The dynamics of octonion-valued neutral type high-order Hopfield neural networks with D operator

Abstract

In this paper, the existence, uniqueness and global exponential stability of pseudo almost periodic solutions for a class of octonion-valued neutral type high-order Hopfield neural network models with D operator are established by using the Banach fixed point theorem and differential inequality techniques. Compared with most existing models, in this class of networks, all connection weights and activation functions are assumed to be octonion-valued functions except for time delays. And unlike most of the existing methods of studying octonion-valued neural networks, our method is a non-decomposition method, that is, the method of directly studying octonion-valued systems. The results and methods in this paper are new. In addition, an example and its numerical simulation are given to illustrate the feasibility of our results.

Keywords

Octonion neutral type neural network pseudo almost periodic solution global exponential stability

1 Introduction

In recent years, the study of various algebra-valued neural networks, such as complex-valued [1 –4], quaternion-valued [5 –8], and Clifford-valued neural networks [9 –13], has aroused increasing interest among researchers. This is because these neural networks are superior to real-valued neural networks in dealing with high dimensional data and geometric transformation [14 –19]. Octonions are a non-commutative and non-associative algebra, a generalization of complex numbers and quaternions, and are not contained by Clifford algebras [20]. Octonions have applications in fields such as geometry [21], physics [22], and signal processing [23]. Octonion-valued neural networks have potential applications in signal processing and high-dimensional and multi-layer data processing. Since Popa [24] first proposed octonion-valued neural networks, some scholars have devoted themselves to the study of the dynamics of octonion-valued neural networks [25 –32]. And some potential applications of octonion-valued neural networks have been continuously discovered [33 –35]. However, octonions are neither commutative nor associative. This makes it difficult to study the dynamics of octonion-valued neural networks. Currently, results from octonion-valued neural networks are rare. Therefore, it has important theoretical significance and potential application value to study various dynamic behaviors of octonion-valued neural networks.

On the one hand, it is well known that time delays are unavoidable in real systems. Therefore, the neural network systems described by functional differential equations are more reasonable than those described by ordinary differential equations. Hopfield neural networks were originally expressed by ordinary differential equations and are an important class of neural network models. Due to the fact that high-order Hopfield neural networks are superior to low-order Hopfield neural networks in terms of storage capacity and fault tolerance, in recent years, high-order Hopfield neural networks described by functional differential equations have gradually become a research hotspot. Consequently, many scholars have conducted extensive research on high-order Hopfield neural networks governed by functional differential equations [36 –42]. Moreover, in many practical applications of neural networks, it is more efficient to use a neutral type neural network with D operator than a neural network without D operator [43 –46]. Therefore, the study of neutral neural network with D operator is of great significance in practice.

On the other hand, the existence and stability of almost periodic solutions are important dynamics of neural networks. But so far, there is no paper published on the existence of almost periodic solutions to octonion-valued neutral type high-order Hopfield neural networks with D operator.

Inspired by the above discussion, the main purpose of this paper is to study the existence and global exponential stability of pseudo almost periodic solutions of octonion-valued neutral type high-order Hopfield neural networks with D operator by direct method.

The main contributions of this paper are as follows:

1. This is the first paper to study the existence and global exponential stability of pseudo almost periodic solutions of octonion-valued neutral type higher-order Hopfield neural networks with D operator by direct methods.

2. The method in this paper can be used to study the existence and stability of almost periodic solutions and almost automorphic solutions of other types of octonion-valued neural networks.

The rest of this paper is organized as follows. In Section 2, we introduce some basic definitions and some known results used in this paper, and give a description of the model. In Section 3, we investigate the existence of pseudo-almost periodic solutions for octonion-valued neutral type high-order Hopfield neural networks with the D operator. In Section 4, we study the global exponential stability of pseudo almost periodic solutions. In Section 5, we give a numerical example to illustrate the feasibility of our results. In Section 6, we give a brief conclusion.

2 Model description and preliminaries

The algebra of octonions is defined as

$O = {x = \sum_{l = 0}^{7} x^{l} e_{l} | x^{0}, x^{1}, x^{2}, x^{3}, x^{4}, x^{5}, x^{6}, x^{7} \in ℝ},$

where e_l, 0 ≤ l ≤ 7, represent the octonion units and meet the following multiplication table [22]:

By Table 1, we can obtain that e_ie_j = - e_je_i ≠ e_je_i for all i ≠ j, 0 < i, j ≤ 7, that is, $O$ is not commutative. And (e_ie_j) e_k = - e_i (e_je_k) ≠ e_i (e_je_k) for i, j, k distinct, 0 < i, j, k ≤ 7, that is, $O$ is also not associative.

Table 1
The octonion units’ multiplication table

× e ₀ e ₁ e ₂ e ₃ e ₄ e ₅ e ₆ e ₇

e ₀ e ₀ e ₁ e ₂ e ₃ e ₄ e ₅ e ₆ e ₇

e ₁ e ₁ -e₀ e ₃ -e₂ e ₅ -e₄ -e₇ e ₆

e ₂ e ₂ -e₃ -e₀ e ₁ e ₆ e ₇ -e₄ -e₅

e ₃ e ₃ e ₂ -e₁ -e₀ e ₇ -e₆ e ₅ -e₄

e ₄ e ₄ -e₅ -e₆ -e₇ -e₀ e ₁ e ₂ e ₃

e ₅ e ₅ e ₄ -e₇ e ₆ -e₁ -e₀ -e₃ e ₂

e ₆ e ₆ e ₇ e ₄ -e₅ -e₂ e ₃ -e₀ -e₁

e ₇ e ₇ -e₆ e ₅ e ₄ -e₃ -e₂ e ₁ -e₀

×	e ₀	e ₁	e ₂	e ₃	e ₄	e ₅	e ₆	e ₇
e ₀	e ₀	e ₁	e ₂	e ₃	e ₄	e ₅	e ₆	e ₇
e ₁	e ₁	-e₀	e ₃	-e₂	e ₅	-e₄	-e₇	e ₆
e ₂	e ₂	-e₃	-e₀	e ₁	e ₆	e ₇	-e₄	-e₅
e ₃	e ₃	e ₂	-e₁	-e₀	e ₇	-e₆	e ₅	-e₄
e ₄	e ₄	-e₅	-e₆	-e₇	-e₀	e ₁	e ₂	e ₃
e ₅	e ₅	e ₄	-e₇	e ₆	-e₁	-e₀	-e₃	e ₂
e ₆	e ₆	e ₇	e ₄	-e₅	-e₂	e ₃	-e₀	-e₁
e ₇	e ₇	-e₆	e ₅	e ₄	-e₃	-e₂	e ₁	-e₀

The addition of octonions is denoted by $a + b = \sum_{l = 0}^{7} (a^{l} + b^{l}) e_{l}$ , where $a = \sum_{l = 0}^{7} a^{l} e_{l}$ , $b = \sum_{l = 0}^{7} b^{l} e_{l}$ and the scalar multiplication is defined as $ma = \sum_{l = 0}^{7} ({ma}^{l}) e_{l}$ , where $m \in ℝ$ . An octonion-valued function $u : r \to O$ is described by $u (t) = \sum_{l = 0}^{7} u^{l} (t) e_{l}$ , where u^l (t) , i = 0, 1, 2, …, 7, are real-valued functions. The derivative of u (t) is given by $\dot{u} (t) = \sum_{l = 0}^{7} {\dot{u}}^{l} (t) e_{l}$ . For $x = \sum_{l = 0}^{7} x^{l} e_{l} \in O$ , the norm of x can be defined as $| x |_{O} = \sqrt{\sum_{l = 0}^{7} (x^{l})^{2}}$ , and for $x = (x_{1}, x_{2}, \dots, x_{n}) \in O^{n}$ , we define $∥ x ∥ = max_{1 \leq i \leq n} {| x_{i} |_{O}}$ , then $(O, | \cdot |_{O})$ and $(O^{n}, ∥ \cdot ∥)$ are Banach spaces.

For readers interested in octonion algebra, see [20].

In this paper, we are concerned with the following octonion-valued neutral type high-order Hopfield neural networks with D operator: $\begin{matrix} [x_{i} (t) - a_{i} (t) x_{i} (t - η_{i} (t))]^{'} \\ = - b_{i} (t) x_{i} (t) + \sum_{j = 1}^{n} h_{ij} (t) f_{j} (x_{j} (t)) \\ + \sum_{j = 1}^{n} v_{ij} (t) g_{j} (x_{j} (t - σ_{ij} (t))) + \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl} (t) \\ k_{j} (x_{j} (t - τ_{ijl} (t))) k_{l} (x_{l} (t - γ_{ijl} (t))) + I_{i} (t), \end{matrix}$ (1) where i = 1, 2, ⋯ , n, $x_{i} (t) \in O$ corresponds to the state of the ith unit at time t; $O$ is an octonion-algebra; $b_{i} (t) \in O$ represents the self feedback coefficient at time t; a_i (t), h_ij (t), $v_{ij} (t) \in O$ are the connection weights; $u_{ijl} (t) \in O$ is the second-order synaptic weight of the neural network; f_j, g_j, $k_{j} : O \to O$ are the activation functions; $I_{i} (t) \in O$ is the external input to the ith unit; $η_{i} (t), σ_{ij} (t), τ_{ijl} (t), γ_{ijl} (t) \in ℝ^{+}$ denote the transmission delays.

Throughout this paper, for any $a, b, c \in O$ , we stipulate that abc = (ab) c, and for convenience, we use the following notations: $\begin{matrix} a_{i}^{+} = sup_{t \in ℝ} | a_{i} (t) |_{O}, b_{i}^{+} = sup_{t \in ℝ} | b_{i} (t) |_{O}, \\ {\overset{ˇ}{c}}_{i} = inf_{t \in ℝ} c_{i} (t), c_{i}^{+} = sup_{t \in ℝ} c_{i} (t), h_{ij}^{+} = sup_{t \in ℝ} | h_{ij} (t) |_{O}, \end{matrix}$ $\begin{matrix} v_{ij}^{+} = sup_{t \in ℝ} | v_{ij} (t) |_{O}, u_{ijl}^{+} = sup_{t \in ℝ} | u_{ijl} (t) |_{O}, \\ η_{i}^{+} = sup_{t \in ℝ} η_{i} (t), σ_{ij}^{+} = sup_{t \in ℝ} σ_{ij} (t), τ_{ijl}^{+} = sup_{t \in ℝ} τ_{ijl} (t), \end{matrix}$ $\begin{matrix} γ_{ijl}^{+} = sup_{t \in ℝ} γ_{ijl} (t), {\dot{η}}_{i}^{+} = sup_{t \in ℝ} {\dot{η}}_{i} (t)_{,} \\ {\dot{σ}}_{ij}^{+} = sup_{t \in ℝ} {\dot{σ}}_{ij} (t), {\dot{τ}}_{ijl}^{+} = sup_{t \in ℝ} {\dot{τ}}_{ijl} (t), {\dot{γ}}_{ijl}^{+} = sup_{t \in ℝ} {\dot{γ}}_{ijl} (t), \end{matrix}$ $\begin{matrix} ι = max_{1 \leq i, j, l \leq n} {η_{i}^{+}, σ_{ij}^{+}, τ_{ijl}^{+}, γ_{ijl}^{+}}, \\ {\dot{α}}^{+} = max_{1 \leq i, j, l \leq n} {{\dot{η}}_{i}^{+}, {\dot{σ}}_{ij}^{+}, {\dot{τ}}_{ijl}^{+}, {\dot{γ}}_{ijl}^{+}} . \end{matrix}$

The initial values of system (2.1) are described by $x_{i} (s) = φ_{i} (s) \in O, s \in [- ι, 0],$ where i = 1, 2, …, n, φ_i are bounded and continuous functions from [- ι, 0] to $O$ .

Let $BC (ℝ, O^{n})$ be the set of all bounded continuous functions from $ℝ$ to $O^{n}$ . Note that $(BC (ℝ, O^{n}), ∥ \cdot ∥_{\infty})$ is a Banach space with the norm: $∥ f ∥_{\infty} = max_{1 \leq i \leq n} {sup_{t \in ℝ} | f_{i} (t) |_{O}},$ where $f = (f_{1}, f_{2}, \dots f_{i})^{T} \in BC (ℝ, O^{n})$ .

Definition 2.1. A function $f \in BC (ℝ, O^{n})$ is said to be almost periodic, if for every ɛ > 0, it is possible to find a real number l = l (ɛ) >0 such that every interval of length l contains a number τ = τ (ɛ) such that ∥f (t + τ) - f (t) ∥ < ɛ for all $t \in ℝ$ . The collection of all such functions will be denoted by $AP (ℝ, O^{n})$ .

Let $\begin{matrix} {PAP}_{0} (ℝ, O^{n}) = \\ {f \in BC (ℝ^{+}, O^{n}) | lim_{T \to + \infty} \frac{1}{2 T} \int_{- T}^{T} ∥ f (t) ∥ dt = 0} . \end{matrix}$

Definition 2.2. [47] A function $f \in BC (ℝ, O^{n})$ is called pseudo almost periodic if it can be expressed as f = f₁ + f₀, where $f_{1} \in AP (ℝ, O^{n})$ and $f_{0} \in {PAP}_{0} (ℝ, O^{n})$ . The collection of such functions will be denoted by $PAP (ℝ, O^{n})$ .

Lemma 2.1. [47] If $α \in ℝ$ and u, $v \in PAP (ℝ, O^{n})$ , then αu, u + v, $u \cdot v \in PAP (ℝ, O^{n})$ .

Lemma 2.2. [48] The space $(PAP (ℝ, O^{n}), ∥ \cdot ∥_{\infty})$ is a Banach space.

Lemma 2.3. [48] Let $f \in C (O, O^{n})$ and satisfy the Lipschitz condition. If $g \in PAP (ℝ, O)$ , then $f (g (\cdot)) \in PAP (ℝ, O^{n}) .$ The following two lemmas can be proved in the same way as Lemma 5 and Lemma 6 in [49], respectively.

Lemma 2.4. Let $f \in PAP (ℝ, O^{n})$ , $σ \in AP (ℝ, [0, \infty)) \cap C^{1} (ℝ, ℝ)$ with |σ (t) | ≤ σ⁺ and $\dot{σ} (t) \leq {\dot{σ}}^{+} < 1$ , then $f (\cdot - σ (\cdot)) \in PAP (ℝ, O^{n})$ .

Lemma 2.5. Let $Γ (t) = \int_{- \infty}^{t} e^{- \int_{s}^{t} a (u) du} Λ (s) ds$ , where $Λ \in PAP (ℝ, O)$ , $a \in AP (ℝ, ℝ)$ with $inf_{t \in ℝ} a (t) > 0$ , then $Γ \in PAP (ℝ, O)$ .

The hypotheses used in this paper are as follows:

Functions $c_{i} \in AP (ℝ, ℝ^{+})$ with ${\overset{ˇ}{c}}_{i} > 0$ , $a_{i}, b_{i}, h_{ij}, v_{ij}, u_{ijl} \in PAP (ℝ, O)$ , η_i, σ_ij, τ_ijl, $γ_{ijl} \in AP (ℝ, [0, \infty)) \cap C^{1} (ℝ, ℝ)$ and ${\dot{α}}^{+} < 1$ , where i, j, l = 1, 2, ⋯ , n.

For $j = 1, 2, \dots, n, f_{j}, g_{j}, k_{j} \in C (O, O)$ and there exist positive constants F_j, G_j, K_j, M_j such that for any $x, y \in O$ , $\begin{matrix} | f_{j} (x) - f_{j} (y) |_{O} \leq F_{j} | x - y |_{O}, \\ | g_{j} (x) - g_{j} (y) |_{O} \leq G_{j} | x - y |_{O}, \\ | k_{j} (x) - k_{j} (y) |_{O} \leq K_{j} | x - y |_{O}, \\ | k_{j} (x) |_{O} \leq M_{j}^{k}; \end{matrix}$ moreover, f_j (0) = g_j (0) = k_j (0) =0.

There exist positive constants β_i, i = 1, 2, …, n such that $\begin{matrix} max_{1 \leq i \leq n} & {a_{i}^{+} + \frac{1}{{\overset{ˇ}{c}}_{i}} (Q_{i}^{+} + c_{i}^{+} a_{i}^{+} + β_{i}^{- 1} \\ \sum_{j = 1}^{n} h_{ij}^{+} F_{j} β_{j} + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij}^{+} G_{j} β_{j} + β_{i}^{- 1} \\ \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl}^{+} (K_{j} M_{l}^{k} β_{j} + K_{l} M_{j}^{k} β_{l}))} \\ : = r < 1, \end{matrix}$ where $Q_{i}^{+} = sup_{t \in R} | - b_{i} (t) + c_{i} (t) |_{O}$ and c_i is mentioned in (A₁).

3 The existence of pseudo almost periodic solutions

Consider the space $V = PAP (ℝ, O^{n})$ with the norm ∥ · ∥ _∞. By Lemma 2.2, $V$ is a Banach space.

Let $v_{I} = (\int_{- \infty}^{t} e^{- \int_{s}^{t} c_{i} (u) du} I_{1} (s) ds, \int_{- \infty}^{t} e^{- \int_{s}^{t} c_{i} (u) du} I_{2} (s) ds, \dots, \int_{- \infty}^{t} e^{- \int_{s}^{t} c_{i} (u) du} I_{n} (s) ds)^{T}$ and choose a constant R such that R > ∥ v_I ∥ _∞. Define $V^{*} = {φ \in V | ∥ φ - v_{I} ∥_{\infty} \leq \frac{rR}{1 - r}} .$ Then, for every $φ \in V^{*}$ , one can easily get $∥ φ ∥_{\infty} \leq \frac{R}{1 - r} .$

Theorem 3.1. Let (A₁)-(A₃) be fulfilled. Then system (1) admits a unique pseudo almost periodic solution in $V^{*}$ . Proof. Let $y_{i} (t) = β_{i}^{- 1} x_{i} (t)$ and Y_i (t) = y_i (t) - a_i (t) y_i (t - η_i (t)) , i = 1, 2, ⋯ , n, then, system (2.1) is transformed into the following form: $\begin{matrix} Y_{i}^{'} (t) = & - b_{i} (t) y_{i} (t) + β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij} (t) f_{j} (β_{j} y_{j} (t)) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij} (t) g_{j} (β_{j} y_{j} (t - σ_{ij} (t))) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl} (t) k_{j} (β_{j} y_{j} (t - τ_{ijl} (t))) \\ k_{l} (β_{l} y_{l} (t - γ_{ijl} (t))) + β_{i}^{- 1} I_{i} (t), \\ i = 1, 2, \dots, n . \end{matrix}$ (3.1.) For the function $c_{i} \in (ℝ, ℝ^{+})$ defined in (A₁), we multiply the both sides of (??) by $e^{\int_{0}^{s} c_{i} (u) du}$ , and integrate it over (- ∞ , t], then, for i = 1, 2, …, n, $\begin{matrix} y_{i} (t) = & a_{i} (t) y_{i} (t - η_{i} (t)) + \int_{- \infty}^{t} e^{- \int_{s}^{t} c_{i} (u) du} \\ [(- b_{i} (s) + c_{i} (s)) y_{i} (s) - c_{i} (s) a_{i} (s) y_{i} (s - η_{i} (s)) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij} (s) f_{j} (β_{j} y_{j} (s)) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij} (s) g_{j} (β_{j} y_{j} (s - σ_{ij} (s))) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl} (s) k_{j} (β_{j} y_{j} (s - τ_{ijl} (s))) k_{l} \\ (β_{l} y_{l} (s - γ_{ijl} (s))) + β_{i}^{- 1} I_{i} (s)] ds . \end{matrix}$ (2) Taking the derivative of (2), we can infer that $\begin{matrix} Y_{i}^{'} (t) = & (- b_{i} (t) + c_{i} (t)) y_{i} (t) - c_{i} (t) a_{i} (t) y_{i} (t - η_{i} (t)) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij} (t) f_{j} (β_{j} y_{j} (t)) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij} (t) g_{j} (β_{j} y_{j} (t - σ_{ij} (t))) + β_{i}^{- 1} \\ \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl} (t) k_{j} (β_{j} y_{j} (t - τ_{ijl} (t))) \\ \times k_{l} (β_{l} y_{l} (t - γ_{ijl} (t))) + β_{i}^{- 1} I_{i} (t) - c_{i} (t) \\ \int_{- \infty}^{t} e^{- \int_{s}^{t} c_{i} (u) du} [(- b_{i} (s) + c_{i} (s)) y_{i} (s) \\ - c_{i} (s) a_{i} (s) y_{i} (s - η_{i} (s)) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij} (s) f_{j} (β_{j} y_{j} (s)) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij} (s) g_{j} (β_{j} y_{j} (s - σ_{ij} (s))) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl} (s) k_{j} (β_{j} y_{j} (s - τ_{ijl} (s))) \\ \times k_{l} (β_{l} y_{l} (s - γ_{ijl} (s))) + β_{i}^{- 1} I_{i} (s)] ds \\ = & - b_{i} (t) y_{i} (t) + β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij} (t) f_{j} (β_{j} y_{j} (t)) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij} (t) g_{j} (β_{j} y_{j} (t - σ_{ij} (t))) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl} (t) k_{j} (β_{j} y_{j} (t - τ_{ijl} (t))) k_{l} \\ (β_{l} y_{l} (t - γ_{ijl} (t))) + β_{i}^{- 1} I_{i} (t) . \end{matrix}$ From the above arguments, we can deduce that if $y^{*} = (y_{1}^{*}, y_{2}^{*}, \dots, y_{n}^{*})$ is a solution of (??), then $x^{*} = (x_{1}^{*}, x_{2}^{*}, \dots, x_{n}^{*}) = (β_{1} y_{1}^{*}, β_{2} y_{2}^{*}, \dots, β_{n} y_{n}^{*})$ is a solution of system (2.1).

Let us consider an operator $S : V \to BC (ℝ, O^{n})$ described by Sφ = (S₁ φ, S₂ φ, ⋯ S_n φ) ^T, where $\begin{matrix} (S_{i} φ) (t) = a_{i} (t) φ_{i} (t - η_{i} (t)) \\ + \int_{- \infty}^{t} e^{- \int_{s}^{t} c_{i} (u) du} G_{i} (s) ds \end{matrix}$ and

$\begin{matrix} G_{i} (s) = & (- b_{i} (s) + c_{i} (s)) φ_{i} (s) - c_{i} (s) a_{i} (s) φ_{i} (s - η_{i} (s)) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij} (s) f_{j} (β_{j} φ_{j} (s)) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij} (s) g_{j} (β_{j} φ_{j} (s - σ_{ij} (s))) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl} (s) k_{j} (β_{j} φ_{j} (s - τ_{ijl} (s))) \\ k_{l} (β_{l} φ_{l} (s - γ_{ijl} (s))) + β_{i}^{- 1} I_{i} (s) . \end{matrix}$

Then by Lemmas 2.1-2.4, we have $G_{i} \in PAP (ℝ, O)$ , i = 1, 2, ⋯ , n. Moreover, by Lemma 2, we infer $S φ \in PAP (ℝ, O^{n})$ . In order to prove this theorem, we divide the rest of the proof into the following two steps.

Step 1. We will show that S maps $V^{*}$ into $V^{*}$ . In fact, for each $φ \in V^{*}$ , we have $\begin{matrix} ∥ S φ - v_{I} ∥_{\infty} \\ \leq & max_{1 \leq i \leq n} sup_{t \in ℝ} {| a_{i} (t) φ_{i} (t - η_{i} (t)) |_{O} \\ + \int_{- \infty}^{t} e^{- \int_{s}^{t} c_{i} (u) du} (| (- b_{i} (s) + c_{i} (s)) φ_{i} (s) |_{O} \\ + | c_{i} (s) a_{i} (s) φ_{i} (s - η_{i} (s)) |_{O} \\ + | β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij} (s) f_{j} (β_{j} φ_{j} (s)) |_{O} \\ + | β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij} (s) g_{j} (β_{j} φ_{j} (s - σ_{ij} (s))) |_{O} \\ + | β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl} (s) k_{j} (β_{j} φ_{j} (s - τ_{ijl} (s))) \\ k_{l} (β_{l} φ_{l} (s - γ_{ijl} (s))) |_{O}) ds} \\ \leq & max_{1 \leq i \leq n} sup_{t \in ℝ} {a_{i}^{+} + \int_{- \infty}^{t} e^{- {\overset{ˇ}{c}}_{i} (t - s)} [Q_{i}^{+} + c_{i}^{+} a_{i}^{+} \\ + β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij}^{+} F_{j} β_{j} + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij}^{+} G_{j} β_{j} + β_{i}^{- 1} \\ \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl}^{+} (K_{j} M_{l}^{k} β_{j} + K_{l} M_{j}^{k} β_{l})] ∥ φ ∥_{\infty} ds} \\ \leq & max_{1 \leq i \leq n} {a_{i}^{+} + \frac{1}{{\overset{ˇ}{c}}_{i}} (Q_{i}^{+} + c_{i}^{+} a_{i}^{+} + β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij}^{+} F_{j} β_{j} \\ + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij}^{+} G_{j} β_{j} + β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl}^{+} (K_{j} M_{l}^{k} β_{j} \\ + K_{l} M_{j}^{k} β_{l}))} ∥ φ ∥_{\infty} \\ \leq & \frac{rR}{1 - r}, \end{matrix}$ which implies that $S (V^{*}) \subset V^{*}$ .

Step 2. We will prove that S is a contracting mapping. Indeed, for any φ, $ψ \in V^{*}$ , we deduce that

$\begin{matrix} ∥ S φ - S ψ ∥_{\infty} \\ \leq & max_{1 \leq i \leq n} {sup_{t \in ℝ} [| a_{i} (t) (φ_{i} (t - η_{i} (t)) \\ - ψ_{i} (t - η_{i} (t))) |_{O} + \int_{- \infty}^{t} e^{- \int_{s}^{t} c_{i} (u) du} \\ \times (| (- b_{i} (s) + c_{i} (s)) (φ_{i} (s) - ψ_{i} (s)) |_{O} \\ + | c_{i} (s) a_{i} (s) (φ_{i} (s - η_{i} (s)) - ψ_{i} (s - η_{i} (s))) |_{O} \\ + | β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij} (s) (f_{j} (β_{j} φ_{j} (s)) - f_{j} (β_{j} ψ_{j} (s))) |_{O} \\ + | β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij} (s) (g_{j} (β_{j} φ_{j} (s - σ_{ij} (s))) \\ - g_{j} (β_{j} ψ_{j} (s - σ_{ij} (s)))) |_{O} + | β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl} (s) \\ [k_{j} (β_{j} φ_{j} (s - τ_{ijl} (s))) k_{l} (β_{l} φ_{l} (s - γ_{ijl} (s))) \\ - k_{j} (β_{j} ψ_{j} (s - τ_{ijl} (s))) k_{l} (β_{l} ψ_{l} (s - γ_{ijl} (s)))] |_{O}) ds]} \\ \leq & max_{1 \leq i \leq n} sup_{t \in ℝ} {a_{i}^{+} + \int_{- \infty}^{t} e^{- {\overset{ˇ}{c}}_{i} (t - s)} [Q_{i}^{+} + c_{i}^{+} a_{i}^{+} \\ + β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij}^{+} F_{j} β_{j} + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij}^{+} G_{j} β_{j} + β_{i}^{- 1} \\ \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl}^{+} (K_{j} M_{l}^{k} β_{j} + K_{l} M_{j}^{k} β_{l})] ∥ φ - ψ ∥_{\infty} ds} \\ \leq & max_{1 \leq i \leq n} {a_{i}^{+} + \frac{1}{{\overset{ˇ}{c}}_{i}} (Q_{i}^{+} + c_{i}^{+} a_{i}^{+} + β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij}^{+} F_{j} β_{j} \\ + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij}^{+} G_{j} β_{j} + β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl}^{+} \\ (K_{j} M_{l}^{k} β_{j} + K_{l} M_{j}^{k} β_{l}))} ∥ φ - ψ ∥_{\infty} \\ < & r ∥ φ - ψ ∥_{\infty}, \end{matrix}$

which means that S is a contraction mapping because of r < 1. Consequently, S possesses a unique fixed point. Subsequently, system (1) admits a unique pseudo almost periodic solution. The proof is complete.

4 Global exponential stability

Definition 4.1. A solution $x^{*} = (x_{1}^{*}, x_{2}^{*}, \dots, x_{n}^{*})^{T}$ of system (1) with the initial value φ = (φ₁, φ₂, ⋯ , φ_n) ^T is called globally exponentially stable if for an arbitrary solution x = (x₁, x₂, ⋯ , x_n) ^T of system (1) with the initial value ψ = (ψ₁, ψ₂, ⋯ , ψ_n) ^T, there exist constants λ > 0 and M > 1 such that $\begin{matrix} ∥ x (t) - x^{*} (t) ∥ \leq M ∥ φ - ψ ∥_{ι} e^{- λ t}, t > 0, \end{matrix}$ where $\begin{matrix} ∥ φ - ψ ∥_{ι} \\ = max_{1 \leq i \leq n} {sup_{t \in (- ι, 0]} β_{i}^{- 1} | φ_{i} (t) - a_{i} (t) φ_{i} (t - η_{i} (t)) \\ - [ψ_{i} (t) - a_{i} (t) ψ_{i} (t - η_{i} (t))] |_{O}} > 0 . \end{matrix}$

Theorem 4.1. Suppose that all of the conditions of Theorem 3 are fulfilled. Then the unique pseudo almost periodic solution of system (1) is globally exponentially stable.

Proof. Let $x^{*} = (x_{1}^{*}, x_{2}^{*}, \dots, x_{n}^{*})^{T}$ be the pseudo almost periodic solution of system (2.1) with the initial value φ = (φ₁, φ₂, ⋯ , φ_n) ^T and x = (x₁, x₂, ⋯ , x_n) ^T an arbitrary solution of system (2.1) with the initial value ψ = (ψ₁, ψ₂, ⋯ , ψ_n) ^T, and denote $y_{i} (t) = β_{i}^{- 1} x_{i} (t), y_{i}^{*} (t) = β_{i}^{- 1} x_{i}^{*} (t), z_{i} (t) = y_{i} (t) - y_{i}^{*} (t), Z_{i} (t) = z_{i} (t) - a_{i} (t) z_{i} (t - η_{i} (t)), i = 1, 2, \dots, n$ . Then, by (2.1), we deduce $\begin{matrix} Z_{i}^{'} (t) = & - b_{i} (t) z_{i} (t) + β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij} (t) (f_{j} (β_{j} y_{j} (t)) \\ - f_{j} (β_{j} y_{j}^{*} (t))) + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij} (t) (g_{j} (β_{j} y_{j} \\ (t - σ_{ij} (t))) - g_{j} (β_{j} y_{j}^{*} (t - σ_{ij} (t)))) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl} (t) (k_{j} (β_{j} y_{j} (t - τ_{ijl} (t))) \\ k_{l} (β_{l} y_{l} (t - γ_{ijl} (t))) - k_{j} (β_{j} y_{j}^{*} (t - τ_{ijl} (t))) \\ k_{l} (β_{l} y_{l}^{*} (t - γ_{ijl} (t)))), i = 1, 2, \dots, n . \end{matrix}$ (4.1) For any ɛ > 0, it is easy to see, from the definition of ∥ · ∥ _ι, that $∥ Z (t) ∥ < M (∥ φ - ψ ∥_{ι} + ɛ) e^{- λ t}, t \in (- ι, 0] .$ We assert that

$∥ Z (t) ∥ < M (∥ φ - ψ ∥_{ι} + ɛ) e^{- λ t}, for all t > 0 .$ (4.2) On the contrary case, if (4.2) is not valid, then there exists a certain t₁ > 0 such that

$∥ Z (t_{1}) ∥ = M (∥ φ - ψ ∥_{ι} + ɛ) e^{- λ t_{1}},$ (4.3) $\begin{matrix} ∥ Z (t) ∥ < M (∥ φ - ψ ∥_{ι} + ɛ) e^{- λ t}, for all t \in (- ι, t_{1}) . \end{matrix}$ Subsequently, we deduce

$\begin{matrix} e^{λ v} ∥ z_{j} (v) |_{O} \\ \leq & e^{λ v} | z_{j} (v) - a_{j} (v) z_{j} (v - η_{j} (v)) |_{O} + e^{λ v} | a_{j} (v) \\ z_{j} (v - η_{j} (v)) |_{O} \\ \leq & e^{λ v} | Z_{j} (v) |_{O} + a_{j}^{+} e^{λ η_{j}^{+}} e^{λ (v - η_{j} (v))} | z_{j} (v - η_{j} (v)) |_{O} \\ \leq & M (∥ φ - ψ ∥_{ι} + ɛ) + a_{j}^{+} e^{λ η_{j}^{+}} sup_{s \in (- ι, t]} e^{λ s} | z_{j} (s) |_{O}, \end{matrix}$ (4.4)

for all v ∈ (- ι, t], t ∈ (- ι, t₁), j = 1, 2, ⋯ , n, these inequalities imply that $\begin{matrix} e^{λ t} | z_{j} (t) |_{O} \leq sup_{s \in (- ι, t]} e^{λ s} | z_{j} (s) |_{O} \\ \leq \frac{M (∥ φ - ψ ∥_{ι} + ɛ)}{1 - a_{j}^{+} e^{λ η_{j}^{+}}} . \end{matrix}$ (4.5) For J = 1, 2, ⋯ , n, let us consider the following continuous functions from [0, ∞) to $ℝ$ : $\begin{matrix} θ_{i} (ω) = & {\overset{ˇ}{c}}_{i} - ω - P_{i} (ω), \end{matrix}$ where $\begin{matrix} P_{i} (ω) = & Q_{i}^{+} \frac{1}{1 - a_{i}^{+} e^{ω η_{i}^{+}}} + c_{i}^{+} a_{i}^{+} \frac{e^{ω η_{i}^{+}}}{1 - a_{i}^{+} e^{ω η_{i}^{+}}} \\ + β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij}^{+} F_{j} β_{j} \frac{1}{1 - a_{j}^{+} e^{ω η_{j}^{+}}} \\ + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij}^{+} G_{j} β_{j} \frac{e^{ω σ_{ij}^{+}}}{1 - a_{j}^{+} e^{ω η_{j}^{+}}} \\ + β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl}^{+} (K_{j} M_{l}^{k} β_{j} \frac{e^{ω τ_{ijl}^{+}}}{1 - a_{j}^{+} e^{ω η_{j}^{+}}} \\ + K_{l} M_{j}^{k} β_{l} \frac{e^{ω γ_{ijl}^{+}}}{1 - a_{j}^{+} e^{ω η_{j}^{+}}}) . \end{matrix}$

By virtue of (A₃), it holds θ_i (ω) >0. Due to the fact that θ_i (ω)→ - ∞ as ω→ + ∞, we infer that there exists a $λ \in (0, min_{1 \leq i \leq n} {{\overset{ˇ}{c}}_{i}})$ such that $a_{i}^{+} e^{λ η_{i}^{+}} < 1$ and θ_i (ω) >0 for i = 1, 2, ⋯ , n. Hence, we obtain that for i = 1, 2, ⋯ , n,

$\frac{U_{i}}{{\overset{ˇ}{c}}_{i} - λ} : = \frac{P_{i} (λ)}{{\overset{ˇ}{c}}_{i} - λ} < 1 .$ (4.6)

Let $M = max_{1 \leq i \leq n} {\frac{{\overset{ˇ}{c}}_{i}}{U_{i}}}$ , then for i = 1, 2, ⋯ , n, one gets

$\frac{1}{M} \leq \frac{P_{i} (λ)}{{\overset{ˇ}{c}}_{i} - λ} .$ (4.7) From (4.1), we can get $\begin{matrix} Z_{i} (t) = & Z_{i} (0) e^{- \int_{0}^{t} c_{i} (u) du} \\ + \int_{0}^{t} e^{- \int_{s}^{t} c_{i} (u) du} [(- b_{i} (s) + c_{i} (s)) z_{i} (s) \\ - c_{i} (s) a_{i} (s) z_{i} (s - η_{i} (s)) + β_{i}^{- 1} \sum_{j = 1}^{n} \\ h_{ij} (s) (f_{j} (β_{j} y_{j} (s)) - f_{j} (β_{j} y_{j}^{*} (s))) \\ + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij} (s) (g_{j} (β_{j} y_{j} (s - σ_{ij} (s))) \\ - g_{j} (β_{j} y_{j}^{*} (s - σ_{ij} (s)))) + β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} \\ u_{ijl} (s) (k_{j} (β_{j} y_{j} (t - τ_{ijl} (s))) k_{l} (β_{l} y_{l} (s - γ_{ijl} (s))) \\ - k_{j} (β_{j} y_{j}^{*} (s - τ_{ijl} (s))) k_{l} (β_{l} y_{l}^{*} (s - γ_{ijl} (s))))] \\ ds, t \in [0, t_{1}], i = 1, 2, \dots, n . \end{matrix}$ This, with the help of (4.3), (4.5), (4.6) and (4.7) for i = 1, 2, …, n, one gains $\begin{matrix} | Z_{i} (t_{1}) |_{O} \\ \leq | Z_{i} (0) |_{O} e^{- \int_{0}^{t_{1}} c_{i} (u) du} \\ + \int_{0}^{t_{1}} e^{- \int_{s}^{t_{1}} c_{i} (u) du} [| (- b_{i} (s) + c_{i} (s)) z_{i} (s) |_{O} \\ + | - c_{i} (s) a_{i} (s) z_{i} (s - η_{i} (s)) |_{O} \\ + | β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij} (s) (f_{j} (β_{j} y_{j} (s)) - f_{j} (β_{j} y_{j}^{*} (s))) |_{O} \\ + | β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij} (s) (g_{j} (β_{j} y_{j} (s - σ_{ij} (s))) \\ - g_{j} (β_{j} y_{j}^{*} (s - σ_{ij} (s)))) |_{O} \\ + | β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl} (s) (k_{j} (β_{j} y_{j} (s - τ_{ijl} (s))) k_{l} \\ (β_{l} y_{l} (s - γ_{ijl} (s))) - k_{j} (β_{j} y_{j}^{*} (s - τ_{ijl} (s))) \\ k_{l} (β_{l} y_{l}^{*} (s - γ_{ijl} (s)))) |_{O}] ds \\ \leq & (∥ φ - ψ ∥_{ι} + ɛ) e^{- {\overset{ˇ}{c}}_{i} t_{1}} + \int_{0}^{t_{1}} e^{- {\overset{ˇ}{c}}_{i} (t_{1} - s)} \\ [Q_{i}^{+} | z_{i} (s) |_{O} + c_{i}^{+} a_{i}^{+} | z_{i} (s - η_{i} (s)) |_{O} \\ + β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij}^{+} F_{j} β_{j} | z_{j} (s) |_{O} + β_{i}^{- 1} \\ \sum_{j = 1}^{n} v_{ij}^{+} G_{j} β_{j} | z_{j} (s - σ_{ij} (s)) |_{O} + β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} \\ u_{ijl}^{+} (K_{j} M_{l}^{k} β_{j} | z_{j} (s - τ_{ijl} (s)) |_{O} \\ + K_{l} M_{j}^{k} β_{l} | z_{j} (s - γ_{ijl} (s)) |_{O})] ds \\ \leq & (∥ φ - ψ ∥_{ι} + ɛ) e^{- {\overset{ˇ}{c}}_{i} t_{1}} + \int_{0}^{t_{1}} e^{- {\overset{ˇ}{c}}_{i} (t_{1} - s)} \\ [Q_{i}^{+} \frac{M (∥ φ - ψ ∥_{ι} + ɛ)}{1 - a_{i}^{+} e^{λ η_{i}^{+}}} e^{- λ s} \\ + c_{i}^{+} a_{i}^{+} \frac{M (∥ φ - ψ ∥_{ι} + ɛ)}{1 - a_{i}^{+} e^{λ η_{i}^{+}}} e^{- λ (s - η_{i} (s))} \\ + β_{i}^{- 1} \sum_{j = 1}^{n} h_{ij}^{+} F_{j} β_{j} \frac{M (∥ φ - ψ ∥_{ι} + ɛ)}{1 - a_{j}^{+} e^{λ η_{j}^{+}}} e^{- λ s} \\ + β_{i}^{- 1} \sum_{j = 1}^{n} v_{ij}^{+} G_{j} β_{j} \frac{M (∥ φ - ψ ∥_{ι} + ɛ)}{1 - a_{j}^{+} e^{λ η_{j}^{+}}} e^{- λ (s - σ_{ij} (s))} \\ + β_{i}^{- 1} \sum_{j = 1}^{n} \sum_{l = 1}^{n} u_{ijl}^{+} (K_{j} M_{l}^{k} β_{j} \frac{M (∥ φ - ψ ∥_{ι} + ɛ)}{1 - a_{j}^{+} e^{λ η_{j}^{+}}} \\ e^{- λ (s - τ_{ijl} (s))} \\ + K_{l} M_{j}^{k} β_{l} \frac{M (∥ φ - ψ ∥_{ι} + ɛ)}{1 - a_{l}^{+} e^{λ η_{l}^{+}}} e^{- λ (s - γ_{ijl} (s))})] ds \\ \leq & M (∥ φ - ψ ∥_{ι} + ɛ) e^{- λ t_{1}} \\ (\frac{e^{(λ - {\overset{ˇ}{c}}_{i}) t_{1}}}{M} + \frac{P_{i} (λ)}{{\overset{ˇ}{c}}_{i} - λ} (1 - e^{(λ - {\overset{ˇ}{c}}_{i}) t_{1}})) \\ \leq & M (∥ φ - ψ ∥_{ι} + ɛ) e^{- λ t_{1}} \\ [e^{(λ - {\overset{ˇ}{c}}_{i}) t_{1}} (\frac{1}{M} - \frac{P_{i} (λ)}{{\overset{ˇ}{c}}_{i} - λ}) + \frac{P_{i} (λ)}{{\overset{ˇ}{c}}_{i} - λ}] \\ \leq & M (∥ φ - ψ ∥_{ι} + ɛ) e^{- λ t_{1}} (\frac{P_{i} (λ)}{{\overset{ˇ}{c}}_{i} - λ}) \\ < & M (∥ φ - φ ∥_{ι} + ɛ) e^{- λ t_{1}}, \end{matrix}$ thus, we arrive at $\begin{matrix} ∥ Z (t_{1}) ∥ < M (∥ φ - ψ ∥_{ι} + ɛ) e^{- λ t_{1}}, \end{matrix}$ invoking of (4.3), this is a contradiction. As a consequent, (4.2) is valid. Passing the limit ɛ → 0⁺, we obtain

$∥ Z (t) ∥ \leq M ∥ φ - ψ ∥_{ι} e^{- λ t}, for all t > 0 .$ (4.8) Then, using a similar derivation in the proofs of (4.4) and (4.5), with the help of (4.8), we can infer that $\begin{matrix} e^{λ t} | z_{j} (t) |_{O} \leq sup_{s \in (- ι, t]} e^{λ s} | z_{j} (s) |_{O} \leq \frac{M ∥ φ - ψ ∥_{ι}}{1 - a_{j}^{+} e^{λ η_{j}^{+}}}, \\ i = 1, 2, \dots, n . \end{matrix}$ Denote $M_{0} = max_{1 \leq i \leq n} {\frac{M}{1 - a_{j}^{+} e^{λ η_{j}^{+}}}}$ , then it holds $∥ z (t) ∥ \leq M_{0} ∥ φ - ψ ∥_{ι} e^{- λ t}, for all t > 0 .$ This ends the proof. □

{From the proof of Theorem 4.1, one can easily see that if we replace condition (A₁) with the following condition:

Functions $c_{i} \in BC (ℝ, ℝ^{+})$ with ${\overset{ˇ}{c}}_{i} > 0$ , $a_{i}, b_{i}, h_{ij}, v_{ij}, u_{ijl} \in BC (ℝ, O)$ , η_i, σ_ij, τ_ijl, $γ_{ijl} \in BC (ℝ, [0, \infty)) \cap C^{1} (ℝ, ℝ)$ and ${\dot{α}}^{+} < 1$ , where i, j, l = 1, 2, ⋯ , n,

then one has∥Corollary 4.1. Suppose that $(A_{1}^{'})$ , (A₂) and (A₃) are fulfilled. Then every solution of system (2.1) is globally exponentially stable.

5 A numerical example

Example 5.1. In system (1), let n = 2, for i, j, l = 1, 2, we choose $\begin{matrix} x_{i} = & x_{i}^{0} e_{0} + x_{i}^{1} e_{1} + x_{i}^{2} e_{2} + x_{i}^{3} e_{3} + x_{i}^{4} e_{4} + x_{i}^{5} e_{5} + x_{i}^{6} e_{6} + x_{i}^{7} e_{7}, \\ f_{j} (x_{j}) = & 0.05 e_{0} sin (x_{j}^{0}) + 0.04 e_{1} sin (x_{j}^{1}) + 0.03 e_{2} sin (x_{j}^{2}) + 0.04 e_{3} tanh (x_{j}^{3}) \\ + 0.037 e_{4} sin (x_{j}^{4}) + 0.042 e_{5} sin (x_{j}^{5}) + 0.037 e_{6} sin (x_{j}^{6}) + 0.041 e_{7} tanh (x_{j}^{7}), \\ g_{j} (x_{j}) = & 0.07 e_{0} sin (x_{j}^{0}) + 0.06 e_{1} sin (x_{j}^{1}) + 0.04 e_{2} tanh (x_{j}^{2}) + 0.05 e_{3} sin (x_{j}^{3}) \\ + 0.05 e_{4} tanh (x_{j}^{4}) + 0.05 e_{5} sin (x_{j}^{5}) + 0.02 e_{6} sin (x_{j}^{6}) + 0.01 e_{7} sin (x_{j}^{7}), \\ k_{j} (x_{j}) = & 0.06 e_{0} sin (x_{j}^{0}) + 0.03 e_{1} sin (x_{j}^{1}) + 0.04 e_{2} sin (x_{j}^{2}) + 0.05 e_{3} sin (x_{j}^{3}) \\ + 0.05 e_{4} tanh (x_{j}^{4}) + 0.05 e_{5} sin (x_{j}^{5}) + 0.02 e_{6} sin (x_{j}^{6}) + 0.02 e_{7} tanh (x_{j}^{7}), \\ I_{1} (t) = & 0.3 e_{0} sin t + 0.6 e_{1} sin t + 0.4 e_{2} cos t + 0.5 e_{3} (sin t + \frac{3}{1 + t^{2}}) \\ + 0.4 e_{4} cos t + 0.5 e_{5} sin t + 0.2 e_{6} (sin t + 3 e^{- | t |}) + 0.7 e_{7} sin t, \\ I_{2} (t) = & 0.4 e_{0} cos t + 1.4 e_{1} sin t + 1.0 e_{2} (sin t - e^{- | t |}) + 0.9 e_{3} sin t \\ + 1.0 e_{4} cos t + 1.3 e_{5} sin t + 0.3 e_{6} sin t + 2 e_{7} (sin t - \frac{2}{1 + t^{2}}), \\ c_{1} (t) = & 2.9 - 0.1 sin t, c_{2} (t) = 2.98 + 0.02 cos t, \\ η_{1} (t) = & 0.2 - 0.1 sin t, η_{2} (t) = 0.4 - 0.1 cos t, \\ b_{1} (t) = & (2.8 - 0.4 sin t) e_{0} + (0.7 + 0.1 cos t) e_{1} + (0.8 + 0.1 sin t) e_{2} + (0.7 + 0.2 cos t) e_{3} \\ + (0.8 - 0.1 sin t) e_{4} + (0.6 + 0.3 sin t) e_{5} + (0.8 + 0.1 sin t) e_{6} + (0.7 + 0.2 cos t) e_{7}, \\ b_{2} (t) = & (2.9 - 0.1 cos t) e_{0} + (0.8 + 0.1 sin t) e_{1} + (0.7 + 0.1 sin t) e_{2} + (0.7 + 0.1 cos t) e_{3} \\ + (0.6 + 0.2 sin t) e_{4} + (0.7 + 0.1 sin t) e_{5} + (0.6 + 0.2 sin t) e_{6} + (0.6 + 0.2 cos t) e_{7}, \\ a_{1} (t) = & (0.01 + 0.005 sin t) e_{0} + (0.01 + 0.002 cos t) e_{1} + (0.01 + 0.004 sin (\sqrt{3} t)) e_{2} \\ + (0.01 + 0.003 sin t) e_{3} + (0.01 + 0.001 cos (\sqrt{5} t)) e_{4} + (0.01 + 0.002 sin (\sqrt{5} t)) e_{5} \\ + (0.01 + 0.002 sin t) e_{6} + (0.01 + 0.004 cos (\sqrt{7} t)) e_{7}, \\ a_{2} (t) = & (0.01 + 0.006 cos t) e_{0} + (0.01 + 0.005 sin t) e_{1} + (0.01 + 0.003 sin (\sqrt{2} t)) e_{2} \\ + (0.01 + 0.003 cos t) e_{3} + (0.01 + 0.004 cos (\sqrt{3} t)) e_{4} + (0.01 + 0.005 sin (\sqrt{5} t)) e_{5} \end{matrix}$

$\begin{matrix} + (0.01 + 0.004 cos (\sqrt{5} t)) e_{6} + (0.01 + 0.002 sin t) e_{7}, \\ h_{11} (t) = & 0.04 sin (\sqrt{2} t) e_{0} + 0.02 sin t e_{1} + 0.015 sin (\sqrt{3} t) e_{3}, \\ h_{12} (t) = & 0.03 cos (\sqrt{2} t) e_{2} + 0.017 cos t e_{6} + 0.025 sin (\sqrt{5} t) e_{7}, \\ h_{21} (t) = & 0.04 cos t e_{0} + 0.03 sin t e_{2} + 0.015 sin (\sqrt{5} t) e_{5}, \\ h_{22} (t) = & 0.04 sin (\sqrt{5} t) e_{0} + 0.02 sin t e_{4} + 0.03 sin (\sqrt{5} t) e_{7}, \\ v_{11} (t) = & 0.03 sin t e_{0} + 0.02 cos t e_{1} + 0.015 sin (\sqrt{3} t) e_{3}, \\ v_{12} (t) = & 0.025 sin (\sqrt{5} t) e_{2} + 0.015 cos t e_{6} + 0.019 sin (\sqrt{3} t) e_{7}, \\ v_{21} (t) = & 0.032 cos t e_{0} + 0.025 cos t e_{2} + 0.013 sin (\sqrt{5} t) e_{5}, \\ v_{22} (t) = & 0.04 sin (\sqrt{2} t) e_{0} + 0.013 cos t e_{4} + 0.016 cos (\sqrt{5} t) e_{7}, \\ u_{111} (t) = & u_{112} (t) = 0.02 sin t e_{0} + 0.011 cos t e_{1} + 0.017 cos (\sqrt{2} t) e_{3} + 0.013 sin 3 t e_{7}, \\ u_{121} (t) = & u_{122} (t) = 0.03 sin t e_{2} + 0.014 sin 4 t e_{4} + 0.019 cos t e_{5} + 0.027 sin t e_{6}, \\ u_{211} (t) = & u_{212} (t) = 0.034 cos t e_{0} + 0.021 cos t e_{1} + 0.027 sin t e_{3} + 0.025 sin t e_{7}, \\ u_{221} (t) = & u_{222} (t) = 0.02 sin t e_{2} + 0.015 sin (\sqrt{3} t) e_{4} + 0.016 cos t e_{5} + 0.009 sin t e_{6}, \\ σ_{11} (t) = & σ_{12} (t) = 0.8 - 0.2 cos t, σ_{21} (t) = σ_{22} (t) = 0.6 + 0.4 sin t, \\ τ_{111} (t) = & τ_{112} (t) = 1.4 - 0.4 cos t, τ_{121} (t) = τ_{122} (t) = 0.9 + 0.2 sin t, \\ τ_{211} (t) = & τ_{212} (t) = 1.3 - 0.2 sin t, τ_{221} (t) = τ_{222} (t) = 1.5 - 0.2 cos t, \\ γ_{111} (t) = & γ_{112} (t) = 1.1 - 0.1 sin t, γ_{121} (t) = γ_{122} (t) = 0.8 + 0.2 sin t, \\ γ_{211} (t) = & γ_{212} (t) = 1.5 - 0.5 sin t, γ_{221} (t) = γ_{222} (t) = 1.2 + 0.1 cos t . \end{matrix}$ Then by calculations, we obtain $\begin{matrix} F_{j} \approx & 0.113, G_{j} \approx 0.135, K_{j} = K_{l} \approx 0.120, M_{j}^{k} = M_{l}^{k} \approx 0.120, {\overset{ˇ}{c}}_{1} = 2.8, {\overset{ˇ}{c}}_{2} = 2.96, \\ c_{1}^{+} = & c_{2}^{+} = 3, a_{1}^{+} \approx 0.037, a_{2}^{+} \approx 0.040, h_{11}^{+} \approx 0.047, h_{12}^{+} \approx 0.043, h_{21}^{+} \approx 0.052, h_{22}^{+} \approx 0.054, \\ v_{11}^{+} \approx & 0.039, v_{12}^{+} \approx 0.035, v_{21}^{+} \approx 0.043, v_{22}^{+} \approx 0.045, u_{111}^{+} = u_{112}^{+} \approx 0.031, u_{121}^{+} = u_{122}^{+} \approx 0.047, \\ u_{211}^{+} = & u_{212}^{+} \approx 0.054, u_{221}^{+} = u_{222}^{+} \approx 0.031, Q_{1} \approx 2.379, Q_{2} \approx 2.166, β_{1} = β_{2} = 1, \\ r = & max {0.93669, 0.98576} < 1 . \end{matrix}$ So conditions (A₁)-(A₃) are verified. Hence, by Theorem 4, we see that system (1) admits a unique pseudo almost periodic solution that is globally exponentially stable (see Figs. 1 –6).

Fig. 1

States $x_{1}^{0}, x_{1}^{1}, x_{1}^{2}, x_{1}^{3}, x_{1}^{4}, x_{1}^{5}, x_{1}^{6}, x_{1}^{7}$ of (2.1) with different initial values.

Fig. 2

States $x_{2}^{0}, x_{2}^{1}, x_{2}^{2}, x_{2}^{3}, x_{2}^{4}, x_{2}^{5}, x_{2}^{6}, x_{2}^{7}$ of (2.1) with different initial values.

Fig. 3

Global exponential stability of states $x_{1}^{0}, x_{1}^{1}, x_{1}^{2}, x_{1}^{3}$ of (2.1) with different initial values.

Fig. 4

Global exponential stability of states $x_{1}^{4}, x_{1}^{5}, x_{1}^{6}, x_{1}^{7}$ of (2.1) with different initial values.

Fig. 5

Global exponential stability of states $x_{2}^{0}, x_{2}^{1}, x_{2}^{2}, x_{2}^{3}$ of (2.1) with different initial values.

Fig. 6

Global exponential stability of states $x_{2}^{4}, x_{2}^{5}, x_{2}^{6}, x_{2}^{7}$ of (1) with different initial values.

6 Conclusion

In this paper, we have investigated the existence and global exponential stability of pseudo almost periodic solutions for octonion-valued neutral type high-order Hopfield neural networks with D operatorby direct method. It is worth mentioning that the parameters of this system are all octonions except the time delay. The results of this paper are new. The method in this paper can be used to study the almost periodic and almost automorphic solutions of octonion-valued neural networks with or without D operator. Finally, it is worth mentioning that studying the almost periodic dynamics of octonion-valued neural networks with random perturbations and fuzzy terms [50, 51], or even octonion-valued neural networks described by fuzzy differential equations [52 –54], is our future direction.

Data availability statement

Not applicable.

Declaration conflict of interest

The authors declare that they have no conflict of interest.

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