,∥B_p\left(x\right)\pm B_p\left(y\right)∥\le L_p^B∥x\pm y∥,\hfill \\ \hfill & ∥g_p\left(x\right)\pm g_p\left(y\right)∥\le L_p^g∥x\pm y∥,A_p\left(0\right)=B_p\left(0\right)=g_p\left(0\right)=0.\hfill \end{array}$$

$\left({\mathrm{H}}_3\right)$

${\gamma }_p:=\frac{1}{a_p}{\displaystyle \sum _{q=1}^n}\left(L_q^A{b}_{pq}+L_q^B{c}_{pq}+L_q^g{d}_{pq}\right)$ and $\gamma :=\underset{1\le p\le n}{max}\left\{{\gamma }_p\right\}<1$.

It is easy to show that

Lemma 1.

If $\zeta ={\left({\zeta }_1,{\zeta }_2,\dots ,{\zeta }_n\right)}^T$ is a bounded solution of system (1), then ζ satisfies the following integral equation:

$$\begin{array}{cc}\hfill {\zeta }_p\left(t\right)=& {\int }_{-\infty }^t{e}^{-a_p(t-s)}[\sum _{q=1}^n{b}_{pq}^0{A}_q\left({\zeta }_q\left(s\right)\right)+\sum _{q=1}^n{c}_{pq}^0{B}_q\left({\zeta }_q\left(s-{\tau }_{pq}\right)\right)\hfill \\ \hfill & +\sum _{q=1}^n{d}_{pq}^0{\int }_{-\infty }^s{O}_{pq}(s-u)g_q\left({\zeta }_q\left(u\right)\right)du+J_p\left(s\right)]ds,p=1,2,\dots ,n,\hfill \end{array}$$

(2)

and vice versa.

3. Almost Anti-Periodic Solutions

Take a positive constant $\omega$ such that $\omega >\parallel {\phi }^0{\parallel }_0$, where

$${\phi }^0\left(t\right)=\left\{{\int }_{-\infty }^t{e}^{-a_1(t-s)}{J}_1\left(s\right)ds,\dots ,{\int }_{-\infty }^t{e}^{-a_n(t-s)}{J}_n\left(s\right)ds\right\}.$$

Then we have

Theorem 1.

Let hypothesises $\left(H_1\right)$–$\left(H_3\right)$ be fulfilled. Then system (1) admits a unique almost anti-periodic solution in

$${\mathbb{C}}_0=\left\{\phi ={\left({\phi }_1,{\phi }_2,\dots ,{\phi }_n\right)}^T\in BC\left(\mathbb{R},{\mathcal{A}}^n\right)|{∥\phi -{\phi }^0∥}_0\le \frac{\gamma \omega }{1-\gamma }\right\}.$$

Proof. 

Consider the operator ${\rm Y}:{\mathbb{C}}_0\to BC\left(\mathbb{R},{\mathcal{A}}^n\right)$ defined by

$$\left({\rm Y}\varphi \right)\left(t\right)={\left({\left({\rm Y}\varphi \right)}_1\left(t\right),{\left({\rm Y}\varphi \right)}_2\left(t\right),\dots ,{\left({\rm Y}\varphi \right)}_n\left(t\right)\right)}^T,$$

where $\varphi =\left({\varphi }_1,{\varphi }_2,\dots ,{\varphi }_n\right)\in {\mathbb{C}}_0$, for any $t\in \mathbb{R}$ and $p=1,2,\dots ,n$,

$$\begin{array}{cc}\hfill {\left({\rm Y}\varphi \right)}_p\left(t\right)=& {\int }_{-\infty }^t{e}^{-a_p(t-s)}[\sum _{q=1}^n{b}_{pq}^0{A}_q\left({\varphi }_q\left(s\right)\right)+\sum _{q=1}^n{c}_{pq}^0{B}_q\left({\varphi }_q\left(s-{\tau }_{pq}\right)\right)\hfill \\ \hfill & +\sum _{q=1}^n{d}_{pq}^0{\int }_{-\infty }^s{O}_{pq}(s-u)g_q\left({\varphi }_q\left(u\right)\right)du+J_p\left(s\right)]ds.\hfill \end{array}$$

Then, by implementing a standard reasoning process, one can show that ${\rm Y}$ maps ${\mathbb{C}}_0$ into ${\mathbb{C}}_0$.

Next, we show that ${\rm Y}$ has a fixed point in ${\mathbb{C}}_0$. Indeed, for any $\phi ,\psi \in {\mathbb{C}}_0$, one infers that

$$\begin{array}{cc}\hfill & {∥\left({\rm Y}\phi \right)-\left({\rm Y}\psi \right)∥}_0\hfill \\ \hfill \le & \underset{1\le p\le n}{max}\left\{{\int }_{-\infty }^t{e}^{-a_p(t-s)}\right[\sum _{q=1}^n{b}_{pq}{L}_q^A{∥\phi -\psi ∥}_0+\sum _{q=1}^n{c}_{pq}{L}_q^B{∥\phi -\psi ∥}_0\hfill \\ \hfill & +\sum _{q=1}^n{d}_{pq}{\int }_{-\infty }^s{O}_{pq}(s-u)L_q^g{∥\phi -\psi ∥}_{0}du\left]ds\right\}\hfill \\ \hfill \le & \underset{1\le p\le n}{max}\left\{\frac{1}{a_p}\sum _{q=1}^n\left(L_q^A{b}_{pq}+L_q^B{c}_{pq}+L_q^g{d}_{pq}\right){∥\phi -\psi ∥}_0\right\}\hfill \\ \hfill \le & \gamma {∥\phi -\psi ∥}_0,\hfill \end{array}$$

by $\left({\mathrm{H}}_3\right)$ and invoking the Banach fixed point theorem, ${\rm Y}$ has a unique fixed point $\widehat{x}=({\widehat{x}}_1,{\widehat{x}}_2,\dots ,{\widehat{x}}_n)$ in ${\mathbb{C}}_0$, hence, $\widehat{x}$ is a unique bounded continuous solution of (1).

Finally, we show that the unique solution $\widehat{x}$ of (1) is almost anti-periodic. We can rewrite (2) as

$$\begin{array}{cc}\hfill z_p\left(t\right)=& {\int }_0^{+\infty }{e}^{-a_{p}h}[\sum _{q=1}^n{b}_{pq}^0{A}_q\left(z_q(t-h)\right)+\sum _{q=1}^n{c}_{pq}^0{B}_q\left(z_q\left(t-h-{\tau }_{pq}\right)\right)\hfill \\ \hfill & +\sum _{q=1}^n{d}_{pq}^0{\int }_0^{+\infty }{O}_{pq}\left(v\right)g_q\left(z_q(t-h-v)\right)dv+J_p(t-h)]dh,p=1,2,\dots ,n.\hfill \end{array}$$

(3)

Since $J_p\in ANP(\mathbb{R},{\mathcal{A}}^n)$ for $p=1,2,\dots ,n$, for each $ϵ>0$, there exists ${\sigma }_p\left(ϵ\right)$ such that

$${∥J_p(t+{\sigma }_p)+J_p\left(t\right)∥}_{\mathcal{A}}<ϵ,t\in \mathbb{R}.$$

(4)

Since x is a solution of (1), by (3), we can obtain

$$\begin{array}{cc}\hfill & {∥{\widehat{x}}_p(t+{\sigma }_p)+{\widehat{x}}_p\left(t\right)∥}_{\mathcal{A}}\hfill \\ \hfill \le & {\int }_0^{+\infty }{e}^{-a_{p}s}[\sum _{q=1}^n{b}_{pq}{∥A_q\left({\widehat{x}}_q(t+{\sigma }_p-s)\right)+A_q\left({\widehat{x}}_q(t-s)\right)∥}_{\mathcal{A}}\hfill \\ \hfill & +\sum _{q=1}^n{c}_{pq}{∥B_q\left({\widehat{x}}_q\left(t+{\sigma }_p-s-{\tau }_{pq}\right)\right)+B_q\left({\widehat{x}}_q\left(t-s-{\tau }_{pq}\right)\right)∥}_{\mathcal{A}}\hfill \\ \hfill & +\sum _{q=1}^n{d}_{pq}{\int }_0^{+\infty }{O}_{pq}\left(v\right){∥g_q\left({\widehat{x}}_q(t+{\sigma }_p-s-v)\right)+g_q\left({\widehat{x}}_q(t-s-v)\right)∥}_{\mathcal{A}}dv\hfill \\ \hfill & +{∥J_p(t-s+{\sigma }_p)+J_p(t-s)∥}_{\mathcal{A}}]ds\hfill \\ \hfill <& \frac{ϵ}{a_p}+{\int }_0^{+\infty }{e}^{-a_{p}s}[\sum _{q=1}^n{b}_{pq}{L}_q^A{∥{\widehat{x}}_q(t+{\sigma }_p-s)+{\widehat{x}}_q(t-s)∥}_{\mathcal{A}}\hfill \\ \hfill & +\sum _{q=1}^n{c}_{pq}{L}_q^B{∥{\widehat{x}}_q\left(t+{\sigma }_p-s-{\tau }_{pq}\right)+{\widehat{x}}_q\left(t-s-{\tau }_{pq}\right)∥}_{\mathcal{A}}\hfill \\ \hfill & +\sum _{q=1}^n{d}_{pq}{\int }_0^{+\infty }{O}_{pq}\left(v\right)L_q^g{∥{\widehat{x}}_q(t+{\sigma }_p-s-v)+{\widehat{x}}_q(t-s-v)∥}_{\mathcal{A}}dv]ds,p=1,2,\dots ,n.\hfill \end{array}$$

Define $V_p=\underset{t\in \mathbb{R}}{sup}{∥{\widehat{x}}_p(t+{\sigma }_p)+{\widehat{x}}_p\left(t\right)∥}_{\mathcal{A}}$, from the above inequality, we have

$$\begin{array}{cc}\hfill V_p\le & \frac{ϵ}{a_p}+V\frac{1}{a_p}\sum _{q=1}^n\left[b_{pq}{L}_q^A+c_{pq}{L}_q^B+d_{pq}{L}_q^g\right]\hfill \\ \hfill \le & \frac{ϵ}{a_p}+{\gamma }_p{V}_p,p=1,2,\dots ,n,\hfill \end{array}$$

as a result, we deduce that

$${∥{\widehat{x}}_p(t+{\sigma }_p)+{\widehat{x}}_p\left(t\right)∥}_{\mathcal{A}}\le V_p\le \frac{ϵ}{a_p}\frac{1}{1-{\gamma }_p},p=1,2,\dots ,n,$$

which implies that $\widehat{x}$ is almost anti-periodic. This ends the proof. □

4. Synchronization

Let us consider the following system as the response system:

$$\begin{array}{cc}\hfill y_p^{\prime }\left(t\right)=& -a_p{y}_p\left(t\right)+\sum _{q=1}^n{b}_{pq}^0{A}_q\left(y_q\left(t\right)\right)+\sum _{q=1}^n{c}_{pq}^0{B}_q\left(y_q\left(t-{\tau }_{pq}\right)\right)\hfill \\ \hfill & +\sum _{q=1}^n{d}_{pq}^0{\int }_{-\infty }^t{O}_{pq}(t-s)g_q\left(y_q\left(s\right)\right)ds+J_p\left(t\right)+{\epsilon }_p\left(t\right),p=1,2,\dots ,n,\hfill \end{array}$$

(5)

where $y_p\left(t\right)\in \mathcal{A}$ presents the state variable of the system, ${\epsilon }_p\left(t\right)\in \mathcal{A}$ presents a controller, the rest of the symbols are the same as those appearing in system (1).

The initial values of system (5) are

$$y_p\left(s\right)={\psi }_p\left(s\right)\in C((-\infty ,0],\mathcal{A}),p=1,2,\dots ,n.$$

Let ${\Lambda }_p=y_p-z_p$, then it follows from systems (1) and (5) that the error system is

$$\begin{array}{cc}\hfill {\Lambda }_p^{\prime }\left(t\right)=& -a_p{\Lambda }_p\left(t\right)+\sum _{q=1}^n{b}_{pq}^0\left[A_q\left({\Lambda }_q\left(t\right)+z_q\left(t\right)\right)-A_q\left(z_q\left(t\right)\right)\right]\hfill \\ \hfill & +\sum _{q=1}^n{c}_{pq}^0\left[B_q\left({\Lambda }_q\left(t-{\tau }_{pq}\right)+z_q\left(t-{\tau }_{pq}\right)\right)-B_q\left(z_q\left(t-{\tau }_{pq}\right)\right)\right]\hfill \\ \hfill & +\sum _{q=1}^n{d}_{pq}^0{\int }_{-\infty }^t{O}_{pq}(t-s)\left[g_q\left({\Lambda }_q\left(s\right)+z_q\left(s\right)\right)-g_q\left(z_q\left(s\right)\right)\right]ds\hfill \\ \hfill & -{\sigma }_p{\Lambda }_p\left(t\right)+\sum _{q=1}^n{\mu }_{pq}^0{h}_q\left({\Lambda }_q\left(t-{\alpha }_{pq}\right)\right),\hfill \end{array}$$

(6)

with the initial values

$${\Lambda }_p\left(s\right)={\psi }_p\left(s\right)-{\phi }_p\left(s\right),s\in (-\infty ,0],p=1,2,\dots ,n.$$

The following nonlinear state-dependent controller is designed:

$${\epsilon }_p\left(t\right)=-{\sigma }_p{\Lambda }_p\left(t\right)+\sum _{q=1}^n{\mu }_{pq}^0{h}_q\left({\Lambda }_q\left(t-{\alpha }_{pq}\right)\right),$$

where $p=1,2,\dots ,n,$${\sigma }_p,{\alpha }_{pq}>0,{\mu }_{pq}^0\in \mathcal{A},h_q:\mathcal{A}\to \mathcal{A}$.

Definition 4.

Systems (1) and (5) are called to be globally exponentially synchronized, if there are positive constants ξ and M such that

$${\parallel y\left(t\right)-x\left(t\right)\parallel }_{{\mathcal{A}}^n}\le M{\parallel \psi -\phi \parallel }_{\tau }{e}^{-\xi t},\forall t>0,$$

where

$${\parallel \psi -\phi \parallel }_{\tau }=\underset{t\in (-\infty ,0]}{sup}{\parallel {\psi }_p\left(t\right)-{\phi }_p\left(t\right)\parallel }_{{\mathcal{A}}^n}.$$

Theorem 2.

Let $\left(H_1\right)$–$\left(H_3\right)$ hold. If the following conditions are fulfilled:

$\left({\mathrm{H}}_4\right)$

For $p,q=1,2,\dots ,n$, ${\sigma }_p,{\alpha }_{pq}>0$.

$\left({\mathrm{H}}_5\right)$

For $p=1,2,\dots ,n$, there exist positive constants $L_p^h$ such that for all $x,y\in \mathcal{A}$,

$$\begin{array}{cc}\hfill & {∥h_p\left(x\right)-h_p\left(y\right)∥}_{\mathcal{A}}\le L_p^h{∥x-y∥}_{\mathcal{A}},h_p\left(0\right)=0.\hfill \end{array}$$

$\left({\mathrm{H}}_6\right)$

For $p,q=1,2,\dots ,n$, denote ${\mu }_{pq}={\parallel {\mu }_{pq}^0\parallel }_{\mathcal{A}}$, then

$$\underset{1\le p\le n}{max}\left\{\frac{1}{a_p+{\sigma }_p}\sum _{q=1}^n\left(b_{pq}{L}_q^A+c_{pq}{L}_q^B+d_{pq}{L}_q^g+{\mu }_{pq}{L}_q^h\right)\right\}<1.$$

Then systems (1) and (5) are globally exponentially synchronized.

Proof. 

From (6), for $p=1,2,\dots ,n$, we find

$$\begin{array}{cc}\hfill {\Lambda }_p\left(t\right)=& {\int }_{-\infty }^t{e}^{-(a_p+{\sigma }_p)(t-s)}[\sum _{q=1}^n{b}_{pq}^0\left[A_q\left({\Lambda }_q\left(s\right)+z_q\left(s\right)\right)-A_q\left(z_q\left(s\right)\right)\right]\hfill \\ \hfill & +\sum _{q=1}^n{c}_{pq}^0\left[B_q\left({\Lambda }_q\left(s-{\tau }_{pq}\right)+z_q\left(s-{\tau }_{pq}\right)\right)-B_q\left(z_q\left(s-{\tau }_{pq}\right)\right)\right]\hfill \\ \hfill & +\sum _{q=1}^n{d}_{pq}^0{\int }_{-\infty }^s{O}_{pq}(s-u)\left[g_q\left({\Lambda }_q\left(u\right)+z_q\left(u\right)\right)-g_q\left(z_q\left(u\right)\right)\right]du\hfill \\ \hfill & +\sum _{q=1}^n{\mu }_{pq}^0{h}_q\left({\Lambda }_q\left(s-{\alpha }_{pq}\right)\right)]ds.\hfill \end{array}$$

(7)

For $p=1,2,\dots ,n$, set

$$N_p\left(\lambda \right)=a_p+{\sigma }_p-\lambda -{\Xi }_p\left(\lambda \right),$$

where ${\Xi }_p\left(\lambda \right)={\displaystyle \sum _{q=1}^n}\left(b_{pq}{L}_q^A+c_{pq}{L}_q^B{e}^{\lambda {\tau }_{pq}}+d_{pq}{L}_q^g{\int }_0^{+\infty }{O}_{pq}\left(s\right)e^{\lambda s}ds+{\mu }_{pq}{L}_q^h{e}^{\lambda {\alpha }_{pq}}\right)$.

For $p=1,2,\dots ,n$, we have $N_p\left(0\right)>0$ by $\left({\mathrm{H}}_6\right)$ and $N_p\left(\lambda \right)\to -\infty$ as $\lambda \to +\infty$ by the definition of $N_p\left(\lambda \right)$. Hence, by the continuity of $N_p\left(\lambda \right)$, we can choose a positive constant $0<\xi <\underset{1\le p\le n}{min}\{a_p+{\sigma }_p\}$ such that $N_p\left(\xi \right)>0,p=1,2,\dots ,n$, which implies that

$$\frac{{\Xi }_p\left(\xi \right)}{a_p+{\sigma }_p-\xi }<1.$$

From $\left({\mathrm{H}}_6\right)$, we obtain

$$M:=\underset{1\le p\le n}{max}\left\{\left(a_p+{\sigma }_p\right){\left[{\Xi }_p\left(0\right)\right]}^{-1}\right\}>1,$$

then

$$\begin{array}{c}\hfill \frac{1}{M}\le \frac{{\Xi }_p\left(0\right)}{a_p+{\sigma }_p}<\frac{{\Xi }_p\left(\xi \right)}{a_p+{\sigma }_p-\xi }<1.\end{array}$$

For any $ϵ>0$, it is obvious that

$${\parallel \Lambda \left(t\right)\parallel }_{{\mathcal{A}}^n}<M\left({\parallel \psi -\phi \parallel }_{\tau }+ϵ\right)e^{-\xi t},t\in (-\infty ,0].$$

We ascertain that

$${\parallel \Lambda \left(t\right)\parallel }_{{\mathcal{A}}^n}<M\left({\parallel \psi -\phi \parallel }_{\tau }+ϵ\right)e^{-\xi t},t\in [0,+\infty ).$$

(8)

If not, then there is a $\widehat{t}>0$ such that

$$\parallel \Lambda \left(\widehat{t}\right){\parallel }_{{\mathcal{A}}^n}=M\left({\parallel \psi -\phi \parallel }_{\tau }+ϵ\right)e^{-\xi \widehat{t}}$$

(9)

and

$${\parallel \Lambda \left(t\right)\parallel }_{{\mathcal{A}}^n}<M\left({\parallel \psi -\phi \parallel }_{\tau }+ϵ\right)e^{-\xi t},t\in (-\infty ,\widehat{t}).$$

Hence, from (7), one obtains

$$\begin{array}{cc}\hfill {∥{\Lambda }_p\left(\widehat{t}\right)∥}_{\mathcal{A}}\le & {\int }_{-\infty }^{\widehat{t}}{e}^{-(a_p+{\sigma }_p)(\widehat{t}-s)}[\sum _{q=1}^n{b}_{pq}{L}_q^A{∥{\Lambda }_q\left(s\right)∥}_{\mathcal{A}}+\sum _{q=1}^n{c}_{pq}{L}_q^B{∥{\Lambda }_q\left(s-{\tau }_{pq}\right)∥}_{\mathcal{A}}\hfill \\ \hfill & +\sum _{q=1}^n{d}_{pq}{L}_q^g{\int }_{-\infty }^s{O}_{pq}(s-u){∥{\Lambda }_q\left(u\right)∥}_{\mathcal{A}}du+\sum _{q=1}^n{\mu }_{pq}{L}_q^h{∥{\Lambda }_q\left(s-{\alpha }_{pq}\right)∥}_{\mathcal{A}}]ds\hfill \\ \hfill \le & M\left({\parallel \psi -\phi \parallel }_{\tau }+ϵ\right)\times {\int }_0^{+\infty }{e}^{-(a_p+{\sigma }_p)h}[\sum _{q=1}^n{b}_{pq}{L}_q^A{e}^{-\xi (\widehat{t}-h)}\hfill \\ \hfill & +\sum _{q=1}^n{c}_{pq}{L}_q^B{e}^{-\xi \left(\widehat{t}-h-{\tau }_{pq}\right)}+\sum _{q=1}^n{d}_{pq}{L}_q^g{\int }_{-\infty }^{\widehat{t}-h}{O}_{pq}(\widehat{t}-h-u)e^{-\xi u}du\hfill \\ \hfill & +\sum _{q=1}^n{\mu }_{pq}{L}_q^h{e}^{-\xi \left(\widehat{t}-h-{\alpha }_{pq}\right)}]dh\hfill \\ \hfill =& M\left({\parallel \psi -\phi \parallel }_{\tau }+ϵ\right)e^{-\xi \widehat{t}}{\int }_0^{+\infty }{e}^{-(a_p+{\sigma }_p-\xi )h}{\Xi }_p\left(\xi \right)dh\hfill \\ \hfill \le & M\left({\parallel \psi -\phi \parallel }_{\tau }+ϵ\right)e^{-\xi \widehat{t}}\frac{{\Xi }_p\left(\xi \right)}{a_p+{\sigma }_p-\xi }\hfill \\ \hfill <& M\left({\parallel \psi -\phi \parallel }_{\tau }+ϵ\right)e^{-\xi \widehat{t}}.\hfill \end{array}$$

Hence,

$${∥\Lambda \left(\widehat{t}\right)∥}_{{\mathcal{A}}^n}<M\left({\parallel \psi -\phi \parallel }_{\tau }+ϵ\right)e^{-\xi \widehat{t}},$$

which contradicts the equality (9), and so (8) holds. Letting $ϵ\to 0^{+}$, then

$${\parallel \Lambda \left(t\right)\parallel }_{{\mathcal{A}}^n}\le M{\parallel \psi -\phi \parallel }_{\tau }{e}^{-\xi t}.$$

This completes the proof. □

5. Illustrative Example

Example 2.

In systems (1) and (5), let $d=n=2$, and for $i,j=1,2$, take

$$\begin{array}{cc}\hfill A_q\left(x_q\right)=& 0.2e_{0}sin\left(x_q^0\right)+0.25e_{1}sin\left(x_q^1\right)+0.25e_{2}arctan\left(x_q^2\right)+0.125e_{12}sin\left(x_q^{12}\right),\hfill \\ \hfill B_q\left(x_q\right)=& 0.1e_0{x}_q^0+0.2e_{1}arctan\left(x_q^1\right)+0.125e_{2}sin\left(x_q^2\right)+0.2e_{12}sin\left(x_q^{12}\right),\hfill \\ \hfill g_q\left(x_q\right)=& 0.5e_{0}sin\left(x_q^0\right)+0.36e_{1}sin\left(x_q^1\right)+0.01e_{2}sin\left(x_q^2\right)+0.26e_{12}sin\left(x_q^{12}\right),\hfill \\ \hfill h_q\left(x_q\right)=& 0.05e_0{x}_q^0+0.1e_{1}sin\left(x_q^1\right)+0.125e_2{x}_q^2+0.04e_{12}arctan\left(x_q^{12}\right),\hfill \\ \hfill b_{pq}^0=& 0.03e_{0}cos\left(\sqrt{2}\right)+0.04e_{1}sin\left(\sqrt{2}\right)+0.05e_{2}sin\left(3\right)+0.02e_{12}cos\left(\sqrt{3}\right),\hfill \\ \hfill c_{pq}^0=& 0.01e_{0}sin\left(\sqrt{3}\right)+0.02e_{1}sin\left(\sqrt{5}\right)+0.01e_{2}cos\left(\sqrt{2}\right)+0.04e_{12}sin\left(\sqrt{6}\right),\hfill \\ \hfill d_{pq}^0=& 0.015e_{0}sin\left(\sqrt{3}\right)+0.022e_{1}sin\left(3\right)+0.03e_{2}cos\left(\sqrt{5}\right)+0.02e_{12}cos\left(\sqrt{2}\right),\hfill \\ \hfill {\mu }_{pq}^0=& 0.02e_{0}cos\left(\sqrt{3}\right)+0.02e_{1}cos\left(2\right)+0.04e_{2}sin\left(\sqrt{2}\right)+0.03e_{12}sin\left(2\right),\hfill \\ \hfill J_1\left(t\right)=& 0.04e_0\left(cos\left(\pi t\right)+cos\left(\sqrt{2}\pi t\right)\right)+0.01e_1\left(cos\left(\pi t\right)+cos\left(\sqrt{2}\pi t\right)\right)\hfill \\ \hfill & +0.03e_2\left(cos\left(\pi t\right)+cos\left(\sqrt{2}\pi t\right)\right)+0.02e_{12}\left(cos\left(\pi t\right)+cos\left(\sqrt{2}\pi t\right)\right),\hfill \\ \hfill J_2\left(t\right)=& 0.02e_0\left(sin\left(\pi t\right)+sin\left(\sqrt{2}\pi t\right)\right)+0.03e_1\left(sin\left(\pi t\right)+sin\left(\sqrt{2}\pi t\right)\right)\hfill \\ \hfill & +0.02e_2\left(sin\left(\pi t\right)+sin\left(\sqrt{2}\pi t\right)\right)+0.05e_{12}\left(sin\left(\pi t\right)+sin\left(\sqrt{2}\pi t\right)\right),\hfill \\ \hfill a_p=& 0.7,{\sigma }_p=0.6,{\tau }_{pq}=3+0.0078|sin\left(\pi \right)|,O_{pq}\left(t\right)=e^{-t},{\alpha }_{pq}=0.3\left|cos\left(\pi \right)\right|.\hfill \end{array}$$

By simple calculations, we have

$$\begin{array}{cc}\hfill & L_q^A=0.25,L_q^B=0.2,L_q^g=0.5,L_q^h=0.125,b_{pq}=0.05,\hfill \\ \hfill & c_{pq}=0.04,d_{pq}=0.03,{\mu }_{pq}=0.04,\gamma \approx 0.062<1.\hfill \end{array}$$

Take $\omega =10$, then, conditions $\left({\mathrm{H}}_1\right)$–$\left({\mathrm{H}}_6\right)$ are verified. From Theorems 1 and 2, system (1) possesses a unique almost anti-periodic solution and (1)–(5) are globally exponentially synchronized (see Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5).

6. Conclusions

In this paper, a novel approach was used to investigate the existence of almost anti-periodic solutions excited by external inputs and the global exponential synchronization for a class of Clifford-valued neural networks. Even though this paper considered usual real-valued neural networks, our results are new. The approach used in this paper can be employed to study the existence of almost anti-periodic solutions to other forms of neural networks.