Fractional-order systems have emerged as a powerful tool in modeling complex real-world dynamics, particularly in fields such as finance, physics, engineering, and biology. The nonlocal properties and memory effects inherent in fractional derivatives provide a more comprehensive framework for capturing system behaviors compared to traditional integer-order models [1] - [4] . In financial systems, these attributes enable the modeling of intricate economic interactions, revealing insights into market dynamics and stability under varying conditions [5] - [7] .

The Caputo derivative has been widely adopted in fractional-order studies due to its ability to handle initial value problems effectively. Numerous studies have employed this derivative to explore financial dynamics, highlighting its flexibility and mathematical rigor [3] [8] . For instance, Diethelm et al. [8] provided a detailed analysis of fractional-order systems using the Caputo derivative, while Podlubny [3] established its foundational role in fractional calculus applications.

The Elzaki transform, another powerful mathematical tool, has been used to simplify and solve fractional differential equations (FDEs), proving particularly effective in handling complex fractional-order systems with memory-dependent behavior [9] - [11] . This method has been instrumental in reducing computational complexity, as demonstrated in applications across engineering and finance [9] [12] . The combination of these methods has led to breakthroughs in understanding and controlling chaotic systems in finance [5] [13] .

In the realm of numerical solutions, the Predictor-Corrector Adams-Bashforth-Moulton (PC-ABM) method has gained prominence for its efficiency and accuracy in solving FDEs. This method enables researchers to obtain high-fidelity solutions, even for systems with strong nonlinearities [7] [8] [14] . Garrappa [14] extended the method’s application to fractional chaotic systems, demonstrating its robustness in predicting long-term dynamics. Recent studies have applied this method to fractional financial systems, offering precise insights into the stability and behavior of such systems under varying parameters [7] [15] .

Synchronization and control of fractional-order financial systems remain active areas of research. Fractional feedback control, in particular, has been shown to be an effective method for stabilizing chaotic systems and guiding them toward desired trajectories. This approach leverages the memory-dependent nature of fractional derivatives to achieve enhanced control precision [4] [16] . Additionally, nonlinear control strategies have been developed to tackle the unique challenges posed by chaotic financial systems, enabling the stabilization and synchronization of these systems under uncertain conditions [10] [15] .

Driven by the advancements in fractional calculus and its applications in finance, this study develops a comprehensive approach to analyze and control a fractional financial system. Key contributions of this research include modeling the financial system using the Caputo derivative, solving the system using the Elzaki transform and PC-ABM method, and applying fractional feedback and nonlinear control to stabilize the system. Furthermore, this work delves into the stability analysis of the system.

This section presents the numerical solution of the fractional differential equation (FDE) system (13) using MATLAB. The Predictor-Corrector Adams-Bashforth-Moulton (PECE) scheme [7] is employed due to its efficiency in solving fractional systems and accounting for memory effects ( Figure 1 ).

The key behaviors observed are as follows:

In contrast, integer-order systems lack these stabilizing memory effects and exhibit persistent oscillations across all variables.

In this section, the Elzaki transform method [9] is applied to solve the fractional differential equation (FDE) system (13) analytically. This approach converts the system into algebraic equations, facilitating efficient series solutions.

Applying the Elzaki transform with the initial conditions $x\left(0\right)=x_0$, $y\left(0\right)=y_0$, $z\left(0\right)=z_0$, and $u\left(0\right)=u_0$, the transformed system is obtained as:

$\{\begin{array}{l}x\left(t\right)=E^{-1}\left\{v^{q}E\left\{z+\left(y-a\right)x+u\right\}\right\},\\ y\left(t\right)=E^{-1}\left\{v^{q}E\left\{1-by-x^2\right\}\right\},\\ z\left(t\right)=E^{-1}\left\{v^{q}E\left\{-x-cz\right\}\right\},\\ u\left(t\right)=E^{-1}\left\{v^{q}E\left\{-dxy-gu\right\}\right\}.\end{array}$(14)

Since nonlinear terms pose a challenge for the Elzaki transform, the Differential Transform Method (DTM) is employed to handle them effectively and compute the solution coefficients recursively [7] . This hybrid approach ensures both accuracy and efficiency in solving fractional nonlinear systems.

The final solution is expressed as a series:

$\{\begin{array}{l}x\left(t\right)=x_0+{\displaystyle {\sum }_{n=1}^{\infty }{c}_n^{\left(x\right)}{t}^{nq}},\hfill \\ y\left(t\right)=y_0+{\displaystyle {\sum }_{n=1}^{\infty }{c}_n^{\left(y\right)}{t}^{nq}},\hfill \\ z\left(t\right)=z_0+{\displaystyle {\sum }_{n=1}^{\infty }{c}_n^{\left(z\right)}{t}^{nq}},\hfill \\ u\left(t\right)=u_0+{\displaystyle {\sum }_{n=1}^{\infty }{c}_n^{\left(u\right)}{t}^{nq}},\hfill \end{array}$

where $c_n^{\left(x\right)}$, $c_n^{\left(y\right)}$, $c_n^{\left(z\right)}$, and $c_n^{\left(u\right)}$ are determined recursively using DTM.

The stability of the fractional-order financial system (13) is analyzed through equilibrium points and eigenvalue conditions. Using numerical and analytical methods, we determine the equilibrium points and evaluate stability criteria based on Proposition (1).

The equilibrium points are obtained by setting ${\text{D}}^{q}x\left(t\right)={\text{D}}^{q}y\left(t\right)={\text{D}}^{q}z\left(t\right)={\text{D}}^{q}u\left(t\right)=0$. Solving the resulting equations yields three equilibrium points:

$\begin{array}{l}{E}_1=\left(0,\frac{1}{b},0,0\right),E_2=\left(\gamma ,\frac{g+acg}{c\left(g-d\right)},\frac{-\gamma }{c},\frac{d\gamma \left(1+ac\right)}{c\left(d-g\right)}\right),\\ E_3=\left(-\gamma ,\frac{g+acg}{c\left(g-d\right)},\frac{\gamma }{c},\frac{-d\gamma \left(1+ac\right)}{c\left(d-g\right)}\right).\end{array}$

Here, $\gamma =\pm \sqrt{1+\frac{gb+acgb}{c\left(d-g\right)}}$.

To analyze the stability of the system (13), we compute the Jacobian matrix:

$J=\left[\begin{array}{cccc}y-a& x& 1& 1\\ -2x& -b& 0& 0\\ -1& 0& -c& 0\\ -dy& -dx& 0& -g\end{array}\right].$

Substituting $E_1=\left(0,\frac{1}{b},0,0\right)$, the Jacobian becomes:

$J\left(E_1\right)=\left[\begin{array}{cccc}\frac{1}{b}-a& 0& 1& 1\\ 0& -b& 0& 0\\ -1& 0& -c& 0\\ -\frac{d}{b}& 0& 0& -g\end{array}\right].$

To attain stability, we calculate the eigenvalues and adjust them to be either negative real numbers or complex numbers, ensuring stability.

We aim to find suitable values for $a$, $b$, $c$, $d$, and $g$ such that:

This ensures that the system is stable at the first equilibrium point using Proposition (1).

The eigenvalues of $J\left(E_1\right)$ are:

${\lambda }_1=-b,{\lambda }_2=\frac{{\sigma }_2}{{\sigma }_1}+{\sigma }_1-\frac{{\sigma }_4}{3b},{\lambda }_{3,4}=-\frac{{\sigma }_2}{2{\sigma }_1}-\frac{{\sigma }_1}{2}-\frac{{\sigma }_4}{3b}\pm i\frac{\sqrt{3}\left(\frac{{\sigma }_2}{{\sigma }_1}-{\sigma }_1\right)}{2}.$

where

${\sigma }_1={\left(\sqrt{{\left(\frac{{\sigma }_4^3}{27b^3}+\frac{cd+bg-cg+abcg}{2b}-{\sigma }_3\right)}^2-{\sigma }_2^3}\right)}^{1/3},$

${\sigma }_2=\frac{{\sigma }_4^2}{9b^2}-\frac{b-c+d-g+abc+abg+bcg}{3b},$

${\sigma }_3=\frac{{\sigma }_4\left(b-c+d-g+abc+abg+bcg\right)}{6b^2},$

${\sigma }_4=ab+bc+bg-1.$

1) Condition for ${\lambda }_1$: ${\lambda }_1=-b<0$, which is satisfied for $b>0$.

2) Condition for ${\lambda }_2$: To ensure ${\lambda }_2<0$, the following inequality must hold:

$\frac{{\sigma }_2}{{\sigma }_1}+{\sigma }_1<\frac{{\sigma }_4}{3b},$

3) Condition for ${\lambda }_{3,4}$: For ${\lambda }_{3,4}$ to be complex, ${\sigma }_2<0$ ensures the square root remains valid:

${\sigma }_4^2<3b\left(b-c+d-g+abc+abg+bcg\right).$

$b^2\left(3+g\left(a+c+g\right)-{\left(a+c\right)}^2\right)<b\left(c-3d+g-2a\right)+1.$

Combining the conditions, the stability criteria become:

$\frac{{\sigma }_2}{{\sigma }_1}+{\sigma }_1<\frac{{\sigma }_4}{3b}$(16)

$b^2\left(3+g\left(a+c+g\right)-{\left(a+c\right)}^2\right)<b\left(c-3d+g-2a\right)+1$(17)

The coefficients are chosen as follows:

$a=0.1,b=0.3,c=0.1,d=0.9,g=\mathrm{0.6.}$

With these values, the equilibrium points are calculated as:

$E_1=\left(0,3.3333,0,0\right),E_2=\left(2.6833,-20.2000,-26.8328,81.3034\right),$

$E_3=\left(-2.6833,-20.2000,26.8328,-81.3034\right).$

Stability Test for $E_1$

The eigenvalues are:

${\lambda }_1=-0.3,{\lambda }_2=-0.2749,{\lambda }_3=1.4041+0.7727i,{\lambda }_4=1.4041-0.7727i.$

Based on Proposition (1), $E_1$ is asymptotically stable when:

$0<q<\mathrm{0.3203.}$

Stability Test for $E_2$ and $E_3$

The eigenvalues for both $E_2$ and $E_3$ are:

${\lambda }_1=-20.4198,{\lambda }_2=-1.0368,{\lambda }_3=0.2412,{\lambda }_4=-\mathrm{0.0846.}$

Both points are unstable as they fail to satisfy the conditions in Proposition (1). Additionally, these points do not meet the derived stability criteria (16) and (17).

Figure 2 illustrates the system’s behavior for $q<0.3203$ (stable) and $q>0.3203$ (unstable).

The stability of a fixed point in a dynamical system is determined by the eigenvalues ( $\lambda$) of the Jacobian matrix. A point is stable if:

1) All eigenvalues are either complex or purely negative real numbers.

2) If any eigenvalue is a positive real numbers, the system becomes unstable.

Here, we analyze the stability of the second equilibrium point ( $E_2$) and demonstrate its instability under the given conditions. The same reasoning applies to $E_3$.

Key Analysis Steps

1) Condition $d>g$: This ensures that $\gamma$ remains real (not complex), which is crucial for analyzing system stability.

2) Analysis of ${\sigma }_2$: If ${\sigma }_2$ is complex, then both eigenvalues ${\lambda }_1$ and ${\lambda }_2$ must also be complex.

3) Analysis of ${\sigma }_1$: If ${\sigma }_2$ is complex, it is impossible for ${\sigma }_1$ to also be complex. This ensures that ${\lambda }_3$ and ${\lambda }_4$ remain real.

4) Analysis of ${\lambda }_3$ and ${\lambda }_4$: It is impossible for both ${\lambda }_3$ and ${\lambda }_4$ to be negative under the given conditions. At least one eigenvalue must be positive, rendering the second equilibrium point unstable.

Jacobian Matrix at $E_2$

The Jacobian matrix at $E_2$ is expressed as:

$J\left(E_2\right)=\left[\begin{array}{cccc}\frac{g+acg}{c\left(g-d\right)}-a& \gamma & 1& 1\\ -2\gamma & -b& 0& 0\\ -1& 0& -c& 0\\ -\frac{d\left(g+acg\right)}{c\left(g-d\right)}& -d\gamma & 0& -g\end{array}\right].$

The eigenvalues are:

${\lambda }_1=-{\sigma }_3-\frac{{\sigma }_{17}}{4{\sigma }_{19}}-{\sigma }_2,{\lambda }_2=-{\sigma }_3-\frac{{\sigma }_{17}}{4{\sigma }_{19}}+{\sigma }_2,$

${\lambda }_3={\sigma }_3-\frac{{\sigma }_{17}}{4{\sigma }_{19}}-{\sigma }_1,{\lambda }_4={\sigma }_3-\frac{{\sigma }_{17}}{4{\sigma }_{19}}+{\sigma }_1.$

Proof of Instability:

We analyze the stability of $E_2$ using the eigenvalues of the Jacobian matrix. The steps are as follows:

1) Condition $d>g$: Ensures positivity of key parameters:

${\sigma }_{19}=c\left(d-g\right),{\sigma }_{17}=g+c^{2}d-c\left(g^2+g\right)+acd+bcd-bcg+cdg.$

This guarantees no undefined or negative values in subsequent eigenvalue analysis.

2) If ${\sigma }_2$ is complex, ${\lambda }_1$ and ${\lambda }_2$ are also complex:

The condition for ${\sigma }_2$ to be complex is:

$12{\sigma }_{12}\sqrt{{\sigma }_9}-{\sigma }_8-{\sigma }_7-{\sigma }_6+{\sigma }_4<0.$

Since ${\lambda }_1$ and ${\lambda }_2$ depend on ${\sigma }_2$:

${\lambda }_1=-{\sigma }_3-\frac{{\sigma }_{17}}{4{\sigma }_{19}}-{\sigma }_2,{\lambda }_2=-{\sigma }_3-\frac{{\sigma }_{17}}{4{\sigma }_{19}}+{\sigma }_2,$

the imaginary component of ${\sigma }_2$ makes both eigenvalues complex.

3) If ${\sigma }_2$ is complex, ${\sigma }_1$ cannot be complex, and ${\lambda }_3$ and ${\lambda }_4$ remain real:

The condition for ${\sigma }_1$ to be complex is:

$12{\sigma }_{12}\sqrt{{\sigma }_9}-{\sigma }_8-{\sigma }_7-{\sigma }_6-{\sigma }_4<0.$

The key difference lies in the sign of ${\sigma }_4$: For ${\sigma }_2$, positive ${\sigma }_4$ pushes the argument towards negativity. For ${\sigma }_1$, negative ${\sigma }_4$ pushes the argument towards positivity.

Thus, ${\sigma }_1$ and ${\sigma }_2$ cannot both be complex, ensuring ${\lambda }_3$ and ${\lambda }_4$ remain real.

4) It is impossible for both ${\lambda }_3$ and ${\lambda }_4$ to be negative:

The expressions for ${\lambda }_3$ and ${\lambda }_4$ are:

${\lambda }_3={\sigma }_3-\frac{{\sigma }_{17}}{4{\sigma }_{19}}-{\sigma }_1,{\lambda }_4={\sigma }_3-\frac{{\sigma }_{17}}{4{\sigma }_{19}}+{\sigma }_1.$

For ${\lambda }_3$ to be negative:

${\sigma }_3-{\sigma }_1<\frac{{\sigma }_{17}}{4{\sigma }_{19}}.$

For ${\lambda }_4$ to be negative:

${\sigma }_3+{\sigma }_1<\frac{{\sigma }_{17}}{4{\sigma }_{19}}.$

Both conditions cannot hold simultaneously because: A larger ${\sigma }_1$ making ${\lambda }_3$ negative causes ${\lambda }_4$ to become positive. A smaller ${\sigma }_1$ making ${\lambda }_4$ negative causes ${\lambda }_3$ to become positive.

At the end, it is impossible for both ${\lambda }_3$ and ${\lambda }_4$ to be negative; at least one must be positive.

We may conclude that, at least one eigenvalue for $E_2$ is positive, violating the stability conditions that require all eigenvalues to be complex or purely negative real numbers. Therefore, $E_2$ is unstable. The same analysis confirms that $E_3$ is also unstable.

This section explores the application of nonlinear feedback control to stabilize a fractional-order financial system (13). Control inputs

$w\left(t\right)={\left[w_1\left(t\right),w_2\left(t\right),w_3\left(t\right),w_4\left(t\right)\right]}^{\text{T}}$ are introduced to drive the system’s state variables $\left(x,y,z,u\right)$ towards a desired equilibrium $\left(\tilde{x},\tilde{y},\tilde{z},\tilde{u}\right)$ [13] .

The controlled system is expressed as:

$\{\begin{array}{l}{}_{t_0}{}^C\text{D}{}_t^{q}x\left(t\right)=z+\left(y-a\right)x+u+w_1,\\ {}_{t_0}{}^C\text{D}{}_t^{q}y\left(t\right)=1-by-x^2+w_2,\\ {}_{t_0}{}^C\text{D}{}_t^{q}z\left(t\right)=-x-cz+w_3,\\ {}_{t_0}{}^C\text{D}{}_t^{q}u\left(t\right)=-dxy-gu+w_4.\end{array}$(18)

Define the error terms $e_i$ = current state – desired state, such that:

$e_1=x-\tilde{x},e_2=y-\tilde{y},e_3=z-\tilde{z},e_4=u-\tilde{u}.$

The error dynamics evolve as:

$\{\begin{array}{l}{\text{D}}^q{e}_1=e_3+e_4-a{e}_1+yx-\tilde{y}\tilde{x}+w_1,\\ {\text{D}}^q{e}_2=-b{e}_2-x^2+{\tilde{x}}^2+w_2,\\ {\text{D}}^q{e}_3=-e_1-c{e}_3+w_3,\\ {\text{D}}^q{e}_4=-g{e}_4-dxy+d\tilde{x}\tilde{y}+w_4.\end{array}$(19)

To cancel nonlinear terms and stabilize the system, the control inputs are defined as:

$\{\begin{array}{l}{w}_1\left(t\right)=\tilde{y}\tilde{x}-yx,\\ w_2\left(t\right)=x^2-{\tilde{x}}^2,\\ w_3\left(t\right)=e_1,\\ w_4\left(t\right)=dxy-d\tilde{x}\tilde{y}.\end{array}$(20)

Substituting these control laws simplifies the error dynamics to:

$\{\begin{array}{l}{\text{D}}^q{e}_1=e_3+e_4-a{e}_1,\\ {\text{D}}^q{e}_2=-b{e}_2,\\ {\text{D}}^q{e}_3=-c{e}_3,\\ {\text{D}}^q{e}_4=-g{e}_4.\end{array}$(21)

The controlled system is as follows:

$\{\begin{array}{l}{}_{t_0}{}^C\text{D}{}_t^{q}x\left(t\right)=z-ax+u+\tilde{y}\tilde{x},\\ {}_{t_0}{}^C\text{D}{}_t^{q}y\left(t\right)=1-by-{\tilde{x}}^2,\\ {}_{t_0}{}^C\text{D}{}_t^{q}z\left(t\right)=-cz-\tilde{x},\\ {}_{t_0}{}^C\text{D}{}_t^{q}u\left(t\right)=-gu-d\tilde{x}\tilde{y}.\end{array}$(22)

Figure 3 illustrates the system behavior before and after applying control.

This study investigated the fractional-order financial system, focusing on its stability and control dynamics. Using the Caputo fractional derivative framework, the system’s equilibrium points were derived, and their stability was rigorously analyzed. For the first equilibrium point $E_1$, the stability was proven under specific conditions. Numerical simulations verified the theoretical results, demonstrating that the system transitions to stability when these conditions are met.

For the second and third equilibrium points ( $E_2$ and $E_3$), the analysis showed that at least one eigenvalue always has a positive real part, rendering these points inherently unstable. These results were validated through eigenvalue computations and verified by numerical simulations, aligning with the theoretical framework presented in [3] [7] .

The study further introduced a nonlinear feedback control strategy to stabilize the fractional financial system. The control inputs were designed to suppress chaotic behavior, ensuring convergence of the system states to a desired equilibrium. The proposed control method effectively dampened oscillations and stabilized the system, as validated by numerical experiments. This demonstrates the practical applicability of fractional-order modeling and control in financial systems [13] [16] .

In conclusion, this work provides a robust framework for analyzing and controlling fractional-order financial systems. Future research can extend these findings by exploring adaptive control strategies to account for parameter uncertainties and dynamic changes in financial systems. Additionally, further studies could investigate the implications of fractional-order dynamics on financial market predictability and risk assessment [5] [15] .