Discrete-Time Stability Analysis of Neural Networks with Piecewise Constant Arguments

Abstract

This paper studies the stability properties of a class of Hopfield-type neural networks involving conformable derivatives and piecewise constant arguments. By constructing an associated discrete-time formulation, the continuous system is expressed in a form that is more suitable for analysis. A Lyapunov-based approach is then developed to investigate the asymptotic and exponential stability of the equilibrium point of the resulting discrete system. The analysis provides conditions that depend on the system parameters and the conformable derivative order, offering insight into the convergence behavior of solutions. The proposed approach treats the discrete formulation as an analytical tool for studying the original model. A numerical example is included to illustrate the theoretical results.

1. Introduction

Artificial intelligence (AI) is a broad scientific field based on the idea of replicating human brain capabilities within a mathematical and computational framework. This field emerged with two key motivations: first, to provide a computational basis for neuroscience by modeling biological brain functions, and second, to develop systems that can outperform the human brain in specific tasks such as image recognition, speech processing, and coordination. Artificial neural networks (ANNs), a crucial subset of AI, are mathematical models inspired by biological neural networks. Neurons receive weighted input through synaptic connections, accumulate these signals, and produce an output when the total input exceeds a threshold value. This mechanism is the fundamental concept of ANNs, which was first mathematically modeled by McCulloch and Pitts in 1943 [1]. Since the computational capacity of a single neuron is highly limited to solving nonlinear problems, the idea of interconnected neurons forming complex networks emerged as a natural extension to overcome these limitations. Significant advances were made in this area with the introduction of Rosenblatt’s Perceptron [2], a multilayer network that can learn patterns, and later Hopfield networks [3], which approached ANNs from the dynamical perspective. Hopfield networks demonstrated remarkable potential in solving optimization problems and pattern recognition tasks, despite their capacity limitations for large-scale data. Their dynamics is governed by differential equations, where the neuron states evolve over time, influenced by connection weights, activation functions, and external inputs. The stability and convergence properties of these networks were widely studied in several studies ([4,5,6]). Such studies laid the foundation for modern neural network research. As neural network models become more complex, understanding their dynamic behavior, stability, and learning capabilities remains important for the advancement of AI systems in both theoretical and practical areas.

Differential Equations with Piecewise Constant Arguments (DEPCAs), which fall into the category of delayed differential equations, first appeared in a contagious disease model established by Busenberg and Cooke [7] in the early 1980s. This topic was later explored by Cooke and Wiener [8], Wiener ([9,10]) and Shah and Wiener [11]. Since DEPCAs exhibit characteristics of both differential and difference equations, they are known as hybrid dynamical systems. Additionally, unlike delay differential equations, solving DEPCAs requires only a finite number of initial points rather than an initial function, making them even more practical. These types of equations have gained significant attention in the literature since they have a wide range of applications in fields such as biology, control theory, and disease modeling ([12,13,14]). The oscillation, stability, and periodicity properties of the solutions of DEPCAs were investigated in ([15,16,17,18,19]). Unsurprisingly, this versatile equation type has also been applied to artificial neural networks. In 2004, Zhu and Huang studied the periodic and convergence properties of a two-neuron system with piecewise continuous arguments [20]. Later, in [21], for the equation

$${\displaystyle \frac{d{x}_i\left(t\right)}{dt}}=-a_i\left(\left[t\right]\right)x_i\left(t\right)+{\displaystyle \sum _{j=1}^n}{b}_{ij}\left(\left[t\right]\right)f_j\left(x_j\left(\left[t\right]\right)\right)+I_i\left(\left[t\right]\right),i=1,2,\dots ,n,$$

sufficient conditions for the existence of almost periodic solutions were obtained. In 2014, Chiu et al. investigated the global exponential stability of periodic solutions for a neural network model with a generalized piecewise continuous argument [22]; then, Akhmet et al. examined the exponential stability of bounded and periodic solutions of neural network models with a generalized piecewise continuous argument [23]. In [24,25], the global exponential stability and periodicity of a neural network model with generalized and alternately delayed and advanced piecewise continuous arguments were established, respectively.

The conformable derivative is a relatively new formulation that was introduced by Khalil et al. [26] around ten years ago to overcome some of the shortcomings of fractional calculus. It maintains several essential properties, such as linearity, product and quotient rules, mean value theorem, Rolle’s theorem, conformable counterparts of the Gronwall inequality, chain rule, integration by parts formula, power series, and Laplace transforms [27]. The geometric interpretation of conformable curves and orthogonal trajectories was explored in [28]. Although the conformable derivative is a local operator, the simplicity of the definition and its reliance on basic differentiation rules make it particularly useful for analytical work. Unlike other fractional operators, it retains the properties of standard differentiation rules and reduces to the classical derivative when $\alpha =1$, thereby enhancing its analytical feasibility ([29,30,31,32,33,34]).

In recent years, artificial neural network models with fractional derivatives have attracted great attention. These types of ANNs have been studied from a signal processing perspective and have been observed to be more flexible and advantageous compared to classical derivative-based networks. Such neural networks have not only been effectively applied to fields like finance and economics but also in various physical systems such as electrical circuits, chemical processes, and biological systems. For instance, in 2009, a fractional-order Hopfield neural network model was examined in [35]. Later, the finite-time stability of a delayed neural network model with fractional derivatives and the stability of the fractional-order Hopfield neural network model were investigated in [36,37], respectively. In [38], the classical derivative was replaced by a conformable derivative in the Hopfield ANNs system, and the fractional exponential stability of the system was examined. Similar results were shown for a BAM-type model in [39].

A review of the literature reveals that studies involving fractional derivatives in neural networks with piecewise constant arguments are quite limited. This subject was first introduced in [40], and some criteria were obtained to guarantee the Mittag-Leffler stability of these systems. Subsequently, in [41], using Lyapunov’s theorem, new sufficient Mittag-Leffler stability conditions were developed for neural networks with piecewise constant arguments and fractional derivatives. Furthermore, in [42], the global Mittag-Leffler stability of quaternion-valued neural networks with fractional order, piecewise constant arguments, and impulses was analyzed.

Motivated by this perspective, the stability of a class of Hopfield-type neural networks involving conformable derivatives and piecewise constant arguments is studied. Within this framework, sufficient conditions ensuring the asymptotic and exponential stability of the equilibrium are derived. In addition, the influence of the conformable order $\alpha$ on the convergence behavior of the solutions is examined.

In this paper, the following initial value problem containing a Hopfield-type neural network with piecewise constant arguments and conformable derivatives of order $\alpha$ is considered:

$${\mathcal{T}}_n^{\alpha }\left(x_i\right)\left(t\right)=-a_i{x}_i\left(t\right)+{\displaystyle \sum _{j=1}^m}{b}_{ij}{f}_j\left(x_j\left(\lfloor t-j\rfloor \right)\right)+{\theta }_i,i=1,2,\dots ,m,t>n\ge 0$$

(1)

with the initial conditions

$$x_i\left(0\right)=c_{0i},x_i(-1)=c_{-1i},\dots ,x_i(-m)=c_{-mi},i=1,2,\dots ,m$$

(2)

where ${\mathcal{T}}_n^{\alpha }$ corresponds to a conformable derivative of order $\alpha ,0<\alpha <1$, $x_i\left(t\right)$ represents the state variable of the ith unit at time t, $f_j$ corresponds to the activation function of the jth neuron, $a_i>0$ denotes the rate with which unit i will reset its potential to the resting state when disconnected from the network, $b_{ij}$ and ${\theta }_i$ are any real constants, $b_{ij}$ denotes the connection weight of unit j on unit i at time t, ${\theta }_i$ represents the input from outside the network to unit i, $\lfloor .\rfloor$ is the floor function, and $c_{0i},c_{-1i},\dots ,c_{-mi},i=1,2,\dots ,m$, are given real constants.

The paper is organized as follows. Section 2 presents the necessary preliminary results. In Section 3, the discrete form of the system (1) and (2) is constructed and its asymptotic and exponential stability is analyzed using a Lyapunov approach. Section 4 provides a numerical example illustrating the theoretical results. The paper concludes with some remarks and open problems.

2. Preliminaries

In this section, the definition and some basic properties of the conformable derivative are recalled.

Definition 1

([26]). The α-order conformable derivative of $f:[0,\infty )\to \mathbb{R}$ is denoted by ${\mathcal{T}}_0^{\alpha }\left(f\right)\left(t\right)$ or $f^{\left(\alpha \right)}\left(t\right)$ and defined as

$${\mathcal{T}}_0^{\alpha }\left(f\right)\left(t\right)=\underset{ϵ\to 0}{\lim }{\displaystyle \frac{f(t+ϵt^{1-\alpha })-f\left(t\right)}{ϵ}}$$

for all $t>0$ and $\alpha \in (0,1]$.

• If f is α-order differentiable in some interval $(0,\delta )$, $\delta >0$, and $li{m}_{t\to 0^{+}}{f}^{\left(\alpha \right)}\left(t\right)$ exists, then

  $$f^{\left(\alpha \right)}\left(0\right)=\underset{t\to 0^{+}}{\lim }{f}^{\left(\alpha \right)}\left(t\right).$$

  On the other hand, the conformable integral of a function $f:[0,\infty )\to \mathbb{R}$ is defined by

  $$I_0^{\alpha }\left(f\right)\left(t\right)={\int }_0^t{s}^{\alpha -1}f\left(s\right)ds$$

  where $\alpha \in (0,1].$

This definition can be generalized to the functions defined on the interval $[\sigma ,\infty )$ as follows:

Definition 2

([27]). The α-order conformable derivative of a function $f:[\sigma ,\infty )\to \mathbb{R}$ can be defined as

$${\mathcal{T}}_{\sigma }^{\alpha }\left(f\right)\left(t\right)=\underset{ϵ\to 0}{\lim }{\displaystyle \frac{f(t+ϵ{(t-\sigma )}^{1-\alpha })-f\left(t\right)}{ϵ}},t>\sigma ,$$

where $\alpha \in (0,1]$. Also, the conformable integral of a function $f:[\sigma ,\infty )\to \mathbb{R}$ is defined as

$$I_{\sigma }^{\alpha }\left(f\right)\left(t\right)={\int }_{\sigma }^t{(s-\sigma )}^{\alpha -1}f\left(s\right)ds$$

where $\alpha \in (0,1].$

Theorem 1

([26]). If $f_1$ and $f_2$ are α-order differentiable, $\alpha \in (0,1]$, at a point $t>\sigma$, then

1. 

${\mathcal{T}}_{\sigma }^{\alpha }(a{f}_1+b{f}_2)=a{\mathcal{T}}_{\sigma }^{\alpha }\left(f_1\right)+b{\mathcal{T}}_{\sigma }^{\alpha }\left(f_2\right)$ for all $a,b\in \mathbb{R}$;

2. 

${\mathcal{T}}_{\sigma }^{\alpha }\left(f_1{f}_2\right)=f_1{\mathcal{T}}_{\sigma }^{\alpha }\left(f_2\right)+f_2{\mathcal{T}}_{\sigma }^{\alpha }\left(f_1\right)$;

3. 

${\mathcal{T}}_{\sigma }^{\alpha }\left({\displaystyle \frac{f_1}{f_2}}\right)={\displaystyle \frac{f_2{\mathcal{T}}_{\sigma }^{\alpha }\left(f_1\right)-f_1{\mathcal{T}}_{\sigma }^{\alpha }\left(f_2\right)}{f_2^2}}$;

4. 

If f is differentiable, then ${\mathcal{T}}_{\sigma }^{\alpha }f\left(t\right)={(t-\sigma )}^{1-\alpha }{f}^{\prime }\left(t\right)$;

5. 

If c is a constant, then ${\mathcal{T}}_{\sigma }^{\alpha }\left(c\right)=0$.

The definition of the solution of the initial value problem (1) and (2) is defined as follows:

Definition 3.

Let $I={\mathbb{R}}^{+}\cup \{-m,\dots ,-1,0\}$. Then, a solution $x:I\to {\mathbb{R}}^m$ of the initial value problem (1) and (2) is a vector function $x\left(t\right)={\left(x_1\left(t\right), x_2\left(t\right),\dots ,x_m\left(t\right)\right)}^T$ that satisfies the following conditions:

(i)   

$x\left(t\right)$ is continuous on I,

(ii)  

$x\left(t\right)$ is differentiable and an α-conformable derivative of $x\left(t\right)$ exists on I, with a possible exception at the points $\lfloor t\rfloor \in I$ where one-sided derivatives exist,

(iii) 

$x\left(t\right)$ satisfies Equation (1) on each interval $[n,n+1)$, $n=0,1,\dots$,

(iv) 

$x\left(t\right)$ satisfies the initial conditions (2).

Differential equations with piecewise constant arguments are often investigated with the aid of associated difference equations. In this setting, Lyapunov–Krasovskii functionals and discrete forms of Halanay’s inequality provide suitable tools for stability analysis, especially in the presence of multiple delays. Various extensions of Halanay’s inequality can be found in [43,44,45,46,47].

A Lyapunov–Krasovskii functional criterion and a discrete version of Halanay’s inequality that will be used in the analysis are recalled below.

Theorem 2

([48]). An autonomous, multivariable discrete-time delay system can be represented by the difference equation

$$x\left(k+1\right)=f\left(x_k\right)$$

(3)

with an associated function of initial state

$$x\left(\theta \right)=\psi \left(\theta \right),\theta \in \{-h,-h+1,\dots ,0\}=\Delta ,$$

(4)

where

$$x\left(k\right)=x(k,\psi )\in {\mathbb{R}}^n$$

$$\forall \theta \in \Delta ,\forall k\in {\mathbb{Z}}^{+}=\{1,2,3,\dots \},x_k=\{x(k-h),x(k-h+1),\dots ,x\left(k\right)\}=x(k+\theta )$$

is a state vector, $A\in {\mathbb{R}}^{n\times n}$ is a constant matrix of appropriate dimension and $h\in {\mathbb{Z}}^{+}$ is an unknown time delay in the general case. Let $\mathbb{D}\left(\Delta ,{\mathbb{R}}^n\right)$ be the space of functions mapping the discrete interval $\Delta$ into ${\mathbb{R}}^n$. Then, $x_k\in \mathbb{D},\varphi \in \mathbb{D},{∥\varphi ∥}_D=\underset{\theta \in \Delta }{\sup }∥\varphi \left(\theta \right)∥$, for $x\in {\mathbb{R}}^n,∥x∥={\sum }_{i=1}^n\left|x_i\right|$ is the norm of an element $\varphi \in \mathbb{D}$ and $f:\mathbb{D}\to {\mathbb{R}}^n$. Let ${\mathbb{D}}^{\gamma }=\{\varphi \in \mathbb{D}:{∥\varphi ∥}_D<\gamma \}\subset \mathbb{D}$, where $\gamma \in \mathbb{R}$.

If there exist a continuous functional $V:\mathbb{D}\to \mathbb{R}$ and continuous non-decreasing functions ν and $w:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with features $\nu \left(0\right)=w\left(0\right)=0$, $\nu \left(s\right)>0$ and $w\left(s\right)>0,\forall s>0,$ such that

$$0<V\left(x_k\right)\le \nu \left({∥x_k∥}_D\right),V\left(0\right)=0,$$

$$\Delta V\left(x_k\right)=V\left(x_{k+1}\right)-V\left(x_k\right)\le -w\left(∥x_k∥\right),$$

$\forall x_k\in \mathbb{D}$ satisfying (3), then the solution $x=0$ of Equations (3) and (4) is asymptotically stable.

Lemma 1

([44]). Suppose that there exist constants $M_a\in (0,1)$ and $M_b\ge 0$ such that $0\le a\left(n\right)\le M_a$ and $0\le b\left(n\right)\le M_b$. Let ${\left\{x_n\right\}}_{n\ge -\tau }$ be a sequence of positive real numbers satisfying the inequality

$$\left\{\begin{array}{c}\Delta x_n\le -a\left(n\right)x_n+b\left(n\right)\underset{0\le s\le \tau }{\sup }\left\{x_{n-s}\right\},\hfill \\ x_s=\phi \left(s\right),s\in \{-\tau ,-\tau +1,\dots ,0\}.\hfill \end{array}\right.$$

If

$$\tau <{\log }_{1-M_a}{\displaystyle \frac{M_{b}T}{M_{b}T+\delta }}$$

and

$${\displaystyle \sum _{k=n_0+nT}^{n_0+(n+1)T-1}}[a\left(k\right)-b\left(k\right)]\ge \delta$$

hold for some $\delta >0$ and integers $n_0\ge 0$, $T>0$, then $x_k\le M\exp (-\lambda k)$ for $k\ge 0$, where

$$\lambda ={\displaystyle \frac{\delta -M_{b}T[{(1-M_a)}^{-\tau }-1]}{T}},$$

and

$$M=\max \left\{x_{n_0}\exp [(M_b{(1-M_a)}^{-\tau }+\lambda )T+\lambda n_0],\underset{s\le n_0}{\sup }\left\{x_s\exp \left(\lambda s\right)\right\}\right\}.$$

An equilibrium point of system (1) is defined as a constant vector $\overline{x}={\left({\overline{x}}_1,{\overline{x}}_2,\dots ,{\overline{x}}_m\right)}^T$ satisfying the equation

$$a_i{\overline{x}}_i={\displaystyle \sum _{j=1}^m}{b}_{ij}{f}_j\left({\overline{x}}_j\right)+{\theta }_i,i=1,2,\dots ,m.$$

(5)

Throughout this paper, the functions $f_j:\mathbb{R}\to \mathbb{R}$, $j=1,2,\dots ,m$ are assumed to be Lipschitz continuous. That is, there exist positive constants $L_j$ such that

$$\left|f_j\left(u\right)-f_j\left(v\right)\right|\le L_j\left|u-v\right|$$

(6)

for all $u,v\in \mathbb{R}$ and $j=1,2,\dots ,m$.

The following lemma gives the condition for the existence and uniqueness of the equilibrium point of (1). Since the proof is the same as in [38], it is omitted here.

Lemma 2.

Let (6) be true and suppose further that $a_i,b_{ji}$ and $L_i$ satisfy

$$L_i{\displaystyle \sum _{j=1}^m}\left|b_{ji}\right|<a_i,i=1,2,\dots ,m.$$

(7)

Then, there exists a unique equilibrium point of the system (1).

3. Stability Analysis of the Discrete System

In this section, first, the corresponding discrete version of (1) and (2) is constructed. Then, the stability analysis of this discrete system is investigated.

To facilitate analysis, the equilibrium point $\overline{x}$ of Equation (1), given as (5), is shifted to the origin with the help of the transformation

$$y_i\left(t\right)=x_i\left(t\right)-\overline{x_i},i=1,2,\dots ,m.$$

(8)

Substituting (8) into Equation (1), we obtain

$${\mathcal{T}}_n^{\alpha }\left(y_i\right)\left(t\right)=-a_i{y}_i\left(t\right)+{\displaystyle \sum _{j=1}^m}{b}_{ij}{g}_j\left(y_j\left(\lfloor t-j\rfloor \right)\right),t>n\ge 0$$

(9)

where

$$g_j\left(s\right)=f_j\left(s+{\overline{x}}_j\right)-f_j\left({\overline{x}}_j\right).$$

It is clear that $g_j\left(0\right)=0$ and $g_j$ satisfy the Lipschitz condition with $L_j$ for $j=1,2,\dots ,m$.

Moreover, the initial conditions (2) are transformed into

$$y_i\left(0\right)=d_{0i},y_i\left(-1\right)=d_{-1i},\dots ,y_i\left(-m\right)=d_{-mi}$$

(10)

where

$$d_{0i}=c_{0i}-\overline{x_i},d_{-1i}=c_{-1i}-\overline{x_i},\dots ,d_{-mi}=c_{-mi}-\overline{x_i},i=1,2,\dots ,m.$$

Henceforth, our investigation focuses on the initial value problem (9) and (10).

Theorem 3.

The unique solution of the initial value problem (9) and (10) is expressed as

$$y_i\left(t\right)={\mathrm{e}}^{-{\displaystyle \frac{a_i}{\alpha }}{\left(t-n\right)}^{\alpha }}{y}_i\left(n\right)+{\displaystyle \frac{1-{\mathrm{e}}^{-\frac{a_i}{\alpha }{\left(t-n\right)}^{\alpha }}}{a_i}}{\displaystyle \sum _{j=1}^m}{b}_{ij}{g}_j\left(y_j\left(n-j\right)\right)$$

(11)

for $i=1,2,\dots ,m,$$n\le t<n+1,n=0,1,\dots ,$ where $y_i\left(n\right)$ satisfy the corresponding difference system

$$y_i\left(n+1\right)={\mathrm{e}}^{-{\displaystyle \frac{a_i}{\alpha }}}{y}_i\left(n\right)+{\displaystyle \frac{1-{\mathrm{e}}^{-\frac{a_i}{\alpha }}}{a_i}}{\displaystyle \sum _{j=1}^m}{b}_{ij}{g}_j\left(y_j\left(n-j\right)\right)$$

(12)

for $i=1,2,\dots ,m$.

Proof. 

Let $y_i\left(t\right)\equiv y_{\left(n\right)i}\left(t\right)$ denote the solution of (9) and (10) for $n\le t<n+1,n=0,1,\dots$ Then, Equation (9) simplifies to

$${\mathcal{T}}_n^{\alpha }\left(y_i\right)\left(t\right)=-a_i{y}_i\left(t\right)+{\displaystyle \sum _{j=1}^m}{b}_{ij}{g}_j\left(y_j\left(n-j\right)\right),n<t<n+1,n=0,1,\dots$$

(13)

Applying the fourth property of Theorem 1 to Equation (13), we obtain the linear differential equation

$${\displaystyle \frac{d{y}_i\left(t\right)}{dt}}+a_i{\left(t-n\right)}^{\alpha -1}{y}_i\left(t\right)={\left(t-n\right)}^{\alpha -1}{\displaystyle \sum _{j=1}^m}{b}_{ij}{g}_j\left(y_j\left(n-j\right)\right),n\le t<n+1,n=0,1,\dots$$

The solution of the above equation is

$$y_i\left(t\right)\equiv y_{\left(n\right)i}\left(t\right)={\mathrm{e}}^{-{\displaystyle \frac{a_i}{\alpha }}{\left(t-n\right)}^{\alpha }}{y}_i\left(n\right)+{\displaystyle \frac{1-{\mathrm{e}}^{-\frac{a_i}{\alpha }{\left(t-n\right)}^{\alpha }}}{a_i}}{\displaystyle \sum _{j=1}^m}{b}_{ij}{g}_j\left(y_j\left(n-j\right)\right).$$

Similarly, on the interval $n+1<t<n+2$, the solution of (9) and (10) has the form

$$y_i\left(t\right)\equiv y_{(n+1)i}\left(t\right)={\mathrm{e}}^{-{\displaystyle \frac{a_i}{\alpha }}{\left(t-n-1\right)}^{\alpha }}{y}_i(n+1)+{\displaystyle \frac{1-{\mathrm{e}}^{-\frac{a_i}{\alpha }{\left(t-n-1\right)}^{\alpha }}}{a_i}}{\displaystyle \sum _{j=1}^m}{b}_{ij}{g}_j\left(y_j\left(n+1-j\right)\right).$$

Since the solution is continuous, we have

$$\underset{t\to {(n+1)}^{-}}{\lim }{y}_{\left(n\right)i}\left(t\right)=\underset{t\to {(n+1)}^{+}}{\lim }{y}_{(n+1)i}\left(t\right)=y_i(n+1)$$

which yields the difference system (12). Evidently, under the initial conditions (10), the solution of the difference system (12) is unique. Consequently, the solution of the initial value problem (9) and (10) becomes unique. □

The equilibrium point of the difference equation system (12) is given by the systems of equations

$$\overline{y_i}={\mathrm{e}}^{-{\displaystyle \frac{a_i}{\alpha }}}\overline{y_i}+{\displaystyle \frac{1-{\mathrm{e}}^{-\frac{a_i}{\alpha }}}{a_i}}{\displaystyle \sum _{j=1}^m}{b}_{ij}{g}_j\left(\overline{y_j}\right),i=1,2,\dots ,m,$$

or equivalently

$$a_i\overline{y_i}={\displaystyle \sum _{j=1}^m}{b}_{ij}{g}_j\left(\overline{y_j}\right),i=1,2,\dots ,m.$$

(14)

It is easy to see that the system (14) has a unique solution $\overline{y}={(\overline{y_1},\dots \overline{y_m})}^T=0\in {\mathbb{R}}^m$ under the conditions (7).

Theorem 4.

In addition to the hypotheses of Lemma 2, assume that

$${\displaystyle \sum _{j=1}^m}\left({\displaystyle \frac{1-{\mathrm{e}}^{-\frac{a_j}{\alpha }}}{a_j}}\right)\left|b_{ji}\right|<{\displaystyle \frac{1-{\mathrm{e}}^{-\frac{a_i}{\alpha }}}{L_i}},i=1,2,\dots ,m.$$

(15)

Then, the unique equilibrium point $\overline{y}=0$ of the discrete system (12) is asymptotically stable.

Proof. 

We will apply Theorem 2 for the proof. Consider the functional

$$V\left(y_n\right)={\displaystyle \sum _{i=1}^m}\left|y_i\left(n\right)\right|+{\displaystyle \sum _{i=1}^m}\left({\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_i}{\alpha }}}{a_i}}\right){\displaystyle \sum _{j=1}^m}{\displaystyle \sum _{k=n-j}^{n-1}}\left|b_{ij}{g}_j\left(y_j\left(k\right)\right)\right|.$$

(16)

It is evident that $V\left(0\right)=0$. Moreover, since ${\displaystyle \frac{1-{\mathrm{e}}^{-a_i/\alpha }}{a_i}}>0$, $V\left(y_n\right)>0$ for $y_n=(y_{1n},y_{2n},\dots ,y_{mn})\ne (0,0,\dots ,0)$.

On the other hand, considering the Lipschitz continuity of $g_j\left(s\right)$ and the notation $y_n\left(\theta \right)=y(n+\theta )$, we get

$$\begin{array}{cc}\hfill V\left(y_n\right)& \le {\displaystyle \sum _{i=1}^m}\left|y_{in}\left(0\right)\right|+M\left[{\displaystyle \sum _{i=1}^m}\left|y_{in}\left(-1\right)\right|+{\displaystyle \sum _{i=2}^m}\left|y_{in}\left(-2\right)\right|+\dots +{\displaystyle \sum _{i=m}^m}\left|y_{in}\left(-m\right)\right|\right]\hfill \\ \hfill & \le M\underset{\xi \in \Delta }{\sup }∥y_n\left(\xi \right)∥=M{∥y_n∥}_D\hfill \end{array}$$

where

$$M=max\left(1,\underset{1\le i\le m}{\max }\left\{{\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_i}{\alpha }}}{a_i}}{\displaystyle \sum _{j=1}^m}\left|b_{ij}\right|L_j\right\}\right),\Delta =-m,-m+1,\dots ,-1,0.$$

Thus, taking $\nu \left(s\right)=Ms$, we see that the conditions on $\nu$ in Theorem 2 are satisfied.

On the other hand, from (16), it is clear that

$$\begin{array}{cc}\hfill \Delta V\left(y_n\right)& =V\left(y_{n+1}\right)-V\left(y_n\right)\hfill \\ \hfill & =\left({\displaystyle \sum _{i=1}^m}\left|y_i\left(n+1\right)\right|+{\displaystyle \sum _{i=1}^m}\left({\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_i}{\alpha }}}{a_i}}\right){\displaystyle \sum _{j=1}^m}{\displaystyle \sum _{k=n+1-j}^n}\left|b_{ij}{g}_j\left(y_j\left(k\right)\right)\right|\right)\hfill \\ \hfill & -\left({\displaystyle \sum _{i=1}^m}\left|y_i\left(n\right)\right|+{\displaystyle \sum _{i=1}^m}\left({\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_i}{\alpha }}}{a_i}}\right){\displaystyle \sum _{j=1}^m}{\displaystyle \sum _{k=n-j}^{n-1}}\left|b_{ij}{g}_j\left(y_j\left(k\right)\right)\right|\right).\hfill \end{array}$$

Substituting the right-hand side of (12) in place of $y_i\left(n+1\right)$ above, we have

$$\begin{array}{cc}\hfill \Delta V\left(y_n\right)& \le {\displaystyle \sum _{i=1}^m}\left|{\mathrm{e}}^{-{\displaystyle \frac{a_i}{\alpha }}}{y}_i\left(n\right)+\left({\displaystyle \frac{1-{\mathrm{e}}^{-\frac{a_i}{\alpha }}}{a_i}}\right){\displaystyle \sum _{j=1}^m}{b}_{ij}{g}_j\left(y_j\left(n-j\right)\right)\right|-{\displaystyle \sum _{i=1}^m}\left|y_i\left(n\right)\right|\hfill \\ \hfill & +{\displaystyle \sum _{i=1}^m}\left({\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_i}{\alpha }}}{a_i}}\right){\displaystyle \sum _{j=1}^m}\left[{\displaystyle \sum _{k=n+1-j}^n}\left|b_{ij}{g}_j\left(y_j\left(k\right)\right)\right|-{\displaystyle \sum _{k=n-j}^{n-1}}\left|b_{ij}{g}_j\left(y_j\left(k\right)\right)\right|\right]\hfill \\ \hfill & \le {\displaystyle \sum _{i=1}^m}-\left(1-{\mathrm{e}}^{{\displaystyle \frac{-a_i}{\alpha }}}\right)\left|y_i\left(n\right)\right|+{\displaystyle \sum _{i=1}^m}\left({\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_i}{\alpha }}}{a_i}}\right){\displaystyle \sum _{j=1}^m}\left|b_{ij}\right|\left|g_j\left(y_j\left(n\right)\right)\right|.\hfill \end{array}$$

Usingthe Lipschitz continuity of $g_j\left(s\right)$, we obtain

$$\begin{array}{cc}\hfill \Delta V\left(y_n\right)& \le {\displaystyle \sum _{i=1}^m}-\left(1-{\mathrm{e}}^{{\displaystyle \frac{-a_i}{\alpha }}}\right)\left|y_i\left(n\right)\right|+{\displaystyle \sum _{i=1}^m}\left({\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_i}{\alpha }}}{a_i}}\right){\displaystyle \sum _{j=1}^m}\left|b_{ij}\right|L_j\left|y_j\left(n\right)\right|\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^m}-\left(1-{\mathrm{e}}^{{\displaystyle \frac{-a_i}{\alpha }}}\right)\left|y_i\left(n\right)\right|\hfill \\ \hfill & +\left({\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_1}{\alpha }}}{a_1}}\left|b_{11}\right|+{\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_2}{\alpha }}}{a_2}}\left|b_{21}\right|+\dots +{\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_m}{\alpha }}}{a_m}}\left|b_{m1}\right|\right)L_1\left|y_1\left(n\right)\right|\hfill \\ \hfill & +\left({\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_1}{\alpha }}}{a_1}}\left|b_{12}\right|+{\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_2}{\alpha }}}{a_2}}\left|b_{22}\right|+\dots +{\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_m}{\alpha }}}{a_m}}\left|b_{m2}\right|\right)L_2\left|y_2\left(n\right)\right|\hfill \\ \hfill & ⋮\hfill \\ \hfill & +\left({\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_1}{\alpha }}}{a_1}}\left|b_{1m}\right|+{\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_2}{\alpha }}}{a_2}}\left|b_{2m}\right|+\dots +{\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_m}{\alpha }}}{a_m}}\left|b_{mm}\right|\right)L_m\left|y_m\left(n\right)\right|\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^m}-\left(1-{\mathrm{e}}^{{\displaystyle \frac{-a_i}{\alpha }}}\right)\left|y_i\left(n\right)\right|+{\displaystyle \sum _{i=1}^m}{L}_i\left|y_i\left(n\right)\right|{\displaystyle \sum _{j=1}^m}{\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_j}{\alpha }}}{a_j}}\left|b_{ji}\right|.\hfill \end{array}$$

Therefore, from the above inequality, we have

$$\begin{array}{cc}\hfill \Delta V\left(y_n\right)& \le {\displaystyle \sum _{i=1}^m}\left[L_i{\displaystyle \sum _{j=1}^m}\left({\displaystyle \frac{1-{\mathrm{e}}^{\frac{-a_j}{\alpha }}}{a_j}}\right)\left|b_{ji}\right|-\left(1-{\mathrm{e}}^{{\displaystyle \frac{-a_i}{\alpha }}}\right)\right]\left|y_i\left(n\right)\right|\hfill \\ \hfill & \le -B^{*}{\displaystyle \sum _{i=1}^m}\left|y_i\left(n\right)\right|=-B^{*}∥y_n∥\hfill \end{array}$$

where

$$B^{*}=\underset{1\le i\le m}{\min }\left\{\left(1-{\mathrm{e}}^{-{\displaystyle \frac{a_i}{\alpha }}}\right)-L_i{\displaystyle \sum _{j=1}^m}\left({\displaystyle \frac{1-{\mathrm{e}}^{-\frac{a_j}{\alpha }}}{a_j}}\right)\left|b_{ji}\right|\right\}.$$

By condition (15), it is clear that $B^{*}>0.$ Hence, if we take $w\left(s\right)=B^{*}s$, the conditions on w in Theorem 2 are true. Therefore, according to Theorem 2, the unique equilibrium point $\overline{y}=0$ is asymptotically stable. □

Now, we shall obtain the sufficient conditions for the exponential stability of the discrete system (12). For this aim, let us consider the following systems of difference inequalities

$$\Delta y_i\left(n\right)\le -(1-e^{-a_i/\alpha })y_i\left(n\right)+\underset{1\le i\le m}{\max }\left[{\displaystyle \sum _{j=1}^m}{\displaystyle \frac{1-e^{-a_j/\alpha }}{a_j}}\left|b_{ji}\right|L_i\right]\left|y_{in}(-i)\right|,i=1,2,\dots ,m,$$

(17)

with the initial conditions (10).

Lemma 3.

Let ${\left\{y_i\left(n\right)\right\}}_{n\ge -m}$ be a sequence of positive real numbers satisfying the inequality (17).

• If for each $i=1,2,\dots ,m$,

  $$b:=\underset{1\le i\le m}{\max }\left[{\displaystyle \sum _{j=1}^m}{\displaystyle \frac{1-e^{-a_j/\alpha }}{a_j}}\left|b_{ji}\right|L_i\right]<(1-e^{-a_i/\alpha })e^{-m{a}_i/\alpha }$$

  (18)

  holds, then

  $$y_i\left(n\right)\le {\overline{M}}_i\exp (-{\lambda }_{i}n),\mathit{for}n\ge 0,$$

  where

  $${\lambda }_i=1-e^{-a_i/\alpha }-b{e}^{m{a}_i/\alpha }>0\mathit{for}\mathit{each}i=1,2,\dots ,m,$$

  (19)

  $${\overline{M}}_i=\max \left\{y_{i0}\exp \left[(1-e^{-a_i/\alpha })T\right],\underset{s\le 0}{\sup }\left\{y_i\left(s\right)e^{{\lambda }_{i}s}\right\}\right\},$$

  (20)

  and $T>0$ is a constant.

Proof. 

For the proof, we shall apply Lemma 1. It is clear that the terms $a\left(n\right)$ and $b\left(n\right)$ in Lemma 1 correspond to the positive constants $1-e^{-a_i/\alpha }$ and ${\max }_{1\le i\le m}{\sum }_{j=1}^m{\displaystyle \frac{1-e^{-a_j/\alpha }}{a_j}}\left|b_{ji}\right|L_i$, respectively. So, let us define

$$M_a=1-e^{-a_i/\alpha }\mathrm{and}{M}_b=b=\underset{1\le i\le m}{\max }{\displaystyle \sum _{j=1}^m}{\displaystyle \frac{1-e^{-a_j/\alpha }}{a_j}}\left|b_{ji}\right|L_i.$$

Considering (18), it can be calculated that

$${\displaystyle \sum _{k=n_0+nT}^{n_0+(n+1)T-1}}[a\left(k\right)-b\left(k\right)]=T(M_a-M_b)=\delta >0.$$

Moreover, by the condition (18), it is observed that

$$e^{-m{a}_i/\alpha }>{\displaystyle \frac{M_b}{M_a}}$$

and so

$$m<{\log }_{1-M_a}\left({\displaystyle \frac{M_b}{M_a}}\right)$$

is also satisfied. Therefore, by Lemma 1, we get

$$y_i\left(n\right)\le \overline{M_i}\exp (-{\lambda }_{i}n),n\ge 0,i=1,2,\dots ,m.$$

□

Theorem 5.

In addition to the hypotheses of Lemma 2, assume that (18) is also true; then, the zero solution of the difference system (12) is exponentially stable. That is, there exist positive constants $M^{*}$ and λ such that

$$\parallel y\left(n\right)\parallel \le M^{*}m\exp \left(-\lambda n\right),i=1,2,\dots ,m,$$

where

$$\lambda =\underset{i=1,2,\dots ,m}{\min }\left\{{\lambda }_i\right\},M^{*}=\underset{1\le i\le m}{\max }\left\{{\overline{M}}_i\right\},$$

${\lambda }_i$ and ${\overline{M}}_i$ are defined in (19) and (20), respectively.

Proof. 

Let $y\left(n\right)={(y_1\left(n\right),y_2\left(n\right),\dots ,y_m\left(n\right))}^T$ be a solution of the system (12). Then, for each $i=1,2,\dots ,m$, we have

$$\Delta y_i\left(n\right)=-(1-e^{-a_i/\alpha })y_i\left(n\right)+{\displaystyle \sum _{j=1}^m}{\displaystyle \frac{1-e^{-a_j/\alpha }}{a_j}}{b}_{ji}{g}_i\left(y_i(n-i)\right),i=0,1,\dots ,m.$$

(21)

This equation can be rewritten as

$$y_i(n+1)=(1-A_i)y_i\left(n\right)+B_i{g}_i\left(y_i(n-i)\right),i=0,1,\dots ,m,$$

(22)

where

$$A_i=1-e^{-a_i/\alpha }\mathrm{and}{B}_i={\displaystyle \sum _{j=1}^m}{\displaystyle \frac{1-e^{-a_j/\alpha }}{a_j}}{b}_{ji}.$$

It is obvious that the solution of (22) is

$$y_i\left(n\right)={(1-A_i)}^n{y}_i\left(0\right)+B_i{\displaystyle \sum _{k=0}^{n-1}}{(1-A_i)}^{n-k-1}{g}_i\left(y_i(k-i)\right),n\ge 0.$$

(23)

Since $g_i$ satisfies the Lipschitz condition for each $i=1,2,\dots ,m$, from (22), we get

$$\begin{array}{cc}\hfill |y_i\left(n\right)|& \le {(1-A_i)}^n|y_i\left(0\right)|+|B_i|L_i{\displaystyle \sum _{k=0}^{n-1}}{(1-A_i)}^{n-k-1}|y_i(k-i)|\hfill \\ \hfill & \le {(1-A_i)}^n|y_i\left(0\right)|+b{\displaystyle \sum _{k=0}^{n-1}}{(1-A_i)}^{n-k-1}\left|y_i(k-i)\right|,\hfill \end{array}$$

where b is defined in (18). Now, let us define

$$z_i\left(n\right)=\left|y_i\left(n\right)\right|,n=-m,-m+1,\dots ,0,$$

and

$$z_i\left(n\right)={(1-A_i)}^n|y_i\left(0\right)|+b{\displaystyle \sum _{k=0}^{n-1}}{(1-A_i)}^{n-k-1}\left|y_i(k-i)\right|,n>0.$$

(24)

It is clear that

$$|y_i\left(n\right)|\le z_i\left(n\right),i=1,2,\dots ,m.$$

(25)

Now, from (24) and (25), we get the inequality

$$\Delta z_i\left(n\right)\le -A_i{z}_i\left(n\right)+b\left|z_i(n-i)\right|,n>0,$$

which is in the form of (17). Hence, from Lemma 3, we have

$$|y_i\left(n\right)|\le z_i\left(n\right)\le {\overline{M}}_i\exp (-{\lambda }_{i}n).$$

From the above inequality, it is observed that

$$\parallel y\left(n\right)\parallel \le M^{*}m\exp \left(-\lambda n\right),i=1,2,\dots ,m,$$

where

$$\lambda =\underset{i=1,2,\dots ,m}{\min }\left\{{\lambda }_i\right\},M^{*}=\underset{1\le i\le m}{\max }\left\{{\overline{M}}_i\right\},$$

${\lambda }_i$ and ${\overline{M}}_i$ are defined in (19) and (20), respectively. Thus, the proof is completed. □

Corollary 1.

Let the conditions of Theorem 5 be satisfied. If for each $i=1,2,\dots ,m$

$$b:=\underset{1\le i\le m}{\max }{L}_i\left[{\displaystyle \sum _{j=1}^m}{\displaystyle \frac{1-e^{-a_j/\alpha }}{a_j}}\left|b_{ji}\right|\right]<{\displaystyle \frac{1}{m}}{e}^{-(m+1)a_i/\alpha },$$

(26)

then the convergence rate λ for the system (12) is inversely proportional to the value of α. That is, as α increases, λ decreases.

Proof. 

Let us consider the function

$$F\left(\alpha \right)=1-e^{-a_i/\alpha }-b{e}^{m{a}_i/\alpha },i=1,2,\dots ,m.$$

Then, by the condition (26), it is clear that

$$F^{\prime }\left(\alpha \right)={\displaystyle \frac{-a_i}{{\alpha }^2}}(e^{-a_i/\alpha }-bm{e}^{m{a}_i/\alpha })<0.$$

So, $F\left(\alpha \right)$ decreases under the condition (26). Since

$${\lambda }_i=1-e^{-a_i/\alpha }-b{e}^{m{a}_i/\alpha },i=1,2,\dots ,m,$$

and $\lambda ={\min }_{1\le i\le m}\left\{{\lambda }_i\right\}$, the proof is completed. □

Corollary 2.

For $\alpha =1$, (9) is transformed to

$${\displaystyle \frac{d{y}_i\left(t\right)}{dt}}=-a_i{y}_i\left(t\right)+{\displaystyle \sum _{j=1}^m}{b}_{ij}{g}_j\left(y_j\left(\lfloor t-j\rfloor \right)\right),t>n\ge 0.$$

(27)

The unique solution of Equation (27) with the initial conditions (10) is expressed as

$$y_i\left(t\right)={\mathrm{e}}^{-a_i\left(t-n\right)}{y}_i\left(n\right)+{\displaystyle \frac{1-{\mathrm{e}}^{-a_i\left(t-n\right)}}{a_i}}{\displaystyle \sum _{j=1}^m}{b}_{ij}{g}_j\left(y_j\left(n-j\right)\right)$$

(28)

on the interval $n\le t<n+1,n=0,1,\dots ,$ where $y_i\left(n\right)$ satisfy the corresponding difference system

$$y_i\left(n+1\right)={\mathrm{e}}^{-a_i}{y}_i\left(n\right)+{\displaystyle \frac{1-{\mathrm{e}}^{-a_i}}{a_i}}{\displaystyle \sum _{j=1}^m}{b}_{ij}{g}_j\left(y_j\left(n-j\right)\right)$$

(29)

for $i=1,2,\dots ,m$.

Corollary 3.

Suppose that (6) and (7) are true and for each $i=1,2,\dots ,m$,

$${\displaystyle \sum _{j=1}^m}\left({\displaystyle \frac{1-{\mathrm{e}}^{-a_j}}{a_j}}\right)\left|b_{ji}\right|<{\displaystyle \frac{1-{\mathrm{e}}^{-a_i}}{L_i}}.$$

Then, the unique equilibrium point $\overline{y}=0$ of the difference system (29) is asymptotically stable.

Corollary 4.

Suppose that (6) and (7) are true and for each $i=1,2,\dots ,m$,

$${\displaystyle \sum _{j=1}^m}\left({\displaystyle \frac{1-{\mathrm{e}}^{-a_j}}{a_j}}\right)\left|b_{ji}\right|<{\displaystyle \frac{1-{\mathrm{e}}^{-a_i}}{L_i}}{\mathrm{e}}^{-m{a}_i}.$$

Then, the unique equilibrium point $\overline{y}=0$ of the difference system (29) is exponentially stable.

Remark 1.

In this study, analyses have been carried out for $0<\alpha \le 1$. It is believed that asymptotic stability and α-synchronization in neural networks can be further explored within the range $n<\alpha \le n+1$, where $n\in \mathbb{N}$.

Remark 2.

In [21], a neural network system with piecewise constant arguments was studied; however, the arguments were restricted to the form $\lfloor t\rfloor$, and the model was formulated using the classical derivative. In contrast, the present work considers a more generalized framework in which the neural network system involves m distinct piecewise constant arguments of the form $\lfloor t-1\rfloor$, $\lfloor t-2\rfloor$, …, $\lfloor t-m\rfloor$, and the dynamics are governed by the conformable derivative.

Remark 3.

The exponential stability of continuous-type Hopfield neural networks with conformable derivatives was addressed in [38]. Additionally, the Mittag-Leffler stability of fractional-order neural networks with generalized piecewise constant arguments was explored in [40,41,42], where the systems involved arguments of the form $x\left(\gamma \right(t\left)\right)$. However, in those works, the inequality $\parallel x\left(\gamma \right(t\left)\right)\parallel \le \eta \parallel x\left(t\right)\parallel$ was employed under certain restrictive conditions. In contrast, the direct involvement of the delay argument in system (3) marks the primary distinction and advantage of our work compared to these previous studies.

4. Numerical Example

This section includes an example that verifies our results.

Example 1.

Let us consider the system

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{T}_n^{\alpha }\left(x_1\right)\left(t\right)\hfill & =-0.5x_1\left(t\right)+0.01\tanh \left(x_1\left(\lfloor t-1\rfloor \right)\right)+0.005\tanh \left(x_2\left(\lfloor t-2\rfloor \right)\right)+0.48\hfill \\ & \\ T_n^{\alpha }\left(x_2\right)\left(t\right)\hfill & =-0.6x_2\left(t\right)-0.01\tanh \left(x_1\left(\lfloor t-1\rfloor \right)\right)+0.005\tanh \left(x_2\left(\lfloor t-2\rfloor \right)\right)+0.6\hfill \end{array}\right.\end{array}$$

(30)

where $a_1=0.5,a_2=0.6,b_{11}=0.01,b_{12}=0.005,b_{21}=-0.01,b_{22}=0.005,{\theta }_1=0.48, and{\theta }_2=0.6$. The activation functions $f_1\left(u\right)=f_2\left(u\right)=\tanh u$ satisfy the Lipschitz condition with $L_1=L_2=1$. Moreover, it can be seen that the condition (7) is also satisfied. So, the system (30) has a unique equilibrium point, which is $({\overline{x}}_1,{\overline{x}}_2)\cong (0.982674,0.993754)$. Using the transformation $y_i\left(t\right)=x_i\left(t\right)-\overline{x_i},i=1,2$, the following system is obtained

$$\left\{\begin{array}{c}{T}_n^{\alpha }\left(y_1\right)\left(t\right)=-0.5y_1\left(t\right)+0.01g_1\left(y_1\left(\lfloor t-1\rfloor \right)\right)+0.005g_2\left(y_2\left(\lfloor t-2\rfloor \right)\right)\hfill \\ T_n^{\alpha }\left(y_2\right)\left(t\right)=-0.6y_2\left(t\right)-0.01g_1\left(y_1\left(\lfloor t-1\rfloor \right)\right)+0.005g_2\left(y_2\left(\lfloor t-2\rfloor \right)\right)\hfill \end{array}\right.$$

(31)

where $g_j\left(u\right)=\tanh (u+{\overline{x}}_j)-\tanh {\overline{x}}_j,j=1,2.$ The corresponding discrete version of the system (31) is obtained as

$$\left\{\begin{array}{c}{y}_1(n+1)=e^{{\displaystyle \frac{-0.5}{\alpha }}}{y}_1\left(n\right)+{\displaystyle \frac{1-e^{\frac{-0.5}{\alpha }}}{0.5}}\left(0.01g_1\left(y_1(n-1)\right)+0.005g_2\left(y_2(n-2)\right)\right)\hfill \\ y_2(n+1)=e^{{\displaystyle \frac{-0.6}{\alpha }}}{y}_2\left(n\right)+{\displaystyle \frac{1-e^{\frac{-0.6}{\alpha }}}{0.6}}\left(-0.01g_1\left(y_1(n-1)\right)+0.005g_2\left(y_2(n-2)\right)\right).\hfill \end{array}\right.$$

(32)

It is observed that the conditions (6), (7) and (15) are fulfilled. Therefore, the unique equilibrium point $({\overline{y}}_1,{\overline{y}}_2)=(0,0)$ is asymptotically stable.

Figure 1 demonstrates the asymptotic stability of the equilibrium point of the system (32) for different initial conditions and fixed $\alpha =0.55$. The initial conditions are taken as

$$\begin{array}{cc}\hfill D_1& =(d_{01},d_{-11},d_{02},d_{-12},d_{-22})=(0.4,0.2,-0.3,-0.1,0.1),\hfill \\ \hfill D_2& =(d_{01},d_{-11},d_{02},d_{-12},d_{-22})=(0.15,0.2,0.3,-0.1,0.1),\hfill \\ \hfill D_3& =(d_{01},d_{-11},d_{02},d_{-12},d_{-22})=(-0.15,0.2,-0.4,-0.1,0.1),\hfill \\ \hfill D_4& =(d_{01},d_{-11},d_{02},d_{-12},d_{-22})=(-0.4,0.2,0.4,-0.1,0.1).\hfill \end{array}$$

Figure 1. Behavior of $y_1\left(n\right)$ and $y_2\left(n\right)$ for the system (32) with different initial conditions.

Figure 2 illustratesthe influence of α on the dynamic behavior of the system (32). The state trajectories $y_1\left(n\right)$ and $y_2\left(n\right)$ are plotted for four distinct values of $\alpha \in \{0.35,0.55,0.75,0.95\}$, while keeping the initial conditions fixed at $(d_{01},d_{-11},d_{02},d_{-12},d_{-22})=(0.3,0.2,-0.3,-0.1,0.1)$. It can be observed that the solutions converge to the equilibrium point for all tested values.

Figure 2. Effect of the parameter $\alpha$ on the state trajectories of the system (32).

Table 1 involves the conditions of the main theorems for varying α values between $0.1$ and $0.9.$ From Table 1, it is seen that condition (15) is satisfied for all values of α, so the equilibrium point $({\overline{y}}_1,{\overline{y}}_2)=(0,0)$ is asymptotically stable. However, the same conclusion cannot be reached for exponential stability, since condition (18) is not satisfied for the first six values of α between $0.1$ and $0.35$. It is seen that the condition (18) is satisfied for $\alpha \in {\Omega }_1=\{0.4,0.45,0.5,\dots ,0.99\}$. So, the equilibrium point $({\overline{y}}_1,{\overline{y}}_2)=(0,0)$ is exponentially stable for $\alpha \in {\Omega }_1$. On the other hand, it is observed that the condition (26) is satisfied for $\alpha \in {\Omega }_2=\{0.6,0.65,0.7,\dots ,0.99\}$. This means that for $\alpha \in {\Omega }_2$, the convergence rate decreases as α increases. In fact, this property is seen from Table 1.

Table 1. Convergence rate for the system (32) for several $\alpha$ values.

$\mathit{\alpha }$	(15) (for i = 1)	(15) (for i = 2)	$\mathit{\lambda }$	(18) (for i = 1)	(18) (for i = 2)	(26) (for i = 1)	(26) (for i = 2)
$0.10$	✓	✓	$-5938.02$	×	×	×	×
$0.15$	✓	✓	$-105.283$	×	×	×	×
$0.20$	✓	✓	$-12.8451$	×	×	×	×
$0.25$	✓	✓	$-3.03349$	×	×	×	×
$0.30$	✓	✓	$-0.807871$	×	×	×	×
$0.35$	✓	✓	$-0.0702934$	✓	×	×	×
$0.40$	✓	✓	$0.230187$	✓	✓	×	×
$0.45$	✓	✓	$0.366681$	✓	✓	×	×
$0.50$	✓	✓	$0.431062$	✓	✓	✓	×
$0.55$	✓	✓	$0.455351$	✓	✓	✓	×
$0.60$	✓	✓	$0.449752$	✓	✓	✓	✓
$0.65$	✓	✓	$0.43986$	✓	✓	✓	✓
$0.70$	✓	✓	$0.427826$	✓	✓	✓	✓
$0.75$	✓	✓	$0.414846$	✓	✓	✓	✓
$0.80$	✓	✓	$0.401603$	✓	✓	✓	✓
$0.85$	✓	✓	$0.388485$	✓	✓	✓	✓
$0.90$	✓	✓	$0.375715$	✓	✓	✓	✓
$0.95$	✓	✓	$0.363412$	✓	✓	✓	✓
$0.99$	✓	✓	$0.353948$	✓	✓	✓	✓

To further illustrate the dynamic behavior of the system, the time responses of the state variables are examined under the same initial conditions $(d_{01},d_{-11},d_{02},d_{-12},d_{-22})=(0.3,0.2,-0.3,-0.1,0.1)$.

To confirm the exponential nature of convergence, let us consider the cases $\alpha =0.75$ and $\alpha =0.95$, both of which satisfy conditions (18) and (26) (see Table 1). The behavior of the state trajectories for $\alpha =0.75$ and $\alpha =0.95$ is given in Figure 2. In addition, Figure 3 presents the log-scale plots of $\log |y_1\left(n\right)|$ and $\log |y_2\left(n\right)|$ for $\alpha =0.75$ and $\alpha =0.95$. In both cases, $\log |y_i\left(n\right)|,i=1,2,$ decreases linearly with n, providing definitive evidence that the convergence is indeed exponential. Moreover, the slope of the log-scale trajectories for $\alpha =0.75$ is steeper than that for $\alpha =0.95$, indicating a faster exponential convergence for the smaller value of α. This observation is in perfect agreement with Corollary 1, which states that the convergence rate λ is inversely proportional to α.

Figure 3. Log-scale plots of $\log |y_i\left(n\right)|$ for the system (32) with $\alpha =0.75$ and $\alpha =0.95$.

Now, let us consider the cases $\alpha =0.10$ and $\alpha =0.35$, both of which only fulfill the condition (15). Figure 4 presents the normal-scale trajectories of $y_1\left(n\right)$ and $y_2\left(n\right)$ for $\alpha \in \{0.10,0.35\}$. It is observed that the solutions converge to the equilibrium point in both cases, confirming that asymptotic stability is preserved. On the other hand, Figure 5 presents the corresponding log-scale plots of $\log |y_i\left(n\right)|,i=1,2$; here, the log-scale trajectories do not exhibit a linear decrease, confirming that the convergence is genuinely not exponential. This demonstrates that falling outside the region where the condition (18) holds does not imply a loss of stability; asymptotic stability is preserved while exponential convergence may be lost.

Figure 4. Normal-scale time responses of $y_1\left(n\right)$ and $y_2\left(n\right)$ for the system (32) with $\alpha =0.10$ and $\alpha =0.35$.

Figure 5. Log-scale plots of $\log |y_i\left(n\right)|$ for the system (32) with $\alpha =0.10$ and $\alpha =0.35$.

5. Discussion

In this paper, the asymptotic stability of the unique equilibrium point of artificial neural networks with piecewise constant arguments and conformable derivatives has been investigated. The existence and uniqueness of the equilibrium point have been established and stability results have been obtained for the associated discrete system using a Lyapunov approach. In addition, sufficient conditions for exponential stability have been derived, providing information on the convergence rate. A numerical example has been presented to illustrate the theoretical findings.

Despite the results obtained, several open problems concerning the dynamics of neural network systems remain to be addressed.

• In real-world problems, an impulse effect may change the asymptotic behavior of the system. So, the dynamics of impulsive neural networks with conformable derivatives and piecewise constant arguments is another problem.
• Since synchronization analysis of the systems could provide a more comprehensive understanding of neural network dynamics, some future work may focus on synchronization analysis.