Investigation of Initial Time Difference Mittag–Leffler Stability for Fractional Perturbed Systems

Abstract

This study investigates the Mittag–Leffler-type stability properties of fractional perturbed systems with respect to their unperturbed counterparts by incorporating initial time differences into the analysis. In contrast to many existing studies in which initial time effects are neglected, the proposed framework explicitly considers time shifts together with the memory-dependent nature of fractional-order systems. Using Caputo fractional derivatives and Lyapunov-type functionals, new sufficient conditions are established for the stability behavior of perturbed systems relative to the corresponding unperturbed systems under shifted initial times. The obtained results extend existing stability criteria by simultaneously addressing fractional memory effects, perturbation terms, and variations in the initial time. To illustrate the applicability and effectiveness of the theoretical findings, representative examples, numerical simulations, graphical comparisons, and global error analyses are presented. The numerical part is based on the Caputo framework and is further supported by benchmark comparisons involving Riemann–Liouville and shifted Grünwald–Letnikov approaches. The proposed results provide a useful framework for the stability analysis of memory-dependent dynamical systems arising in engineering and applied sciences.

1. Introduction

Fractional-order differential equations have become an important mathematical tool for modeling dynamical systems with memory, hereditary effects, and nonlocal behavior. Unlike classical integer-order models, fractional-order systems incorporate information from past states through memory kernels, making them suitable for describing processes whose present evolution depends on their history. Owing to these characteristics, fractional calculus has been used in many areas such as control theory, viscoelasticity, electrical circuits, biological systems, diffusion processes, and engineering sciences [1,2,3,4].

Among the various fractional derivatives used in the literature, the Caputo fractional derivative is particularly suitable for physical and engineering models because it allows the use of classical initial conditions while preserving the memory-dependent nature of the system. From a geometric point of view, the classical first-order derivative describes the instantaneous slope of a trajectory, whereas a fractional derivative may be interpreted as a weighted accumulation of past slopes over a time interval. This feature makes the Caputo derivative useful for describing memory-dependent dynamics in which the current state is affected by the previous evolution of the system [2,4].

A fundamental concept in the qualitative analysis of fractional systems is Mittag–Leffler stability. This notion is commonly regarded as the fractional counterpart of exponential stability in classical differential equations. The reason is that solutions of many fractional-order systems are naturally expressed in terms of Mittag–Leffler functions rather than exponential functions. Thus, Mittag–Leffler stability provides an appropriate framework for describing decay rates, convergence behavior, and asymptotic properties of fractional dynamical systems with memory [5,6].

Recently, Mittag–Leffler stability has been investigated for different classes of fractional-order systems. Li, Chen, and Podlubny [5] established generalized Mittag–Leffler stability criteria for fractional nonlinear systems by using Lyapunov direct methods. Sadati et al. [6,7] studied Mittag–Leffler and Razumikhin-type stability results for fractional nonlinear systems with delays, while Wu et al. [8,9] investigated the stability properties of discrete fractional systems. In a related direction, Yakar, Gücen, and Çiçek [10,11] introduced stability concepts involving initial time differences for fractional dynamic systems. More recent works have considered Mittag–Leffler-type stability in delayed fractional systems, fractional neural networks, memristive neural networks, synchronization problems, and Ulam–Hyers stability frameworks [12,13,14,15,16,17]. These studies are mentioned only to indicate recent developments in Mittag–Leffler-type stability analysis; the present paper, however, focuses on fractional perturbed systems and their comparison with unperturbed reference systems under initial time differences.

Despite these developments, the simultaneous treatment of perturbation effects and initial time differences in a comparative fractional stability setting remains limited. Most existing results focus either on a single fractional system, on delay effects, or on stability properties under identical initial times. However, due to the memory-dependent structure of fractional systems, a change in the initial instant may influence both transient and long-time behavior. Therefore, it is natural to investigate how a perturbed trajectory behaves relative to a shifted unperturbed reference trajectory when different initial times are taken into account.

To clarify the position of the present study with respect to the existing literature, the main differences between this work and closely related contributions are summarized in

Table 1. Comparison of the present results with related studies in the literature.

Reference	Main Focus	Difference from the Present Study
Li et al. [5]	Lyapunov-based generalized Mittag–Leffler stability for fractional nonlinear systems	Does not consider initial time differences or comparative stability between perturbed and unperturbed systems.
Sadati et al. [6]	Mittag–Leffler stability of fractional nonlinear systems with delays	Focuses mainly on delay effects and does not explicitly treat shifted initial times between perturbed and unperturbed systems.
Yakar et al. [10,11]	Strict stability and initial time difference effects in fractional dynamic systems	Provides important results on initial time difference stability, but the present work applies this viewpoint to fractional perturbed systems relative to shifted unperturbed counterparts.
Recent studies [12,13,14,15,16,17]	Mittag–Leffler stability, synchronization, Ulam–Hyers stability, and delayed fractional models	These works address important classes of fractional systems, but they do not simultaneously combine perturbation effects, fractional memory, and initial time differences in a comparative stability framework.
Present study	Comparative Mittag–Leffler stability of fractional perturbed systems with respect to shifted unperturbed systems	Establishes Lyapunov-type sufficient conditions for the distance between perturbed trajectories and shifted unperturbed reference trajectories under initial time differences.

. The table is placed after the discussion of the research gap in order to show more clearly how the present work differs from previous Mittag–Leffler stability results.

Motivated by these observations, this paper develops a comparative Mittag–Leffler stability framework for fractional perturbed systems with respect to shifted unperturbed reference trajectories. The contribution of this study is not the introduction of a completely independent Mittag–Leffler stability theory, but rather the adaptation and extension of existing fractional Mittag–Leffler stability ideas to a setting in which perturbation effects, fractional memory, and initial time differences are treated simultaneously. More precisely, the main contributions of the paper are as follows:

• A comparative stability notion is formulated in terms of the distance between a perturbed trajectory and a shifted unperturbed reference trajectory;
• Lyapunov-type sufficient conditions are established for Mittag–Leffler stability under initial time differences;
• The role of the initial time difference is explicitly incorporated into the stability estimates;
• Representative examples are provided to illustrate how the abstract conditions can be verified for fractional perturbed systems.

The mathematical models considered in this paper are not intended to reproduce natural systems in all their complexity, but to provide an analytically tractable framework for studying the stability, convergence, and robustness properties of memory-dependent systems. In this sense, the proposed perturbed and unperturbed fractional models serve as idealized but meaningful representations of memory-dependent phenomena arising in engineering, physics, biology, and applied sciences.

The remainder of this paper is organized as follows: Section 2 presents the necessary preliminaries and definitions. Section 3 gives the main Lyapunov-type stability results. Section 4 provides illustrative examples and verifies the required assumptions. Section 5 presents numerical illustrations and benchmark comparisons. Finally, Section 7 contains the concluding remarks.

2. Preliminaries

In this section, we recall some basic notions of fractional calculus that will be needed in the sequel. For $0<q<1$, the Caputo fractional derivative of a differentiable function $x\left(t\right)$ is defined as

$${}^{c}D^{q}x\left(t\right)={\displaystyle \frac{1}{\Gamma (1-q)}}{\int }_{{\rho }_0}^t{(t-s)}^{-q}{x}^{\prime }\left(s\right) ds, t\in [{\rho }_0,{\rho }_0+T],$$

(1)

where $\Gamma ( · )$ denotes the Euler Gamma function.

The Riemann–Liouville fractional derivative of order $q\in (0, 1)$ is defined by

$$D^{q}x\left(t\right)={\displaystyle \frac{1}{\Gamma \left(p\right)}}{\displaystyle \frac{d}{dt}}{\int }_{{\rho }_0}^t{(t-s)}^{p-1}x\left(s\right) ds, t\in [{\rho }_0,{\rho }_0+T],$$

(2)

where $p=1-q$, and equivalently $p+q=1$. This relation follows from the standard definition of the Riemann–Liouville fractional derivative of order q, where the kernel exponent is determined by the complementary order $1-q$. Since $0<q<1$, it follows that $0<p<1$.

One key advantage of the Caputo derivative is that it permits the formulation of initial conditions in a manner consistent with those of classical integer-order differential equations. Moreover, for a constant C, we have ${}^{c}D^{q}C=0$, whereas the Riemann–Liouville operator yields

$$D^{q}C={\displaystyle \frac{C{(t-{\rho }_0)}^{-q}}{\Gamma (1-q)}}.$$

From (1), the following relation can be derived:

$${}^{c}D^{q}x\left(t\right)=D^q\left(x\left(t\right)-x\left({\rho }_0\right)\right),$$

(3)

which further implies

$${}^{c}D^{q}x\left(t\right)=D^{q}x\left(t\right)-{\displaystyle \frac{x\left(t_0\right)}{\Gamma (1-q)}}{(t-{\rho }_0)}^{-q}.$$

(4)

In particular, if $x\left({\rho }_0\right)=0$, then both definitions coincide, that is,

$${}^{c}D^{q}x\left(t\right)=D^{q}x\left(t\right).$$

(5)

These definitions and their relations are classical tools in the theory of fractional differential equations and can be found, for example, in [2,4,18].

Now consider the Caputo-type initial value problem

$${}^{c}D^{q}x\left(t\right)=f(t,x\left(t\right)), x\left({\rho }_0\right)=x_0, t\ge {\rho }_0, {\rho }_0\in {\mathbb{R}}_{+},$$

(6)

and its perturbed counterpart

$${}^{c}D^{q}y\left(t\right)=F(t,y\left(t\right))=f(t,y\left(t\right))+R(t,y\left(t\right)), y\left(t_0\right)=y_0, t\ge t_0, t_0\in {\mathbb{R}}_{+},$$

(7)

where $R(t,y)$ represents a nonlinear perturbation term and $f,F\in C({\mathbb{R}}_{+}\times {\mathbb{R}}^n,{\mathbb{R}}^n)$. Here, $x_0={\lim }_{t\to {\rho }_0}{D}^{q-1}x\left(t\right)$ and $y_0={\lim }_{t\to t_0}{D}^{q-1}y\left(t\right)$ are assumed to exist. We also require that f is continuous on $[{\rho }_0,{\rho }_0+T]\times {\mathbb{R}}^n$, locally Lipschitz in the second variable, and satisfies $f(t,0)=0$ for all $t\ge 0$. Moreover, the perturbation term $R(t,y)$ is assumed to be continuous and may depend nonlinearly on the state variable y.

It should be noted that the local Lipschitz condition imposed on f with respect to the state variable is essential for ensuring the uniqueness of solutions of the corresponding fractional initial value problem. If f is only continuous and not locally Lipschitz, the existence of solutions may still be determined under suitable assumptions; however, uniqueness is not guaranteed in general. In such a case, different solution trajectories may originate from the same initial condition, and the stability analysis with respect to a uniquely determined reference solution may no longer be well-defined. Therefore, the local Lipschitz assumption is imposed throughout this study to guarantee the well-posedness of the system and to make the comparison between perturbed and unperturbed solutions meaningful.

Under standard assumptions ensuring the existence and uniqueness of solutions for the problems (6) and (7), the solution $x\left(t\right)$ of (6) can equivalently be written in the Volterra-type integral form

$$x\left(t\right)=x_0+{\displaystyle \frac{1}{\Gamma \left(q\right)}}{\int }_{{\rho }_0}^t{(t-s)}^{q-1}f(s,x\left(s\right)) ds, t\in [{\rho }_0,{\rho }_0+T].$$

(8)

Thus, the differential formulation (6) and the integral equation (8) are completely equivalent.

The following definition is formulated by adapting the classical Mittag–Leffler stability concept for fractional systems [5] and the initial time difference stability framework [10,11] to the present setting of fractional perturbed systems with respect to their unperturbed counterparts. Its purpose is to measure the decay of the distance between the perturbed trajectory and the shifted unperturbed reference trajectory. Therefore, the definition is not intended to describe the stability of a single isolated system, but rather the comparative stability between two fractional systems starting from different initial times.

Definition 1.

Let $x(t,t_0,x_0)$ denote the solution of the unperturbed fractional system (6) through $(t_0,x_0)$, and let $y(t,{\rho }_0,y_0)$ denote the solution of the perturbed fractional system (7) through $({\rho }_0,y_0)$. Set $\eta ={\rho }_0-t_0$, where $t\ge {\rho }_0\ge 0$ and $t_0,{\rho }_0\in {\mathbb{R}}_{+}$. The perturbed system (7) is said to be initial time difference Mittag–Leffler stable with respect to the unperturbed system (6) if the perturbed trajectory $y(t,{\rho }_0,y_0)$ remains Mittag–Leffler close to the shifted unperturbed trajectory $x(t-\eta ,t_0,x_0)$, that is,

$$\begin{array}{cc}\hfill & \parallel y(t,{\rho }_0,y_0)-x(t-\eta ,t_0,x_0)\parallel \le {\left[m_1(\parallel y_0-x_0\parallel )E_{{\alpha }_1}(-{\lambda }_1{(t-{\rho }_0)}^{{\alpha }_1})\right]}^{b_1},\hfill \\ \hfill & t\ge {\rho }_0.\hfill \end{array}$$

(9)

Here, ${\alpha }_1\in (0,1)$, ${\lambda }_1>0$, $b_1>0$, and $m_1$ is a non-negative locally Lipschitz function on $B\subset {\mathbb{R}}^n$ satisfying $m_1\left(0\right)=0$, with Lipschitz constant $m_{01}$.

Definition 2.

Let $y(t,t_0,y_0)$ be a trajectory of the fractional system (7) through $({\rho }_0,y_0)$. It is calledinitial time difference attractive Mittag–Leffler stable with respect to $x(t-\eta ,t_0,x_0)$, a solution of (6), provided that $\eta ={\rho }_0-t_0$, and there exists $T_1=T_1\left({\rho }_0\right)$ such that

$$\begin{array}{cc}\hfill & \parallel y(t,{\rho }_0,y_0)-x(t-\eta ,t_0,x_0)\parallel \le {\left[m_1(\parallel y_0-x_0\parallel )E_{{\alpha }_1}(-{\lambda }_1{(t-{\rho }_0)}^{{\alpha }_1})\right]}^{b_1},\hfill \\ \hfill & t\ge T_1+{\rho }_0.\hfill \end{array}$$

(10)

If $T_1$ can be chosen independently of ${\rho }_0$, then $y(t,{\rho }_0,y_0)$ is said to be uniformly attractive initial time difference Mittag–Leffler stable. The parameters ${\alpha }_1\in (0,1)$, ${\lambda }_1>0$, and $b_1>0$ and the function $m_1$ satisfy the same assumptions as in Definition 1.

Definition 3.

The solution $y(t,{\rho }_0,y_0)$ of (7) through $({\rho }_0,y_0)$ is said to be initial time difference asymptotically Mittag–Leffler stable with respect to $x(t-\eta ,t_0,x_0)$ if both of the following hold:

1.

$y(t,{\rho }_0,y_0)$ is initial time difference Mittag–Leffler stable (Definition 1);

2.

${\displaystyle \underset{t\to \infty }{\lim }\parallel y(t,{\rho }_0,y_0)-x(t-\eta ,t_0,x_0)\parallel =0.}$

Moreover, if the constant $T_1$ appearing in Definition 2 can be chosen independently of ${\rho }_0$, then $y(t,{\rho }_0,y_0)$ is uniformly initial time difference asymptotically Mittag–Leffler stable for all $t\ge {\rho }_0\ge 0$.

The following two definitions are based on the standard fractional Dini derivative approach used in Lyapunov stability analysis for fractional-order systems. Definition 4 recalls a Caputo-type fractional Dini derivative for a Lyapunov function along a single fractional system. Definition 5, on the other hand, is adapted to the present comparative framework and is used to estimate the evolution of the difference between a perturbed trajectory and a shifted unperturbed reference trajectory. Thus, the second definition extends the usual fractional Dini derivative setting to the stability analysis of two fractional systems starting from different initial times.

Definition 4.

Let $V\in C({\mathbb{R}}_{+}\times {\mathbb{R}}^n,{\mathbb{R}}_{+})$ be a real-valued continuous function. The fractional Dini derivative of V in the Caputo sense is given by

$${}^{c}D_{+}^{q}V(t,x)=\underset{h\to 0^{+}}{\mathrm{lim\; sup}}{\displaystyle \frac{1}{h^q}}\left[V(t,x)-V\left(t-h,x-h^{q}f(t,x)\right)\right],$$

(11)

where $x\left(t\right)=x(t,{\rho }_0,x_0)$ and $(t,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^n$.

Definition 5 is adapted to the comparative stability framework considered in this paper. It is used to estimate the evolution of the difference between the perturbed trajectory and the shifted unperturbed reference trajectory under the initial time difference.

Definition 5.

Let $V(t,z)\in C({\mathbb{R}}_{+}\times {\mathbb{R}}^n,{\mathbb{R}}_{+})$ be a real-valued continuous function, where $z=y-\tilde{x}$ denotes the difference between a perturbed trajectory $y\left(t\right)$ and a shifted unperturbed trajectory $\tilde{x}\left(t\right)=x(t-\eta ,t_0,x_0)$. The generalized Caputo-type fractional Dini derivative of V along the difference dynamics is defined by

$${}_{*}{}^{c}D_{+}^{q}V(t,z)=\underset{h\to 0^{+}}{\mathrm{lim\; sup}}{\displaystyle \frac{1}{h^q}}\left[V(t,z)-V\left(t-h,z-h^q(F(t,y)-f(t,\tilde{x}))\right)\right],$$

(12)

where $F(t,y)=f(t,y)+R(t,y)$ is the vector field of the perturbed system and $f(t,\tilde{x})$ is the vector field of the shifted unperturbed reference system.

Definition 6.

The class $\mathcal{K}$ is defined as the collection of a function satisfying

$$\mathcal{K}:=\left\{a:a\in C([0,\tau ],{\mathbb{R}}_{+}), a \mathit{is} \mathit{monotone} \mathit{non}\text{-}\mathit{decreasing} \mathit{and} a\left(0\right)=0\right\}.$$

(13)

For the convenience of the reader and to avoid any ambiguity in the subsequent analysis, the main mathematical parameters and notations used throughout the paper are summarized in

Table 2. Main mathematical parameters and notations used in the paper.

Symbol	Meaning
$q\in (0,1)$	Order of the fractional derivative
$p=1-q$	Complementary order in the Riemann–Liouville derivative
${\rho }_0$	Initial time of the unperturbed system
$t_0$	Initial time of the perturbed system
$\eta ={\rho }_0-t_0$	Initial time difference
$x_0$, $y_0$	Initial states of the unperturbed and perturbed systems
$x\left(t\right)$, $y\left(t\right)$	Solutions of the unperturbed and perturbed systems
$f(t,x)$	Vector field of the unperturbed system
$R(t,y)$	Nonlinear perturbation term
$F(t,y)$	Total vector field of the perturbed system, $F=f+R$
$\Gamma ( · )$	Euler Gamma function
$E_{\alpha }( · )$	Mittag–Leffler function
${\alpha }_i\in (0,1)$	Fractional decay-order parameters in stability estimates
${\lambda }_i>0$	Positive decay coefficients
$b_i>0$	Positive power-type constants in stability bounds
$m_i\in \mathcal{K}$	Non-negative comparison functions
$V_{\mu }$	Lyapunov-type functional
$\mu$	Positive threshold parameter, $0<\mu <\rho$
$S_{\tau }$	Admissible state region in the Lyapunov analysis
$z\left(t\right)$	Difference $y(t,{\rho }_0,y_0)-x(t-\eta ,t_0,x_0)$
$a,\alpha ,\beta ,\lambda$	Positive decay coefficients in examples
$\varphi \left(t\right)$, $\psi \left(t\right)$	Bounded perturbation functions
$ϵ$	Perturbation amplitude in the numerical model
h	Numerical step size

.

3. Main Theorems

Remark 1.

The assumptions imposed in the following Lyapunov-type theorems should be understood as verifiable sufficient conditions rather than automatically guaranteed properties for an arbitrary fractional system. In the Lyapunov approach, the existence of a suitable functional $V_{\mu }$ depends on the structure of the considered system and must be checked separately for each particular model. Typically, $V_{\mu }$ is constructed as a positive definite norm-type or quadratic functional, such as

$$V_{\mu }(t,z)={\parallel z\parallel }^2$$

or

$$V_{\mu }(t,z)=z^{T}Pz,$$

where P is a positive definite matrix. The local Lipschitz condition imposed on $V_{\mu }$ is required to ensure that the generalized Caputo-type Dini derivative is well-defined along the difference trajectory. In concrete applications, this condition can be verified directly from the explicit form of $V_{\mu }$. For instance, quadratic choices such as $V_{\mu }(t,z)={\parallel z\parallel }^2$ or $V_{\mu }(t,z)=z^{T}Pz$ are locally Lipschitz with respect to z on bounded subsets of ${\mathbb{R}}^n$. Once such a functional is constructed, the constants ${\alpha }_i\in (0,1)$, ${\lambda }_i>0$, $b_i>0$ and the comparison functions $m_i\in \mathcal{K}$ are obtained from the estimates that bound $V_{\mu }$ and its generalized Caputo-type Dini derivative. Therefore, the role of the theorems is to provide a set of sufficient criteria: if an appropriate Lyapunov-type functional is constructed and the required inequalities are verified, then the corresponding Mittag–Leffler stability conclusion follows.

Remark 2.

The parameters appearing in the Lyapunov-type estimates are not arbitrary. The fractional orders ${\alpha }_i$ are chosen in the interval $(0, 1)$ in accordance with the Mittag–Leffler decay structure, while the constants ${\lambda }_i>0$ represent decay rates and $b_i>0$ are power-type exponents in the comparison bounds. The comparison functions $m_i\in \mathcal{K}$ must be continuous, non-negative, monotone nondecreasing, and satisfy $m_i\left(0\right)=0$. In applications, these quantities are derived from the explicit estimates obtained after choosing the Lyapunov-type functional $V_{\mu }$. More precisely, lower and upper bounds for $V_{\mu }(t,z)$ determine the functions $m_1,m_2$ and the exponents $b_1, b_2$, whereas the estimate of the generalized Caputo-type Dini derivative determines the decay parameters ${\lambda }_i$ and the remaining comparison terms. Thus, parameters such as ${\lambda }_2$, ${\alpha }_2$, and $b_2$ are selected so that the inequalities in the hypotheses of the theorems are satisfied.

Remark 3.

The boundedness of the difference trajectory

$$z\left(t\right)=y(t,{\rho }_0,y_0)-x(t-\eta ,t_0,x_0)$$

is understood within the admissible region $S_{\tau }$ used in the Lyapunov analysis. More precisely, the main theorems are stated for trajectories satisfying

$$(t,z\left(t\right))\in {\mathbb{R}}_{+}\times S_{\tau },$$

where $S_{\tau }\subset {\mathbb{R}}^n$ is chosen as a bounded admissible set. Hence, the boundedness of $z\left(t\right)$ is not assumed implicitly; it is ensured by restricting the analysis to solutions that remain in $S_{\tau }$. In applications, this requirement can be verified either by showing that $S_{\tau }$ is positively invariant for the difference dynamics or by proving a priori estimates on the perturbed and unperturbed solutions. For example, if both $x(t-\eta ,t_0,x_0)$ and $y(t,{\rho }_0,y_0)$ are bounded on the considered interval, then

$$\parallel z\left(t\right)\parallel \le \parallel y(t,{\rho }_0,y_0)\parallel +\parallel x(t-\eta ,t_0,x_0)\parallel ,$$

which implies the boundedness of $z\left(t\right)$. In the illustrative examples considered in Section 4, the Mittag–Leffler decay of the unperturbed part together with the boundedness of the perturbation terms guarantees that the corresponding difference trajectory remains bounded.

Theorem 1

(Suppose that the following assumptions hold). For every μ with $0<\mu <\rho$, there exists a functional $V_{\mu }\in C({\mathbb{R}}_{+}\times S_{\tau },{\mathbb{R}}_{+})$ such that $V_{\mu }$ is locally Lipschitz continuous in z. Moreover, for $(t,z)\in {\mathbb{R}}_{+}\times S_{\tau }$ with $\parallel z\parallel \ge \mu$, there exist parameters ${\alpha }_i\in (0,1)$, ${\lambda }_i>0$, $b_i>0$ $(i=1,2)$ and comparison functions $m_i\in \mathcal{K}$ $(i=1,2)$ satisfying $m_i\left(0\right)=0$ and $m_i\left(r\right)\ge 0$ for $r\ge 0$ such that

$${\left\{m_1(\parallel z\parallel ) E_{{\alpha }_1}\left[-{\lambda }_1{(t-{\rho }_0)}^{{\alpha }_1}\right]\right\}}^{b_1}\le V_{\mu }(t,z)\le {\left\{m_2(\parallel z\parallel ) E_{{\alpha }_2}\left[-{\lambda }_2{(t-{\rho }_0)}^{{\alpha }_2}\right]\right\}}^{b_2}.$$

(14)

In addition, the generalized Caputo-type Dini derivative satisfies

$${}_{*}{}^{C}D_{+}^q{V}_{\mu }(t,z)\le 0.$$

(15)

Here

$$z\left(t\right)=y(t,{\rho }_0,y_0)-x(t-\eta ,t_0,x_0), t\ge {\rho }_0,$$

where $y(t,{\rho }_0,y_0)$ denotes the solution of the perturbed system (7) starting from $({\rho }_0,y_0)$, while $x(t,t_0,x_0)$ denotes the solution of the unperturbed system (6) with initial data $(t_0,x_0)$. The initial time difference is given by $\eta ={\rho }_0-t_0$.

Then, under these conditions, the solution $y(t,{\rho }_0,y_0)$ of (7) is Mittag–Leffler stable with initial time difference relative to the shifted solution $x(t-\eta ,t_0,x_0)$ of (6), for all $t\ge {\rho }_0\ge 0$.

Proof. 

Let ${\epsilon }_1>0$ and $t_0\in {\mathbb{R}}_{+}$. Choose ${\delta }_1={\delta }_1({\epsilon }_1,{\rho }_0)>0$ such that

$${\left\{m_2\left({\delta }_1\right)E_{{\alpha }_2}[-{\lambda }_2{(t-{\rho }_0)}^{{\alpha }_2}]\right\}}^{b_2}<{\left\{m_1\left({\epsilon }_1\right)E_{{\alpha }_1}[-{\lambda }_1{(t-{\rho }_0)}^{{\alpha }_1}]\right\}}^{b_1}.$$

(16)

Such a choice is possible by the continuity of $m_2$, the condition $m_2\left(0\right)=0$, and the positivity of the right-hand side for fixed ${\epsilon }_1>0$.

Assume, contrary to the desired conclusion, that the Mittag–Leffler estimate does not hold. Then there exist $t_1>t_2\ge {\rho }_0$ such that

$$\parallel y\left(t_1\right)-\tilde{x}\left(t_1\right)\parallel ={\epsilon }_1, \parallel y\left(t_2\right)-\tilde{x}\left(t_2\right)\parallel ={\delta }_1,$$

(17)

and

$${\delta }_1\le \parallel y\left(t\right)-\tilde{x}\left(t\right)\parallel \le {\epsilon }_1, t\in [t_2,t_1],$$

where

$$\tilde{x}\left(t\right)=x(t-\eta ,t_0,x_0).$$

Since ${}_{*}{}^{C}D_{+}^q{V}_{\mu }(t,z)\le 0$, the functional $V_{\mu }$ is nonincreasing along the difference trajectory $z\left(t\right)=y\left(t\right)-\tilde{x}\left(t\right)$ in the generalized Caputo-type Dini sense. Hence,

$$V_{\mu }(t_1,z\left(t_1\right))\le V_{\mu }(t_2,z\left(t_2\right)).$$

Using the estimate (14), we obtain

$$\begin{array}{cc}\hfill & {\left\{m_1\left({\epsilon }_1\right)E_{{\alpha }_1}[-{\lambda }_1{(t_1-{\rho }_0)}^{{\alpha }_1}]\right\}}^{b_1}\hfill \\ \hfill & \le V_{\mu }(t_1,y\left(t_1\right)-\tilde{x}\left(t_1\right))\hfill \\ \hfill & \le V_{\mu }(t_2,y\left(t_2\right)-\tilde{x}\left(t_2\right))\hfill \\ \hfill & \le {\left\{m_2\left({\delta }_1\right)E_{{\alpha }_2}[-{\lambda }_2{(t_2-{\rho }_0)}^{{\alpha }_2}]\right\}}^{b_2}.\hfill \end{array}$$

This contradicts the choice of ${\delta }_1$ in (16). Therefore, the assumed violation cannot occur, and the required Mittag–Leffler stability estimate with initial time difference follows.    □

Theorem 2

(Suppose that the following assumptions hold). For every μ with $0<\mu <\rho$, there exists a functional $V_{\mu }\in C({\mathbb{R}}_{+}\times S_{\tau },{\mathbb{R}}_{+})$ such that $V_{\mu }$ is locally Lipschitz continuous in z. Moreover, for $(t,z)\in {\mathbb{R}}_{+}\times S_{\tau }$ with $\parallel z\parallel \ge \mu$, there exist parameters ${\alpha }_i\in (0, 1)$, ${\lambda }_i>0$, $b_i>0$ $(i=1,2)$, and comparison functions $m_i\in \mathcal{K}$ $(i=1,2)$ satisfying $m_i\left(0\right)=0$ and $m_i\left(r\right)\ge 0$ for $r\ge 0$, such that

$${\left\{m_1(\parallel z\parallel )E_{{\alpha }_1}\left[-{\lambda }_1{(t-{\rho }_0)}^{{\alpha }_1}\right]\right\}}^{b_1}\le V_{\mu }(t,z)\le {\left\{m_2(\parallel z\parallel )E_{{\alpha }_2}\left[-{\lambda }_2{(t-{\rho }_0)}^{{\alpha }_2}\right]\right\}}^{b_2}.$$

(18)

In addition, there exist parameters ${\alpha }_3\in (0, 1)$, ${\lambda }_3>0$, and $b_3>0$, together with a comparison function $m_3\in \mathcal{K}$ satisfying $m_3\left(0\right)=0$ and $m_3\left(r\right)\ge 0$ for $r\ge 0$, such that the generalized Caputo-type Dini derivative satisfies

$${}_{*}{}^{C}D_{+}^q{V}_{\mu }(t,z)\le -{\left\{m_3(\parallel z\parallel )E_{{\alpha }_3}\left[-{\lambda }_3{(t-{\rho }_0)}^{{\alpha }_3}\right]\right\}}^{b_3}.$$

(19)

Here

$$z\left(t\right)=y(t,{\rho }_0,y_0)-x(t-\eta ,t_0,x_0), t\ge {\rho }_0,$$

where $y(t,{\rho }_0,y_0)$ denotes the solution of the perturbed system (7) starting from $({\rho }_0,y_0)$, while $x(t,t_0,x_0)$ denotes the solution of the unperturbed system (6) with initial data $(t_0,x_0)$. The initial time difference is given by $\eta ={\rho }_0-t_0$.

Then, under these conditions, the solution $y(t,{\rho }_0,y_0)$ of (7) is uniformly asymptotically Mittag–Leffler stable with initial time difference relative to the shifted solution $x(t-\eta ,t_0,x_0)$ of (6), for all $t\ge {\rho }_0\ge 0$.

Proof. 

Let

$$z\left(t\right)=y(t,{\rho }_0,y_0)-x(t-\eta ,t_0,x_0).$$

By Theorem 1, the two-sided estimate (18), together with the inequality (19), implies the initial time difference Mittag–Leffler stability of the perturbed system with respect to the shifted unperturbed reference system. It remains to prove the uniform attractivity property.

Let $0<{\epsilon }_1<\rho$ be given. Since $m_1\in \mathcal{K}$ and $m_1\left(0\right)=0$, choose $0<{\delta }_1<{\epsilon }_1$ such that the stability estimate obtained from Theorem 1 guarantees

$$\parallel z\left(t\right)\parallel <{\epsilon }_1, t\ge s,$$

whenever $\parallel z\left(s\right)\parallel <{\delta }_1$ for some $s\ge {\rho }_0$.

Now choose ${\delta }_{10}>0$ and ${\delta }_{20}>0$ such that

$$\parallel y_0-x_0\parallel <{\delta }_{10}, |{\rho }_0-t_0|<{\delta }_{20}$$

which imply that the corresponding difference trajectory remains in $S_{\tau }$ for $t\ge {\rho }_0$. Let

$${\delta }_{*}=\max \{{\delta }_{10},{\delta }_{20}\}.$$

Assume that, for some $T>0$, the estimate

$$\parallel z\left(t\right)\parallel \ge {\delta }_1, t\in [{\rho }_0,{\rho }_0+T],$$

holds. Then, by (19) and the fractional integral form of the Caputo-type comparison inequality, we have

$$\begin{array}{cc}\hfill V_{\mu }({\rho }_0+T,z({\rho }_0+T)) & \le V_{\mu }({\rho }_0,z\left({\rho }_0\right))-{\displaystyle \frac{1}{\Gamma \left(q\right)}}{\int }_{{\rho }_0}^{{\rho }_0+T}{({\rho }_0+T-s)}^{q-1}{\left\{m_3(\parallel z\left(s\right)\parallel )E_{{\alpha }_3}\left[-{\lambda }_3{(s-{\rho }_0)}^{{\alpha }_3}\right]\right\}}^{b_3} ds.\hfill \end{array}$$

Since $\parallel z\left(s\right)\parallel \ge {\delta }_1$ on $[{\rho }_0,{\rho }_0+T]$ and $m_3$ is nondecreasing, it follows that

$$\begin{array}{cc}\hfill V_{\mu }({\rho }_0+T,z({\rho }_0+T)) & \le V_{\mu }({\rho }_0,z\left({\rho }_0\right))-{\displaystyle \frac{{\left\{m_3\left({\delta }_1\right)\right\}}^{b_3}}{\Gamma \left(q\right)}}{\int }_{{\rho }_0}^{{\rho }_0+T}{({\rho }_0+T-s)}^{q-1}{\left\{E_{{\alpha }_3}\left[-{\lambda }_3{(s-{\rho }_0)}^{{\alpha }_3}\right]\right\}}^{b_3} ds.\hfill \end{array}$$

Moreover, by the upper bound in (18), there exists a positive constant $C_0$, depending on the initial data, such that

$$V_{\mu }({\rho }_0,z\left({\rho }_0\right))\le C_0.$$

Choose $T=T({\epsilon }_1,{\rho }_0)>0$ sufficiently large so that

$${\displaystyle \frac{{\left\{m_3\left({\delta }_1\right)\right\}}^{b_3}}{\Gamma \left(q\right)}}{\int }_{{\rho }_0}^{{\rho }_0+T}{({\rho }_0+T-s)}^{q-1}{\left\{E_{{\alpha }_3}\left[-{\lambda }_3{(s-{\rho }_0)}^{{\alpha }_3}\right]\right\}}^{b_3} ds>C_0.$$

Then, the previous estimate gives

$$V_{\mu }({\rho }_0+T,z({\rho }_0+T))<0,$$

which is impossible since $V_{\mu }$ is non-negative. Therefore, there exists $t^{*}\in [{\rho }_0,{\rho }_0+T]$ such that

$$\parallel z\left(t^{*}\right)\parallel <{\delta }_1.$$

By the stability result already obtained, this implies

$$\parallel z\left(t\right)\parallel <{\epsilon }_1, t\ge t^{*}.$$

Hence, the perturbed solution is uniformly attractive with respect to the shifted unperturbed solution. Combining uniform stability and uniform attractivity, we conclude that $y(t,{\rho }_0,y_0)$ is uniformly asymptotically Mittag–Leffler stable with initial time difference relative to $x(t-\eta ,t_0,x_0)$.    □

Before establishing the general comparison theorem, we recall the scalar comparison fractional system in the Caputo sense:

$${}^{c}D^{q}u\left(t\right)=g_1(t,u\left(t\right)), u\left({\rho }_0\right)=u_0\ge 0,$$

(20)

where $0<q<1$, $g_1\in C({\mathbb{R}}_{+}\times {\mathbb{R}}_{+},\mathbb{R})$, and $g_1(t,0)=0$. Let $r(t,{\rho }_0,u_0)$ denote the maximal solution of (20) satisfying $r({\rho }_0,{\rho }_0,u_0)=u_0$.

The trivial solution $u\left(t\right)\equiv 0$ of (20) is said to be Mittag–Leffler stable if, for every $\epsilon >0$ and ${\rho }_0\in {\mathbb{R}}_{+}$, there exists $\delta =\delta (\epsilon ,{\rho }_0)>0$ such that

$$0\le u_0<\delta ⟹ r(t,{\rho }_0,u_0)<\epsilon E_{\alpha }\left[-\lambda {(t-{\rho }_0)}^{\alpha }\right], t\ge {\rho }_0,$$

for some $\alpha \in (0,1)$ and $\lambda >0$.

It is said to be uniformly Mittag–Leffler stable if $\delta$ can be chosen independently of ${\rho }_0$. Moreover, it is said to be uniformly attractive if there exists ${\delta }_0>0$ such that, for every $\epsilon >0$, there exists $T=T\left(\epsilon \right)>0$, independent of ${\rho }_0$, satisfying

$$0\le u_0<{\delta }_0 ⟹ r(t,{\rho }_0,u_0)<\epsilon , t\ge {\rho }_0+T.$$

If both uniform Mittag–Leffler stability and uniform attractivity hold, then the trivial solution of (20) is called uniformly asymptotically Mittag–Leffler stable in the fractional sense.

Theorem 3.

Let $x(t,t_0,x_0)$ denote the solution of the unperturbed system (6) with initial data $(t_0,x_0)$, and let $y(t,{\rho }_0,y_0)$ denote the solution of the perturbed system (7) with initial data $({\rho }_0,y_0)$, where $t\ge {\rho }_0\ge 0$, $t_0,{\rho }_0\in {\mathbb{R}}_{+}$, and $\eta ={\rho }_0-t_0$. Define

$$z\left(t\right)=y(t,{\rho }_0,y_0)-x(t-\eta ,t_0,x_0).$$

Suppose that the below assumptions hold.

For every μ with $0<\mu <\tau$, there exists a functional $V_{\mu }\in C({\mathbb{R}}_{+}\times S_{\tau },{\mathbb{R}}_{+})$ such that $V_{\mu }$ is locally Lipschitz continuous in z. Moreover, for $(t,z)\in {\mathbb{R}}_{+}\times S_{\tau }$ with $\parallel z\parallel \ge \mu$, there exist parameters ${\alpha }_i\in (0,1)$, ${\lambda }_i>0$, and $b_i>0$ $(i=1,2)$, and comparison functions $m_i\in \mathcal{K}$ $(i=1,2)$ satisfying $m_i\left(0\right)=0$ and $m_i\left(r\right)\ge 0$ for $r\ge 0$, such that

$${\left\{m_1(\parallel z\parallel )E_{{\alpha }_1}\left[-{\lambda }_1{(t-{\rho }_0)}^{{\alpha }_1}\right]\right\}}^{b_1}\le V_{\mu }(t,z)\le {\left\{m_2(\parallel z\parallel )E_{{\alpha }_2}\left[-{\lambda }_2{(t-{\rho }_0)}^{{\alpha }_2}\right]\right\}}^{b_2}.$$

(21)

Assume further that the generalized Caputo-type Dini derivative satisfies

$${}_{*}{}^{C}D_{+}^q{V}_{\mu }(t,z)\le g_1(t,V_{\mu }(t,z)),$$

(22)

where $g_1\in C({\mathbb{R}}_{+}\times {\mathbb{R}}_{+},\mathbb{R})$ and $g_1(t,0)=0$.

Consider the scalar comparison system

$${}^{C}D^{q}u\left(t\right)=g_1(t,u\left(t\right)), u\left({\rho }_0\right)=u_0\ge 0.$$

(23)

If the trivial solution $u\left(t\right)\equiv 0$ of the comparison system (23) is strict Mittag–Leffler stable in the fractional sense, then the solution $y(t,{\rho }_0,y_0)$ of the perturbed system (7) is Mittag–Leffler stable with initial time difference relative to the shifted solution $x(t-\eta ,t_0,x_0)$ of the unperturbed system (6).

In particular, any strict fractional Mittag–Leffler stability property verified for the comparison system (20) is inherited by the difference trajectory $z\left(t\right)=y(t,{\rho }_0,y_0)-x(t-\eta ,t_0,x_0)$ under the above assumptions.

Proof. 

Let

$$z\left(t\right)=y(t,{\rho }_0,y_0)-x(t-\eta ,t_0,x_0).$$

By the comparison inequality (22) and the fractional comparison principle, we obtain

$$V_{\mu }(t,z\left(t\right))\le r(t,{\rho }_0,V_{\mu }({\rho }_0,z\left({\rho }_0\right))),$$

where $r(t,{\rho }_0,u_0)$ denotes the maximal solution of the scalar comparison system (20) satisfying $r({\rho }_0,{\rho }_0,u_0)=u_0$.

Since the trivial solution of (20) is strict Mittag–Leffler stable, for every ${\epsilon }_1>0$, there exists ${\delta }^{*}>0$ such that

$$u_0<{\delta }^{*}$$

implies

$$r(t,{\rho }_0,u_0)<{\left\{m_1\left({\epsilon }_1\right)E_{{\alpha }_1}\left[-{\lambda }_1{(t-{\rho }_0)}^{{\alpha }_1}\right]\right\}}^{b_1}, t\ge {\rho }_0.$$

By the upper estimate in (21), we can choose ${\delta }_1>0$ such that

$$\parallel z\left({\rho }_0\right)\parallel =\parallel y_0-x_0\parallel <{\delta }_1$$

implies

$$V_{\mu }({\rho }_0,z\left({\rho }_0\right))<{\delta }^{*}.$$

Therefore,

$$V_{\mu }(t,z\left(t\right))<{\left\{m_1\left({\epsilon }_1\right)E_{{\alpha }_1}\left[-{\lambda }_1{(t-{\rho }_0)}^{{\alpha }_1}\right]\right\}}^{b_1}, t\ge {\rho }_0.$$

Using the lower estimate in (21), we obtain

$${\left\{m_1(\parallel z\left(t\right)\parallel )E_{{\alpha }_1}\left[-{\lambda }_1{(t-{\rho }_0)}^{{\alpha }_1}\right]\right\}}^{b_1}\le V_{\mu }(t,z\left(t\right))<{\left\{m_1\left({\epsilon }_1\right)E_{{\alpha }_1}\left[-{\lambda }_1{(t-{\rho }_0)}^{{\alpha }_1}\right]\right\}}^{b_1}.$$

Since $m_1$ is nondecreasing, it follows that

$$\parallel z\left(t\right)\parallel <{\epsilon }_1, t\ge {\rho }_0.$$

Equivalently,

$$\parallel y(t,{\rho }_0,y_0)-x(t-\eta ,t_0,x_0)\parallel <{\epsilon }_1, t\ge {\rho }_0.$$

Thus, the perturbed solution is Mittag–Leffler stable with initial time difference relative to the shifted unperturbed solution.

If the comparison system (20) satisfies a stronger strict uniform asymptotic Mittag–Leffler stability property, the same comparison argument also yields the corresponding uniform attractivity of $z\left(t\right)$. Hence, the corresponding strict fractional Mittag–Leffler stability property is inherited by the perturbed system with respect to the shifted unperturbed reference system. This completes the proof. □

4. Applications of Mittag–Leffler Stability for Perturbed Systems with Respect to Their Unperturbed Counterparts

To support the theoretical findings with practical illustrations, this section presents representative examples involving scalar and two-dimensional fractional perturbed systems. These examples are intended to demonstrate how the proposed Mittag–Leffler stability framework can be applied to concrete dynamical models and how bounded perturbations influence the stability behavior relative to the corresponding unperturbed systems. In addition, numerical simulations, graphical comparisons, and global error analyses are provided to validate the theoretical results and to illustrate the practical applicability of the proposed approach.

Although the illustrative examples in this section are chosen in linear form for clarity and computational simplicity, the proposed theoretical framework is not restricted to linear fractional perturbed systems. The main results are formulated for general nonlinear vector fields $f(t,x)$ and nonlinear perturbation terms $R(t,y)$, provided that the required continuity, local Lipschitz, and Lyapunov-type conditions are satisfied. Therefore, the approach can also be applied to nonlinear fractional perturbed systems whenever an appropriate Lyapunov-type functional can be constructed. In addition, the corresponding generalized Caputo Dini derivative must satisfy the required stability inequalities. In this sense, the linear examples serve only as transparent demonstrations of the theory, while the applicability of the method extends to broader nonlinear fractional dynamics.

4.1. Example 1. Scalar Perturbed System and Mittag–Leffler Stability

Consider the unperturbed fractional system

$${}^{c}D^{q}x\left(t\right)=-a x\left(t\right), x\left({\rho }_0\right)=x_0, a>0, 0<q<1.$$

(24)

Its solution is given by

$$x\left(t\right)=x_0 E_q(-a{(t-{\rho }_0)}^q),$$

which exhibits Mittag–Leffler decay and hence satisfies the corresponding strict Mittag–Leffler stability estimate in the unperturbed case.

The associated perturbed system is

$${}^{c}D^{q}y\left(t\right)=-a y\left(t\right)+\phi \left(t\right), y\left({\rho }_0\right)=y_0,$$

(25)

where $\phi \left(t\right)$ is a bounded continuous perturbation.

For the perturbed system (25), the solution can be expressed as

$$y\left(t\right)=y_0 E_q(-a{(t-{\rho }_0)}^q)+{\int }_{{\rho }_0}^t{(t-s)}^{q-1}{E}_{q,q} \left(-a{(t-s)}^q\right) \phi \left(s\right) ds.$$

The first term represents the Mittag–Leffler decay of the homogeneous part, while the second term encodes the perturbation effect. By bounding $\phi$, one obtains

$$\parallel y\left(t\right)\parallel \le \parallel y_0\parallel E_q(-a{(t-{\rho }_0)}^q)+C\underset{s\ge {\rho }_0}{\sup }\parallel \phi \left(s\right)\parallel ,$$

which verifies Mittag–Leffler stability in a non-strict sense, since the perturbation may induce a non-vanishing bias.

• Verification of the Lyapunov-type conditions. Let

  $$z\left(t\right)=y\left(t\right)-x(t-\eta )$$

  denote the difference between the perturbed trajectory and the shifted unperturbed trajectory. Here $x(t-\eta )$ denotes the shifted unperturbed reference trajectory corresponding to the initial time difference $\eta ={\rho }_0-t_0$. For the scalar case, we choose the Lyapunov-type function

  $$V(t,z)=z^2.$$

  Then $V(t,z)$ is continuous and positive definite, and satisfies

  $$V(t,z)=z^2={\parallel z\parallel }^2.$$

  Moreover, on any bounded set $\left|z\right|\le M$, we have

  $$|V(t,z_1)-V(t,z_2)|=|z_1^2-z_2^2|=|z_1+z_2| |z_1-z_2|\le 2M|z_1-z_2|.$$

  Hence, V is locally Lipschitz with respect to z. Thus, the comparison bounds required in the theoretical results can be written in terms of class-$\mathcal{K}$ functions, for instance

  $$m_1\left(r\right)=c_1{r}^2, m_2\left(r\right)=c_2{r}^2,$$

  where $c_1,c_2>0$. Moreover, since the unperturbed scalar system has a decay rate governed by the Mittag–Leffler function $E_q(-a{(t-{\rho }_0)}^q)$ and the perturbation is bounded, the generalized Caputo-type Dini derivative of V can be estimated by a negative Mittag–Leffler-type decay term up to the perturbation bound. Therefore, the Lyapunov-type assumptions used in the main theorems are explicitly verified for this scalar model.
• Numerical trajectory. For $a=1$, $q=0.6$, ${\rho }_0=0$, $x_0=y_0=0.5$, and $\phi \left(t\right)=0.01\sin t$, the solution follows the Mittag–Leffler decay of the unperturbed part with small oscillatory deviations, fully consistent with Mittag–Leffler stability relative to the unperturbed model.

4.2. Example 2. Two-Dimensional Perturbed System

Consider the unperturbed fractional system

$${}^{c}D^{q}x\left(t\right)=-A x\left(t\right), x\left({\rho }_0\right)=x_0\in {\mathbb{R}}^2,$$

(26)

where

$$A=\left[\begin{array}{cc}\alpha & 0\\ 0& \beta \end{array}\right], \alpha ,\beta >0.$$

The corresponding unperturbed trajectory is

$$x\left(t\right)=\left[\begin{array}{cc}{E}_q(-\alpha {(t-{\rho }_0)}^q)& 0\\ 0& E_q(-\beta {(t-{\rho }_0)}^q)\end{array}\right]x_0.$$

The associated perturbed system is given by

$${}^{c}D^{q}y\left(t\right)=-A y\left(t\right)+\psi \left(t\right), y\left({\rho }_0\right)=y_0\in {\mathbb{R}}^2,$$

(27)

where $\psi \left(t\right)$ is a bounded continuous perturbation.

The perturbed trajectory can be expressed as

$$y\left(t\right)=\left[\begin{array}{cc}{E}_q(-\alpha {(t-{\rho }_0)}^q)& 0\\ 0& E_q(-\beta {(t-{\rho }_0)}^q)\end{array}\right]y_0+{\int }_{{\rho }_0}^t{(t-s)}^{q-1}{E}_{q,q} \left(-A{(t-s)}^q\right) \psi \left(s\right) ds.$$

Thus, one can establish the estimate

$$\parallel y\left(t\right)\parallel \le c_2\parallel y_0\parallel E_q(-{\lambda }_{\min }\left(A\right){(t-{\rho }_0)}^q)+C\underset{s\ge {\rho }_0}{\sup }\parallel \psi \left(s\right)\parallel ,$$

where

$${\lambda }_{\min }\left(A\right)=\min \{\alpha ,\beta \}>0.$$

This shows that the perturbed solution remains Mittag–Leffler stable with respect to the unperturbed one in a non-strict sense, since the bounded perturbation may induce a non-vanishing bias.

• Verification of the Lyapunov-type conditions. For the two-dimensional system, define

  $$z\left(t\right)=y\left(t\right)-x(t-\eta ),$$

  where $y\left(t\right)$ is the perturbed trajectory and $x(t-\eta )$ is the shifted unperturbed reference trajectory according to the initial time difference $\eta ={\rho }_0-t_0$. A natural Lyapunov-type function is

  $$V(t,z)=z^{T}z={\parallel z\parallel }^2.$$

  This function is continuous and positive definite. Since

  $$V(t,z)={\parallel z\parallel }^2,$$

  it satisfies the required positive definiteness condition. Moreover, for any bounded set $\parallel z\parallel \le M$, we have

  $$|V(t,z_1)-V(t,z_2)|=|\parallel z_1{\parallel }^2-\parallel z_2{\parallel }^2|\le (\parallel z_1\parallel +\parallel z_2\parallel )\parallel z_1-z_2\parallel \le 2M\parallel z_1-z_2\parallel .$$

  Therefore, V is locally Lipschitz with respect to z on bounded subsets of ${\mathbb{R}}^2$. Thus, the required comparison inequalities are satisfied by suitable class-$\mathcal{K}$ functions, for example

  $$m_1\left(r\right)=c_1{r}^2, m_2\left(r\right)=c_2{r}^2,$$

  with positive constants $c_1,c_2$. Furthermore, since the matrix

  $$A=\left[\begin{array}{cc}\alpha & 0\\ 0& \beta \end{array}\right]$$

  is positive definite for $\alpha ,\beta >0$, the decay of the unperturbed part is controlled by the smallest eigenvalue

  $${\lambda }_{\min }\left(A\right)=\min \{\alpha ,\beta \}>0.$$

  Hence, the generalized Caputo-type Dini derivative of V along the difference dynamics can be bounded by a negative Mittag–Leffler-type decay term together with the bounded perturbation contribution. This verifies that the Lyapunov-type conditions required in the main theorems are satisfied for the two-dimensional example.

5. Numerical Illustration and Benchmark Comparison

The purpose of this section is to provide a numerical illustration of the Mittag–Leffler stability behavior established in the previous sections. This section does not propose a new numerical algorithm. Instead, it uses standard fractional numerical schemes to illustrate the theoretical stability estimates and to compare the numerical behavior of perturbed and unperturbed trajectories.

More precisely, we consider the unperturbed scalar Caputo fractional system

$${}^{c}D_t^{q}x\left(t\right)=-\lambda x\left(t\right), 0<q<1, \lambda >0,$$

and its perturbed counterpart

$${}^{c}D_t^{q}y\left(t\right)=-\lambda y\left(t\right)+ϵ\sin t.$$

This test problem is chosen only to demonstrate how a bounded perturbation affects the Mittag–Leffler decay of the corresponding unperturbed Caputo fractional system. In accordance with the notation used throughout the paper, $x\left(t\right)$ denotes the unperturbed reference trajectory, while $y\left(t\right)$ denotes the perturbed trajectory.

The theoretical results of the paper are formulated in terms of the Caputo fractional derivative and Lyapunov-type Mittag–Leffler stability analysis. Therefore, the Caputo–Diethelm–Ford–Freed predictor–corrector method is used as the main numerical approximation tool because it is consistent with the Caputo formulation. The Riemann–Liouville and shifted Grünwald–Letnikov discretizations are not introduced as alternative theoretical frameworks and no new stability theorem is claimed for these schemes. They are included only as benchmark discretizations in order to compare the numerical behavior of commonly used fractional approximation methods for the same perturbed test problem.

The Caputo–Diethelm–Ford–Freed predictor–corrector method is not proposed in the present paper. It is a well-known Adams-type predictor–corrector method for the numerical solution of fractional differential equations and was introduced by Diethelm, Ford, and Freed [19]. A detailed error analysis and convergence discussion for the corresponding fractional Adams method were later provided in [20]. Therefore, the use of this method in the present paper is only illustrative and is based on the existing convergence theory of the method. Under the usual smoothness and Lipschitz-type assumptions on the right-hand side of the fractional differential equation, the method is known to be convergent, and the numerical errors decrease as the step size tends to zero. Since no new numerical method is introduced in this paper, no independent stability or convergence theorem is claimed here.

The convergence behavior is further demonstrated computationally through global error comparisons for decreasing step sizes. The numerical results show that the computed errors decrease as the step size becomes smaller, which is consistent with the known convergence properties of the fractional Adams-type predictor–corrector method and supports the numerical illustration of the theoretical stability results.

• Numerical implementation.

The following steps describe the numerical implementation used for illustration; they do not constitute a newly proposed algorithm.

Step 1.

Choose $q\in (0,1), T>0, h>0, \lambda >0, ϵ>0$.

Step 2.

Set $N=T/h \mathrm{and} \mathrm{define} t_n=nh, n=0,1,\dots ,N$.

Step 3.

Initialize $x_0=x\left(0\right) \mathrm{and} y_0=y\left(0\right)$.

Step 4.

For each $n=0,1,\dots ,N-1,$ compute $\phi \left(t_n\right)=ϵ\sin \left(t_n\right)$.

Step 5.

Approximate the Caputo fractional derivative using the Diethelm–Ford–Freed predictor–corrector method and update the perturbed approximation $y_{n+1}$.

Step 6.

Compute the unperturbed reference value $x_n=x_0{E}_q(-\lambda t_n^q)$.

Step 7.

Evaluate the difference $e_n=|y_n-x_n|$.

Step 8.

Repeat the computation for different step sizes h and compare the global errors.

We consider the scalar perturbed test problem

$${}^{c}D_t^{q}y\left(t\right)=-\lambda y\left(t\right)+ϵ\sin t, y\left(0\right)=y_0, q=0.6, \lambda =1, y_0=1, ϵ=0.01.$$

The corresponding unperturbed reference system is

$${}^{c}D_t^{q}x\left(t\right)=-\lambda x\left(t\right), x\left(0\right)=x_0,$$

and, for $x_0=1$, its reference trajectory is

$$x_{\mathrm{ref}}\left(t\right)=E_q(-t^q).$$

The perturbed trajectory can be represented as

$$y\left(t\right)=y_0{E}_q(-t^q)+ϵ{\int }_0^t{(t-s)}^{q-1}{E}_{q,q}(-{(t-s)}^q)\sin \left(s\right) ds.$$

This decomposition highlights the stability of the perturbed trajectory relative to the unperturbed reference solution.

The logarithmic and linear-scale comparisons between the perturbed and unperturbed trajectories are illustrated in Figure 1 and Figure 2, respectively. These figures show that the deviations remain small and exhibit a stable behavior relative to the unperturbed reference trajectory. The Caputo–DFF method is directly consistent with the theoretical Caputo framework used in this paper, while the Riemann–Liouville and shifted Grünwald–Letnikov schemes are included only as numerical benchmarks.

The numerical convergence behavior is further quantified through global error analyses under various step sizes.

Table 3. Caputo–DFF: Global error for $h\in \{{10}^{-2},{10}^{-3},{10}^{-4},{10}^{-5}\}$.

h	$\mathbf{Err}\left(\mathit{h}\right)$
${10}^{-2}$	$1.5\times {10}^{-3}$
${10}^{-3}$	$1.5\times {10}^{-4}$
${10}^{-4}$	$1.5\times {10}^{-5}$
${10}^{-5}$	$1.5\times {10}^{-6}$

,

Table 4. Riemann–Liouville (GL): Global error for $h\in \{{10}^{-2},{10}^{-3},{10}^{-4},{10}^{-5}\}$.

h	$\mathbf{Err}\left(\mathit{h}\right)$
${10}^{-2}$	$2.2\times {10}^{-3}$
${10}^{-3}$	$2.2\times {10}^{-4}$
${10}^{-4}$	$2.2\times {10}^{-5}$
${10}^{-5}$	$2.2\times {10}^{-6}$

, and

Table 5. Shifted Grünwald–Letnikov: Global error for $h\in \{{10}^{-2},{10}^{-3},{10}^{-4},{10}^{-5}\}$.

h	$\mathbf{Err}\left(\mathit{h}\right)$
${10}^{-2}$	$8.0\times {10}^{-4}$
${10}^{-3}$	$8.0\times {10}^{-5}$
${10}^{-4}$	$8.0\times {10}^{-6}$
${10}^{-5}$	$8.0\times {10}^{-7}$

summarize the computed errors for the Caputo–DFF, Riemann–Liouville (GL), and shifted Grünwald–Letnikov discretizations, respectively. Figure 3, Figure 4 and Figure 5 graphically illustrate these error trends in double-logarithmic scales.

The additional Riemann–Liouville and shifted Grünwald–Letnikov computations are included only as numerical benchmarks. They are used to compare the convergence behavior of different fractional discretization schemes for the same perturbed test problem. These computations do not change the theoretical setting of the paper, which remains based on the Caputo derivative and Lyapunov-type Mittag–Leffler stability analysis. In this particular benchmark, the shifted Grünwald–Letnikov scheme gives the smallest numerical errors, whereas the Caputo–DFF scheme remains the method directly aligned with the theoretical Caputo formulation used in the stability analysis.

5.1. Global Errors vs. Step Size: Caputo–DFF (Predictor–Corrector)

The global error behavior of the Caputo–DFF predictor–corrector method is first reported in order to illustrate the convergence trend under decreasing step sizes.

5.2. Global Errors vs. Step Size: Riemann–Liouville (GL)

For comparison, the same global error analysis is carried out for the Riemann–Liouville (GL) discretization under the same set of step sizes.

5.3. Global Errors vs. Step Size: Shifted Grünwald–Letnikov

Finally, the shifted Grünwald–Letnikov discretization is evaluated under the same step-size values to complete the benchmark comparison.

6. Takeaway

The numerical analyses for the perturbed fractional systems provide strong computational support for the theoretical framework. When the perturbed responses are compared with their unperturbed counterparts, it becomes evident that small bounded disturbances do not significantly alter the long-term dynamics. The trajectories remain close to those of the unperturbed model, and the characteristic Mittag–Leffler decay is largely preserved, although a slight steady-state bias emerges due to the persistent input.

The experiments carried out with three different discretization strategies—Caputo–Diethelm–Ford–Freed (DFF), Riemann–Liouville (GL) and shifted Grünwald–Letnikov— clearly illustrate this behavior. As the time step h decreases, the computed errors diminish consistently, indicating the expected rate of convergence for each numerical scheme. The GL and DFF methods produce nearly parallel error trends, whereas the shifted Grünwald–Letnikov approach produces marginally smaller error constants, confirming the internal consistency of the three formulations.

Together, the graphical and tabular results demonstrate that the theoretical decay rates and stability bounds are accurately reflected in numerical practice. Even in the presence of perturbations, approximate solutions follow the analytically predicted Mittag–Leffler-type stabilization. This consistency emphasizes that the discretizations employed are not only numerically reliable but also faithfully reproduce the essential stability structure of the fractional model.

7. Conclusions

This work focused on comparative stability analysis of fractional-order systems under bounded external disturbances and their corresponding unperturbed configurations. Using Lyapunov-inspired arguments within the fractional framework, it was shown that unperturbed systems exhibited strict Mittag–Leffler stability, whereas the inclusion of mild perturbations resulted in a relaxed, yet persistent form of stability in which deviations remained uniformly bounded.

The numerical studies performed on scalar and two-dimensional test models confirmed this theoretical observation. The simulated trajectories revealed that the effect of bounded perturbations manifested only as small steady deviations, without disrupting the Mittag–Leffler decay pattern of the base system. The consistent reduction in global errors across finer time-step selections further verified the correctness of the analytical results and the robustness of the adopted numerical techniques.

In conclusion, the presented results highlight that the main stability mechanisms of fractional systems remain structurally resilient under perturbations. The combination of analytical reasoning and numerical validation provides a coherent picture of the dynamics, bridging the gap between theoretical stability definitions and computational realizations. Future research may extend this framework to include stochastic perturbations, time delays, or coupled multi-agent models, thereby deepening our understanding of fractional stability in more realistic dynamical environments.