Global exponential stability and existence of almost periodic solutions in distribution for Clifford-valued stochastic high-order Hopfield neural networks with time-varying delays

1.   Introduction

Clifford-valued neural networks (NNs) are the NNs whose state variables, connection weights and external inputs are Clifford numbers. They are generalizations of real-valued, complex-valued and quaternion-valued neural networks. In recent years, due to their advantages over real-valued networks and their potential application values in many fields, they have attracted the attention of many researchers ^{[1,2,3,4,5,6,7,8,9,10,11,12]}. However, because the multiplication of Clifford numbers does not satisfy the commutative law, it is difficult to study the dynamics of Clifford-valued NNs. At this stage, there are few results on the dynamics of Clifford-valued NNs ^{[8,9,10,11,12,13,14]}. In addition, it is worth noting that in most of the existing results ^[5,7,8,9,14], the coefficients of the leakage terms in neural networks are assumed to be real numbers.

On the one hand, it is well known that high-order Hopfield NNs have more advantages than low-order Hopfield NNs. Therefore, in the past few decades, many scholars have done a lot of research on the dynamics of high-order Hopfield NN ^{[13,15,16,17,18]}. This is because the application of neural networks in various fields largely depends on their dynamic performance. Moreover, the use of neural networks with complex or even chaotic dynamic behaviors in information processing is expected to improve the efficiency and flexibility of information processing.

In addition, noise interference is the main source of neural network instability, which can lead to poor neural network performance. In the real nervous system, synaptic transmission is a noisy process, caused by random fluctuations in neurotransmitter release and other probabilistic reasons. As we all know, neural networks can be stable or unstable through some random inputs ^[19]. For this reason, stochastic neural networks are widely studied ^{[20,21,22,23,24,25,26]}.

Besides, we know that the existence and stability of equilibrium points are important dynamics of autonomous neural networks. For nonautonomous neural networks, there are generally no equilibrium points. Therefore, the existence and stability of periodic or almost periodic solutions are important dynamics. Since almost periodicity is more common than periodicity, in the past few decades, many scholars have studied the almost periodic solutions of deterministic neural networks ^[7,27,28,29]. However, the existing results on the existence of almost periodic solutions of stochastic neural networks are almost all about mean-square almost periodic solutions. Unfortunately, in ^[30], some counterexamples show that the nontrivial solutions of some stochastic differential equations with almost periodic coefficients cannot be mean-square almost periodic. Therefore, it is more reasonable to study the almost periodic solutions in distribution of stochastic differential equations. Random almost periodic oscillation is a complex oscillation phenomenon. However, so far, no papers have been published on almost periodic solutions in distribution of Clifford-valued stochastic high-order Hopfield NNs. Therefore, it is necessary to study this issue.

Inspired by the above discussion, and considering the fact that time delay is inevitable, in this work, we consider the following Clifford-valued stochastic high-order Hopfield NN with time varying-delays:

where $p\in \{1, 2, \ldots, n\}: = \mathcal{D}$, $n$ is the number of neurons in layers; $x_p(t)\in \mathcal{A}$ is the state variable of the $p$th unit at time $t$ and $\mathcal{A}$ is a Clifford algebra; $c_p(t)\in\mathcal{A}$ is the coefficient of the leakage term, which represents the rate with which the $p$th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs; $a_{pq}(t), b_{pql}(t)\in\mathcal{A}$ are the first-order and second-order connection weights of the neural network; $\tau_{pq}(t)\geq0$, $\sigma_{pql}(t)\geq0$, $\nu_{pql}(t)\geq0$ and $\gamma_{pq}(t)\geq 0$ correspond to the transmission delays; $I_p(t)\in \mathcal{A}$ denotes the external inputs at time $t$; $f_q, g_{q}:\mathcal{A}\rightarrow \mathcal{A}$ are the activation functions of signal transmission; $\omega(t) = (\omega_1(t), \omega_2(t), \ldots, \omega_n(t))^T$ is an $n$-dimensional Brownian motion defined on a complete probability space; $\delta_{pq}: \mathcal{A}\rightarrow \mathcal{A}$ is a Borel measurable function.

Let $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\geq0}, P)$ be a complete probability space with a natural filtration $\{\mathcal{F}_t\}_{t\geq0}$ satisfying the usual conditions. Denote by $CB_{\mathcal{F}_0}([-\varrho, 0], \mathcal{A}^n)$ the family of all bounded, $\mathcal{F}_0$-measurable, $C([-\varrho, 0], \mathcal{A}^n)$-valued random variables $\phi$.

The initial values of system (1.1) are given by

where $\phi_{p}\in CB_{\mathcal{F}_0}([-\varrho, 0], \mathcal{A}^n)$.

The main purpose of this paper is to study the existence and global exponential stability of almost periodic solutions in distribution of system (1.1). The innovations of this paper are as follows: (1) This is the first paper that uses a non-decomposition method to study stochastic NNs whose coefficients are all Clifford numbers except for time delays. (2) This is the first time to study almost periodic solutions in distribution of Clifford-valued stochastic high-order Hopfield NNs. (3) The method of dealing with time-varying delays in this paper can be used to study the corresponding problems of other types of stochastic NNs with time-varying delays. (4) When the system we consider degenerates into real-valued system, complex-valued system or quaternion-valued system, the results of this paper are also new.

The rest of this paper is organized as follows. In Sect. 2, we recollect some basic definitions and lemmas. In Sect. 3, based on the principle of contractive mapping, we establish the existence of almost periodic solutions in distribution for system (1.1). In Sect. 4, we study the global exponential stability of the almost periodic solution in distribution of system (1.1) by inequality techniques. In Sect. 5, we give an example to illustrate the feasibility of the theoretical results obtained in this paper. In Sect. 6, we give a concise conclusion to end this paper.

2.   Preliminaries

The real Clifford algebra over $\mathbb{R}^m$ is defined as

where $\Pi = \{\emptyset, 1, 2, \ldots, A, \ldots, 12\cdots m\}$, $e_\emptyset = e_0 = 1$ and $e_p, p = 1, 2, \ldots, m$ are called the Clifford generators and satisfy there exits an $\iota \, (0\leq\iota < m)$ such that

For $x = \sum\limits_{A}x^Ae_A\in \mathcal{A}$, let $\|x\|_{\mathcal{A}} = \max\limits_{A}\{\mid x^A\mid\}$, where $\sum\limits_{A}$ and $\max\limits_{A}$ are short for $\sum\limits_{A\in\Pi}$ and $\max\limits_{A\in\Pi}$, respectively. For $y = (y_1, y_2, \dots, y_n)^T\in \mathcal{A}^n$, we define $\|y\|_n = \max\{\|y_p\|_\mathcal{A}\}$. Then $(\mathcal{A}, \|\cdot\|_{n})$ is a Banach space. For more information on Clifford analysis, see ^[31].

Throughout this paper, for $x = \sum\limits_{A}x^Ae_A\in\mathcal{A}$, we denote $x^{\eth} = \sum\limits_{A\neq\emptyset}x^Ae_{A}$ and $x^{\emptyset} = x-x^{\eth}$.

Definition 2.1. ^[6] Let $BC(\mathbb{R}, \mathcal{A}^n)$ denote the set of all bounded continuous functions from $\mathbb{R}$ to $\mathcal{A}^n$. A function $f\in BC(\mathbb{R}, \mathcal{A}^n)$ is said to be almost periodic, if for every $\varepsilon > 0$ there exists a positive number $\ell$ such that every interval of length $\ell$ contains a number $\tau$ such that

The $\tau$ is called the $\varepsilon$-translation number of $f$. Denote by $AP(\mathbb{R}, \mathcal{A}^n)$ the set of all such functions.

Let $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\geq0}, P)$ be a complete probability space with a natural filtration $\{\mathcal{F}_t\}_{t\geq0}$ satisfying the usual conditions.

A stochastic process $X = \{X(t):t\geq0\}$ (or, simply $X_t$) is called adapted to the filtration $\{\mathcal {F}_t\}_{t\geq0}$ if $X_t$ is $\mathcal F_t$-measurable for all $t\geq0$.

Let $(\mathbb E, d)$ be a separable, complete metric space and $\mathcal B(\mathbb E)$ be the $\sigma$-algebra of Borel sets of $\mathbb E$. We denote by $\mathcal P(\mathbb E)$ the set of all probability measures defined on $\mathcal B(\mathbb E)$ and by $CB(\mathbb{E})$ the set of all bounded continuous functions $f:\mathbb E \rightarrow \mathbb R$ with $\|f\|_\infty: = \sup_{x\in\mathbb E}\mid f(x)\mid < \infty$.

For $f\in CB(\mathbb E)$, $\mu, \nu\in \mathcal{P}(\mathbb E)$, we define

It is well known that the metric space $(\mathcal P(\mathbb E), d_{BL})$ is a Polish space ^[32]. For a random variable $X:(\Omega, \mathcal{F}, P)\rightarrow \mathbb E$, we will denote by $\mu(X): = P\circ X^{-1}$ its law and by $E(X)$ its expectation.

Let $\mathcal L^2(\Omega, \mathcal A^n)$ be the space of all $\mathcal A^n$-valued random variables such that

For $X\in\mathcal L^2(\Omega, \mathcal A^n)$, we denote

Definition 2.2. ^[33] A stochastic process $X:\mathbb R\rightarrow \mathcal L^2(\Omega, \mathcal A^n)$ is said to be $\mathcal L^2$-continuous if for any $t_0\in \mathbb R$,

It is said to be $\mathcal L^2$-bounded if $\sup\limits_{t\in \mathbb R}E\| X(t)\|^2_n < \infty$.

Definition 2.3. ^[34] An $\mathcal{L}^2$-continuous stochastic process $X:\mathbb R\rightarrow \mathcal L^2(\Omega, \mathcal A^n)$ is said to be square-mean almost periodic if for every $\varepsilon > 0$, there exist a positive number $\ell$ such that every interval of length $\ell$ contains a number $\tau$ such that

Definition 2.4. ^[35] A stochastic process $X:\mathbb R \rightarrow \mathcal A^n$ is said to be almost periodic in distribution if the mapping

is almost periodic, where $\mu (X(t)) = P\circ[X(t)]^{-1}$ is the law of $X(t)$ under $P$, that is to say, if for every $\varepsilon > 0$, there exists a positive number $\ell$ such that every interval of length $\ell$ contains a number $\tau$ such that

From Remark 2.12 in ^[33], one can deduce that

Lemma 2.1. If an $\mathcal{L}^2$-continuous stochastic process $X(t)$ is square-mean almost periodic, then $X(t)$ is almost periodic in distribution; but the converse is not true.

Lemma 2.2. ^[36] Let $g:\mathbb R\rightarrow \mathbb R$ be a continuous function such that, for every $t\in\mathbb R$,

where $\alpha, \beta, \delta\geq0$ are constants and $\delta > \beta$. Then we have $g(t)\leq\alpha\frac{\delta}{\delta-\beta}$.

In the rest part of this paper, we will adopt the following notation:

Throughout this paper, we assume that

$(H_1)$ For $p, q, l\in\mathcal{D}$, $c_p^{\emptyset}\in AP(\mathbb{R}, \mathbb{R}^+)$ with $\underline{c}_p^{\emptyset} > 0$, $c_p^{\eth}, a_{pq}, b_{pql}, I_{p}\in AP(\mathbb{R}, \mathcal{A})$, $\tau_{pq}, \sigma_{pql}, \nu_{pql}, \gamma_{pq}\in AP(\mathbb{R}, \mathbb{R^{+}})\cap C^{1}(\mathbb{R}, \mathbb{R})$, there exist positive constants $\dot{\tau}_{pq}^{+}, \dot{\sigma}_{pql}^{+}, \dot{\nu}_{pql}^{+}, \dot{\gamma}_{pq}^{+}$ such that $\dot{\tau}_{pq}(t)\leq\dot{\tau}_{pq}^{+} < 1$, $\dot{\sigma}_{pql}(t)\leq\dot{\sigma}_{pql}^{+} < 1$, $\dot{\nu}_{pql}(t)\leq\dot{\nu}_{pql}^{+} < 1$, $\dot{\gamma}_{pq}(t)\leq\dot{\gamma}_{pq}^{+} < 1$.

$(H_2)$ For $q\in\mathcal{D}$, $f_{q}, h_{q}\in C(\mathcal{A}, \mathcal{A})$, there exist constants $L^{f}_{q}, L^{g}_{q}, M_q^g$ such that

for all $x, y\in \mathcal{A}$, and $f_{q}(0) = g_{q}(0) = \delta_{pq}(0) = 0.$

$(H_3)$

$(H_{4})$

3.   The existence of almost periodic solutions in distribution

We denote by $UCB(\mathbb R, \mathcal L^{2}(\Omega, \mathcal A^n))$ the space of all $\mathcal{L}^2$-bounded and uniformly $\mathcal{L}^2$-continuous functions from $\mathbb R$ to $\mathcal L^{2}(\Omega, \mathcal A^n)$. Let $\mathbb{B} = UCB(\mathbb{R}, \mathcal{L}^2(\Omega, \mathcal{A}^n))$ with the norm $\|x\|_{\mathbb{B}} = \bigg(\sup\limits_{t\in\mathbb{R}}E(\|x(t)\|^2_n)\bigg)^{\frac{1}{2}}$, then $\mathbb{B}$ is a Banach space.

Set $x^0 = (x_1^0, x_2^0, \ldots, x_n^0)^T$, where $x_p^0(t) = \int_{-\infty}^te^{-\int_s^tc_p^{\emptyset}(u)du}I_p(s)ds$, $t\in\mathbb{R}$, $p\in\mathcal{D}$ and take a constant $\kappa$ such that $\|x^0\|_{\mathbb{B}}\leq \kappa$.

Definition 3.1. An $\mathcal F_t$-progressively measurable stochastic process $x(t)$ is called a mild solution of system $(1.1)$, if $x(t)$ satisfies the following stochastic integral equation

where $t\geq t_0, p\in\mathcal{D},$

Theorem 3.1. Assume that $(H_1)$-$(H_4)$ hold, then system $(1.1)$ has a unique almost periodic solution in distribution in the closed ball $\mathbb{B}_{\kappa} = \{x\mid x\in \mathbb{B}, \|x-x^0\|_{\mathbb{B}}\leq \kappa\}$.

Proof. According to Definition 3.1, taking the limit as ${t_{0}}\longrightarrow -\infty$, we obtain

which is a mild solution of system (1.1).

Define an operator $\Phi:\mathbb{B}_{\kappa}\rightarrow \mathbb{B}_{\kappa}$ by

where for $t\in\mathbb{R}, p\in \mathcal{D},$

Firstly, we will prove that the operator $\Phi$ is well defined.

In fact, for $x = (x_1, x_2, \ldots, x_n)^T\in \mathbb{B}_{\kappa}$, one has

and

By the Cauchy-Schwarz inequality, we have

and

Similarly, we have

Moreover, by the Itô isometry, we obtain

Substituting $(3.4)$–$(3.7)$ into $(3.3)$, by (3.2) and $(H_3)$, we have

For any $x\in \mathbb{B}_\kappa$ and $t_1, t_2\in\mathbb{R}$ with $t_1 > t_2$, we derive that

where

Consequently, we deduce that $E\|(\Phi x)(t_1)-(\Phi x)(t_2)\|_{0}^2\rightarrow0$ as $t_1\rightarrow t_2$, which implies $\Phi x$ is uniformly $\mathcal{L}^2$-continuous. Therefore, we gain $\Phi(\mathbb{B}_{\kappa})\subset\mathbb{B}_{\kappa}$, that is, $\Phi$ is well defined.

Next, we will show that $\Phi$ is a contraction operator. Actually, for every $x, y\in\mathbb{B}_{\kappa}$, one has

Hence,

which implies that $\Phi$ is a contraction mapping. Therefore, $\Phi$ has a unique fixed point $x$ in $\mathbb{B}_{\kappa}$, that is, system (1.1) possesses a unique solution $x$ in $\mathbb{B}_{\kappa}$.

Finally, we will show that the unique solution $x$ of system (1.1) in $\mathbb{B}_{\kappa}$ is almost periodic in distribution.

From the above discussion, we know that $x\in UCB(\mathbb{R}, \mathcal{L}^{2}(\Omega, \mathcal{A}^n))$. Hence, for given $\epsilon > 0$, there exists $\delta\in(0, \epsilon)$ such that, when $\mid t_1-t_2 \mid < \delta$, we have $E\|x(t_1)-x(t_2)\|_0^2 < \epsilon$. According to $(H_1)$, we see that, for the $\delta$ above, there exists $l > 0$ such that in every interval of length $l$ of $\mathbb{R}$, we can find a number $\varsigma$ such that for $t\in\mathbb{R}, \, p, q\in\mathcal{D},$

and hence, we have

By (3.1), we get

where $p\in\mathcal{D}$, $\omega_q(s+\varsigma)-\omega_q(\varsigma)$ is a Brownian motion with the same distribution as $\omega_q(s)$.

Let us consider the process

Then

According to the Cauchy-Schwarz inequality, we have that

Similarly, we can get

Note that

let $u = s-\tau_{pq}(s+\varsigma)$, we obtain

By a same method, one gets

Similar to (3.18) and by the It$\hat{o}$ isometry, one has

and

Substituting (3.9)–(3.21) into (3.8), we get

where $c^- = \min\limits_{p\in \mathcal{D}}\{\underline{c}_p^{\emptyset}\}$,

Thus, by Lemma 2.2, we conclude that

which implies that ${x}(t)$ is square-mean almost periodic. According to Lemma 2.1, we deduce that $x(t)$ is almost periodic in distribution. The proof is complete.

4.   Global exponential stability in mean square

Definition 4.1. ^[37] Let $x$ be an almost periodic solution in distribution of (1.1) with the initial value $\varphi$. If there exist positive constants $\lambda > 0$ and $M > 0$ such that for every solution $y$ with initial value $\psi$ satisfies

where $\| \varphi-\phi\|_\theta^2 = \sup\limits_{s\in[-\theta, 0]}\| \varphi(s)-\phi(s)\|_{n}^2,$ then the almost periodic solution $x(t)$ in distribution of (1.1) is said to be globally exponentially stable.

Theorem 4.1. Assume that $(H_1)$–$(H_4)$ hold, then the unique almost periodic solution in distribution of system $(1.1)$ is globally exponentially stable.

Proof. From Theorem 3.1, we know that system (1.1) has a unique almost periodic solution $x$ in distribution with the initial value $\varphi$. Suppose that $y$ is an arbitrary solution of (1.1) with initial value $\psi$. Set $z = x-y$, then from (1.1), we get

Let $\Upsilon_p: \mathbb{R}\rightarrow \mathbb{R}$ be defined as follows:

By $(H_4)$, we have $\Upsilon_p(0) > 0$. Due to the continuity of $\Upsilon_p$ on $[0, +\infty)$ and the fact that $\Upsilon_p(\varpi)\rightarrow -\infty$, as $\varpi \rightarrow +\infty$, there exists $\varsigma_p > 0$ such that $\Upsilon_p(\varsigma_p) = 0$ and $\Upsilon_p(\varpi) > 0$ for $\varpi\in(0, \varsigma_p)$. Consequently, one can take a positive constant $0 < \lambda < \min_{1\leq p\leq n}\{\varsigma_p, \underline{c}_p^{\emptyset}\}$ such that $\Upsilon(\lambda) > 0, \, \, p\in\mathcal{D}$, which implies that

Let $M = \max\limits_{p\in \mathcal{D}}\Big\{\frac{(\underline{c}_p^{\emptyset})^2}{\gamma_p}\Big\}$, where

then by $(H_4)$, we know $M > 1$, and further, we can deduce that

Hence, for any $\epsilon > 0$, it is easy to see that

and for any $t\in[-\varrho, 0]$,

We assert that

On the contrary, there exists a certain $t_1 > 0$ such that

Multiplying (4.1) by $e^{\int_0^sc_p^{\emptyset}(u)du}$ and integrating it over the interval $[0, t]$, we deduce that, for $p\in\mathcal{D}$,

From the above equation, we have

By the Cauchy-Schwarz inequality, (4.3) and (4.4), one can get

By the Itô isometry, one gets

Substituting (4.6)–(4.9) into (4.5), one has

Therefore,

which contradicts (4.3). Thus, (4.2) holds. Letting $\epsilon\rightarrow0^+$, by (4.2), one has

Consequently, the almost periodic solution in distribution of system (1.1) is globally exponentially stable. This completes the proof.

Remark 4.1. In ^[37], a class of Clifford-valued stochastic recurrent neural networks whose leakage term coefficients are real numbers are considered. In this paper, we consider a class of Clifford-valued stochastic high-order Hopfield neural networks whose leakage term coefficients are also Clifford numbers. Therefore, even if the $c_p(t)$ in (1.1) is a real-valued function, the conclusions of theorems 3.1 and 4.1 in this paper cannot be obtained from the corresponding results in ^[37].

5.   A numerical example

Our example is as follow.

Example 5.1. In system $(1.1)$, let $m = n = 2$, $\iota = 1$, and for $p, q, l = 1, 2$, take

By a simple calculation, we have

Therefore, all of the conditions of Theorem 4.1 are satisfied. Hence, system $(1.1)$ has a unique almost periodic solution in distribution that is globally exponentially stable (see Figures 1–10).

Remark 5.1. The results of Example 5.1 cannot be deduced from the existing results.

6.   Conclusions

In this work, we use a direct method to prove the existence and global exponential stability of almost periodic solutions in distribution for Clifford-valued stochastic higher-order Hopfield neural networks with all parameters being Clifford numbers except time delays. Even when the neural network considered in this paper degenerates into real-valued, complex-valued and quaternion-valued neural networks, our results are new. In addition, the method proposed in this paper can be used to study the almost periodic solutions in distribution for other types of Clifford-valued stochastic neural networks with time-varying delays.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant No. 11861072, the Natural Science Foundation of Anhui Province under Grant No. 2108085QA10 and the Applied Basic Research Foundation of Yunnan Province under Grant No. 2019FB003.

Conflict of interest

The authors declare that they have no competing interests.