Dynamic Analysis of a Fractional-Order Economic Model: Chaos and Control

Abstract

Fractional calculus in discrete-time is a recent field that has drawn much interest for dealing with multidisciplinary systems. A result of this tremendous potential, researchers have been using constant and variable-order fractional discrete calculus in the modelling of financial and economic systems. This paper explores the emergence of chaotic and regular patterns of the fractional four-dimensional (4D) discrete economic system with constant and variable orders. The primary aim is to compare and investigate the impact of two types of fractional order through numerical solutions and simulation, demonstrating how modifications to the order affect the behavior of a system. Phase space orbits, the 0-1 test, time series, bifurcation charts, and Lyapunov exponent analysis for different orders all illustrate the constant and variable-order systems’ behavior. Moreover, the spectral entropy (SE) and $C_0$ complexity exhibit fractional-order effects with variations in the degree of complexity. The results provide new insights into the influence of fractional-order types on dynamical systems and highlight their role in promoting chaotic behavior. Additionally, two types of control strategies are devised to guide the states of a 4D fractional discrete economic system with constant and variable orders to the origin within a specified amount of time. MATLAB simulations are presented to demonstrate the efficacy of the findings.

1. Introduction

Economic systems behave erratically, sometimes in a chaotic form [1,2,3]. The reason for and character of these irregularities have a major impact on how irregular behavior is employed for economic prediction and stabilization. Unusual occurrences, such as strikes, civil unrest, currency and oil shocks, major bankruptcies, epidemics, and floods, could cause irregular economic fluctuations [4,5]. One way to conceptualize these occurrences is as external shocks. The very nature of economic systems, however, can involve endogenous forces that produce erratic swings. Therefore, there are two methods for analyzing economic irregularities. Random processes that are regarded as exogenous shocks in the model are taken into consideration in the first method. Finding a deterministic endogenous mechanism for the emergence of irregular fluctuations—which could be chaotic—is the foundation of the second one. Views regarding the causes of irregular fluctuations were hotly debated as a result of the parallel development of these two methods in the economics literature.

Chaotic discrete systems serve as significant models for understanding natural events and can occur in a variety of settings. Because of their inherent characteristics, these systems provide excellent examples of chaos and are crucial for studying dynamical systems. Through their use, chaotic behavior has been successfully characterized in a variety of fields, including fluid dynamics, semiconductor processes, secure communication [6], economics [7], and finance [8]. The chaotic regime and attractor variety of a discrete economic system are revealed by the introduction of state-dependent nonlinearity.

In economic modelling through mathematics, the fundamentally language used to address non-integer-order integrals and derivatives, as well as difference and differential equations, is actively applied to systems [9,10]. Because of these operators and equations, economists were able to create mathematical models that described a variety of economic procedures and circumstances. The integer-order derivatives of the functions are known to depend on their properties in the infinitely small region where the derivatives are under consideration. Economic models, which rely on differential equations of integer orders, are therefore unable to explain memory-based and non-local processes [11,12].

In recent years, scientists from all around the world have worked hard to establish a general framework for the stability theory of fractional discrete calculus. In [13], the rich dynamics of a fractional discrete Lotka–Volterra system were investigated in the setting of fractional operators. The complexity behavior under the Caputo-left operator was examined by Fečkan et al. [14]. In [15], Xu et al. used a fractional Caputo-left operator to analyze a logistic map. Despite its more than 300-year history, fractional calculus has only been used in a number of scientific and engineering fields since the beginning of the twentieth century. The primary benefit of this type of calculus is the memory effect that results from fractional differentiation and integration, which allows for the more flexible modelling of natural occurrences. The dynamics, applications, and control of chaos have drawn the interest of researchers investigating discrete fractional calculus. Recent developments in fractional-order modeling further highlight its wide applicability in various fields, including the analysis and control of chaotic supply chain systems [16], the study of nonlinear distributed-order models with adaptive synchronization and encryption [17], and complex physical systems described by nonlinear Schrödinger equations [18]. These developments underscore the versatility of fractional-order modeling in capturing complex dynamics across disciplinary boundaries. The evolution of the fractional forms of various conventional discrete chaotic systems, as well as the stabilization of such discrete fractional systems, have been the subject of relatively recent investigations, for example, the bifurcation dynamics in discretized fractional predator–prey models [19], theoretical formulations of delta and nabla Caputo fractional differences [20], and the study of the dynamics and complexity of discrete-time fractional systems [21,22,23,24]. The monograph on discrete fractional calculus by Goodrich and Peterson [25] provides a comprehensive theoretical foundation that supports the methodology adopted in the present work.In addition, in [26], proposed a financial risk map using a Caputo-left operator. In [27], an artificial macroeconomic fractional system in discrete-time has been proposed. In contrast, the issue of incommensurate orders on the features of the macroeconomic system was examined in [28]. A discrete financial model with fractional orders and its stability is examined in [29].

As variable-order derivatives are more adaptable and adjustable as opposed to being of a constant order, they have increased suitability for modelling systems with time-varying diseases. In contrast to assuming a constant memory effect during system evolution, variable-order derivatives enable us to obtain a memory effect that shifts depending on the external situation or the system status. This type of adaptability improves system stability and prevents chaos by dynamically controlling the prior-state influence [30]. The main advantage of adopting a variable fractional order lies in its ability to represent time-dependent memory effects in complex systems. In economic dynamics, the degree of dependence on past states is rarely constant; rather, it evolves with market shocks, learning behavior, or policy interventions. Thus, the variable-order framework provides a more realistic and adaptive tool to capture such evolving memory characteristics compared with constant fractional-order models. In applications such as financial and economic applications, this capacity allows systems to improve their reactivity and durability while adapting their performance to various scenarios. In addition, when the order of the nonlocality varies over time, variable-order derivatives can explain real-world occurrences. This makes them very helpful in economic, financial, and other systems that require a great deal of control over chaotic dynamics. For instance, in [31], Jahanshahi et al. created a predictive controller using a nonlinear model for a variable fractional-order economic model. A symmetric chaotic financial system for variable fractional derivatives is examined in [32] through the application of intelligent parameter identification and prediction. Almatroud et al. [33] developed a variable fractional discrete system. Additionally, ref. [34] explored the chaotic dynamics of a variable fractional map.

The dynamics of the proposed fractional four-dimensional (4D) discrete economic system with constant and variable order are more complex. In this research, we use the Caputo delta difference derivative to investigate the nonlinear dynamic interaction between the parameters and fractional orders. The structure of this paper is as follows: the Caputo delta difference derivative is shown in Section 2. A mathematical model of a discrete economic system under constant order is presented in Section 3. Section 4 examines the dynamic analysis of a 4D chaotic economic model with variable-order derivatives. In Section 5, two control strategies of the system are investigated. The main discussion and conclusion are finally presented in Section 6.

2. Fractional Discrete Operators

A non-integer generalization of an operator’s order can be identified as discrete fractional calculus. Its application is expanding across various fields, including statistics, control theory, informatics, diffusion, and viscoelasticity. Difference operators take on a variety of forms, including Riemann–Liouville, Grunwald–Letnikov, and Caputo-left delta operators. This study focuses on a Caputo-left difference operator in a fractional 4D discrete economic system. The definition of the ${}^C\Delta _t^{\alpha }$-Caputo-left difference operator of the function $D\left(x\right):{\mathbb{N}}_t\to \mathbb{R}$ by ${}^C\Delta _t^{\alpha }D\left(x\right)$, with ${\mathbb{N}}_t=\{t,t+1,t+2,\dots \}$, and $t\in \mathbb{R}$ fixed, is displayed as follows [35]:

$${}^C\Delta _t^{\alpha }D\left(x\right)={\Delta }_t^{-(\mathfrak{l}-\alpha )}{\Delta }^{\mathfrak{l}}D\left(x\right)={\displaystyle \frac{1}{\Gamma (\mathfrak{l}-\alpha )}}\sum _{v=t}^{x-(\mathfrak{l}-\alpha )}{(x-v-1)}^{(\mathfrak{l}-1-t)}{\Delta }^{\mathfrak{l}}D\left(v\right),$$

(1)

where $\mathfrak{l}=\lfloor \alpha \rfloor +1$, $x\in {\mathbb{N}}_{t+\mathfrak{l}-\alpha }$, and $\alpha \notin \mathbb{N}$.

The fractional sum ${\alpha }^{th}$ of $D\left(x\right)$ can be written as [36]

$${\Delta }_t^{-\alpha }D\left(x\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\sum _{v=t}^{x-\alpha }{(x-1-v)}^{(\alpha -1)}D\left(v\right),$$

(2)

with $x\in {\mathbb{N}}_{t+\alpha }$ and $\alpha >0$.

The falling function $y^{\left(\alpha \right)}$ is as follows:

$$x^{\left(\alpha \right)}={\displaystyle \frac{\Gamma (x+1)}{\Gamma (x-\alpha +1)}}.$$

(3)

Also,

$${\Delta }^{\mathfrak{l}}D\left(x\right)=\Delta \left({\Delta }^{\mathfrak{l}-1}D\left(x\right)\right)=\sum _{\mathfrak{n}=0}^{\mathfrak{l}}\left({\displaystyle \genfrac{}{}{0pt}{}{\mathfrak{l}}{\mathfrak{n}}}\right){(-1)}^{\mathfrak{l}-\mathfrak{n}}D(x+\mathfrak{n}),\mathfrak{n}\in {\mathbb{N}}_t.$$

(4)

The following theorem now allows us to determine the numerical expression for the fractional 4D discrete economic system.

Theorem 1

([37]). The delta fractional difference formula is as follows:

$$\left\{\begin{array}{c}{}^C\Delta _t^{\alpha }D\left(x\right)=\mathfrak{F}(x+\alpha -1,D(x+\alpha -1)),\hfill \\ {\Delta }^{j}D\left(t\right)=D_j,\mathfrak{l}=\lfloor \alpha \rfloor +1,\hfill \end{array}\right.$$

(5)

The related discrete integral equation is as follows:

$$D\left(x\right)=D_0\left(t\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\sum _{\mathfrak{e}=t-1-\alpha }^{x-\alpha }{(x-1-\mathfrak{e})}^{(\alpha -1)}\mathfrak{F}(\mathfrak{e}+\alpha -1,D(\mathfrak{e}+\alpha -1)),x\in {\mathbb{N}}_{t+\mathfrak{l}},$$

(6)

where

$$D_0\left(t\right)=\sum _{\mathfrak{e}=0}^{\mathfrak{l}-1}{\displaystyle \frac{{(x-t)}^{\mathfrak{e}}}{\Gamma (\mathfrak{e}+1)}}{\Delta }^{\mathfrak{e}}D\left(t\right).$$

(7)

The stability of discrete fractional systems with constant and variable fractional orders, respectively, can be determined by the following theorems:

Theorem 2

([38]). For the discrete system with a constant commensurate fractional order represented by ${}^C\Delta _t^{\alpha }D\left(x\right)=\mathfrak{Q}\left(\mathfrak{Y}\left(v-1+\alpha \right)\right)$, if all ${\lambda }_j$, $j=\overline{1,m}$ eigenvalues of $J_{\mathfrak{r}}$ are as follows:

$${\lambda }_j\in \left\{\mathfrak{h}\in \mathbb{C}:\left|\mathfrak{h}\right|<{\left(2\cos {\displaystyle \frac{\left|arg\mathfrak{h}\right|-\pi }{2-\alpha }}\right)}^{\alpha }\mathit{\text{and}}\left|arg\mathfrak{h}\right|>{\displaystyle \frac{\alpha \pi }{2}}\right\}.$$

(8)

then the system has a fixed point that is asymptotically stable, where $\mathfrak{Y}\left(v\right)={\left({\mathfrak{Y}}_1\left(v\right),{\mathfrak{Y}}_2\left(v\right),\dots ,{\mathfrak{Y}}_m\left(v\right)\right)}^T$, and $J_{\mathfrak{r}}\left(\mathfrak{Y}\right)={\left.{\displaystyle \frac{\partial D\left(x\right)}{\partial \mathfrak{Y}}}\right|}_{\mathfrak{Y}={\mathfrak{Y}}_{\mathfrak{p}}}$ is the Jacobian matrix at ${\mathfrak{Y}}_{\mathfrak{p}}$.

Theorem 3

([39]). The stability of a variable fractional discrete system is therefore determined if any eigenvalues ${\lambda }_j$ of $J\mathfrak{r}$ satisfy the following:

$$\mathit{d}\left({\lambda }_j,\mathbb{C}\setminus {\mathcal{S}}^{\alpha \left(0\right)}\right)>\eta .$$

(9)

Thus, the variable fractional system is locally asymptotically stable, where

$$\eta :=\max \left\{\left(1-{\displaystyle \frac{{\alpha }_{\min }}{{\alpha }_{\max }}}+{\alpha }_{\min }-\alpha \left(0\right)\right),\left({\displaystyle \frac{{\alpha }_{\max }}{{\alpha }_{\min }}}-{\alpha }_{\max }-1+\alpha \left(0\right)\right)\right\},$$

(10)

$${\alpha }_{\min }=\inf \alpha \left(v\right),{\alpha }_{\max }=\sup \alpha \left(v\right),$$

(11)

which

$${\mathcal{S}}^{\alpha \left(0\right)}=\left\{\mathfrak{h}\in \mathbb{C}:\left|\mathfrak{h}\right|<{\left(2\cos {\displaystyle \frac{\left|\arg \mathfrak{h}\right|-\pi }{2-\alpha \left(0\right)}}\right)}^{\alpha \left(0\right)}\mathit{\text{and}}\left|\arg \mathfrak{h}\right|>{\displaystyle \frac{\alpha \left(0\right)\pi }{2}}\right\}.$$

(12)

and

$$\mathit{d}\left({\lambda }_j,\mathbb{C}\setminus {\mathcal{S}}^{\alpha \left(0\right)}\right)=\inf \left\{|{\lambda }_j-\mathfrak{h}|,\mathfrak{h}\in \mathbb{C}\setminus {\mathcal{S}}^{\alpha \left(0\right)}\right\}.$$

(13)

3. The 4D Fractional Discrete Economic System with Constant Fractional Order

A continuous-time hyper-chaotic 4D economic system with fractional order was proposed in [40]. In this work, the 4D economic system can be discretized as follows:

$$\left\{\begin{array}{c}{y}_1(v+1)=y_1\left(v\right)+y_3\left(v\right)+y_1\left(v\right)(y_2\left(v\right)-\delta )+y_4\left(v\right),\hfill \\ y_2(v+1)=y_2\left(v\right)+1-\gamma y_2\left(v\right)-y_1^2\left(v\right),\hfill \\ y_3(v+1)=y_3\left(v\right)-y_1\left(v\right)-\mu y_3\left(v\right),\hfill \\ y_4(v+1)=y_4\left(v\right)-\sigma y_1\left(v\right)y_2\left(v\right)-\kappa y_4\left(v\right).\hfill \end{array}\right.$$

(14)

where the state variables $y_1$ denote the aggregate investment rate, $y_2$ represents the consumer demand or output index, $y_3$ corresponds to the price index, and $y_4$ describes the government expenditure feedback. The discrete iteration step is indicated by v, and the real parameters $\delta ,\gamma ,\mu ,\sigma ,$ and $\beta$ were selected based on established ranges from previous research on chaotic economic systems [40], ensuring comparability with the existing literature. These values represent typical ranges where economic systems exhibit complex dynamics, including chaos and bifurcations, while maintaining mathematical tractability for analysis. Using the 0-1 test [41], Figure 1 displays the translation components $p-q$. A Brownian-like (unbounded) trajectory was discovered, suggesting the presence of a chaotic attractor.

The alternative version of system (14) can be described using the usual forward difference operator, as follows: $\Delta y_n\left(v\right)=y_n(v+1)-y_n\left(v\right)$

$$\left\{\begin{array}{c}\Delta y_1\left(v\right)=y_3\left(v\right)+y_1\left(v\right)(y_2\left(v\right)-\delta )+y_4\left(v\right),\hfill \\ \Delta y_2\left(v\right)=1-\gamma y_2\left(v\right)-y_1^2\left(v\right),\hfill \\ \Delta y_3\left(v\right)=-y_1\left(v\right)-\mu y_3\left(v\right),\hfill \\ \Delta y_4\left(v\right)=-\sigma y_1\left(v\right)y_2\left(v\right)-\kappa y_4\left(v\right).\hfill \end{array}\right.$$

(15)

Once v is changed to $v-1+\alpha$ and $\Delta$ is changed to the Caputo-like operator ${}^C{\Delta }_t^{\alpha }$, the resulting system is a fractional-order difference system:

$$\left\{\begin{array}{c}{}^C{\Delta }_t^{\alpha }{y}_1\left(v\right)=y_3\left(\mathcal{W}\right)+y_1\left(\mathcal{W}\right)(y_2\left(\mathcal{W}\right)-\delta )+y_4\left(\mathcal{W}\right),\hfill \\ {}^C{\Delta }_t^{\alpha }{y}_2\left(v\right)=1-\gamma y_2\left(\mathcal{W}\right)-y_1^2\left(\mathcal{W}\right),\hfill \\ {}^C{\Delta }_t^{\alpha }{y}_3\left(v\right)=-y_1\left(\mathcal{W}\right)-\mu y_3\left(\mathcal{W}\right),\hfill \\ {}^C{\Delta }_t^{\alpha }{y}_4\left(v\right)=-\sigma y_1\left(\mathcal{W}\right)y_2\left(\mathcal{W}\right)-\kappa y_4\left(\mathcal{W}\right).\hfill \end{array}\right.$$

(16)

where $\mathcal{W}=v-1+\alpha$$v\in {\mathbb{N}}_{t-\alpha +1}$, $\alpha \in (0,1]$ and t is the starting point. Our main goal now is to investigate the dynamic features of the 4D fractional discrete economic system (16) using phase space orbits, the 0-1 test, time series, bifurcation charts, and the Lyapunov exponent [42], while taking into account the effects of fractional constant order and $\sigma$ system parameters. The 4D fractional discrete economic system (16) with constant order is numerically described in our simulations using Matlab R2025a, based on Theorem 1, which can be derived using the following:

$$\left\{\begin{array}{c}{\displaystyle y_1\left(\mathfrak{e}\right)=y_1\left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\sum _{j=0}^{\mathfrak{e}}{\displaystyle \frac{\Gamma (\mathfrak{e}-j+\alpha )}{\Gamma (\mathfrak{e}-j+1)}}\left\{y_3\left(j\right)+y_1\left(j\right)(y_2\left(j\right)-\delta )+y_4\left(j\right)\right\},}\hfill \\ {\displaystyle y_2\left(\mathfrak{e}\right)=y_2\left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\sum _{j=0}^{\mathfrak{e}}{\displaystyle \frac{\Gamma (\mathfrak{e}-j+\alpha )}{\Gamma (\mathfrak{e}-j+1)}}\left\{1-\gamma y_2\left(j\right)-y_1^2\left(j\right)\right\},}\hfill \\ {\displaystyle y_3\left(\mathfrak{e}\right)=y_3\left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\sum _{j=0}^{\mathfrak{e}}{\displaystyle \frac{\Gamma (\mathfrak{e}-j+\alpha )}{\Gamma (\mathfrak{e}-j+1)}}\left\{-y_1\left(j\right)-\mu y_3\left(j\right)\right\},}\hfill \\ {\displaystyle y_4\left(\mathfrak{e}\right)=y_4\left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\sum _{j=0}^{\mathfrak{e}}{\displaystyle \frac{\Gamma (\mathfrak{e}-j+\alpha )}{\Gamma (\mathfrak{e}-j+1)}}\left\{-\sigma y_1\left(j\right)y_2\left(j\right)-\kappa y_4\left(j\right)\right\}.}\hfill \end{array}\right.$$

(17)

for instance, when selecting parameters $\delta =0.09,\gamma =1.5,\mu =1.5,\sigma =0.5$ and $\kappa =0.1$ and initial values $y_1\left(0\right)=1.5,y_2\left(0\right)=1,y_3\left(0\right)=0.5,y_4\left(0\right)=0.5$ (I.C). The discrete time development of the states $y_1$, $y_2$, $y_3$, and $y_4$ in the proposed constant fractional-order system is shown in Figure 2 to provide an extensive overview of these features. For all numerical simulations, the discrete iteration length was set to e = 3000 steps, with the initial 1000 iterations discarded as transients to ensure convergence with the attractor. It is evident to us that the trajectories are irregular and unpredictable. They exhibit erratic patterns, which are a defining characteristic of chaotic behavior, in which slight variations in the original setting result in wildly disparate movements. In Figure 3a,c, the stable closed invariant curve is shown. The extended closed invariant curve is depicted in Figure 1. A chaotic attractor is seen at $\alpha =0.95$ in Figure 3e, confirming that the closed invariant curve is impacted by the constant fractional order $\alpha$. As previously mentioned, the system parameters are maintained, and the constant fractional order $\alpha$ is adjusted within the range $[0.7,1]$ in order to further observe the dynamical behavior. We will provide more detail by discussing the chaos of the 4D fractional discrete economic system (16) for $\sigma =0.5$, where $\alpha$ varies from 0.7 to 1 by the step size $\Delta \alpha =0.8\times {10}^{-4}$. Both the highest Lyapunov exponent $LLE$ diagram and the bifurcation diagram of $y_1$ vs $\alpha$ are displayed in Figure 4. As illustrated in this Figure, there is a clear correspondence between the bifurcation diagram and the Largest Lyapunov Exponent (LLE) curve. The negative values of the LLE ($\mathrm{LLE}<0$) indicate the stable region where the trajectories converge toward a steady equilibrium. The narrow intervals where the LLE approaches zero ($\mathrm{LLE}\approx 0$) represent periodic or quasi-periodic oscillations, while the positive values ($\mathrm{LLE}>0$) identify the chaotic regime characterized by irregular and unpredictable fluctuations. This correspondence highlights the transitions between stability, periodicity, and chaos in the fractional-order economic system. It shows that as $\alpha$ decreases, the extensive memory system’s values change. The 4D fractional discrete economic system (16) loses stability through Neimark–Sacker bifurcation whenever we increase the value of $\alpha$. Furthermore, we find that the chaotic zone disappears as the fractional-order value decreases. At $\alpha \in [0.95,0.973]\cup [1.99,1]$, the system (16) behaves chaotically. Figure 5 illustrates the bifurcation charts pertaining to parameter $\sigma$, since $\delta =0.09,\gamma =1.5,\mu =1.5,$ and $\kappa =0.1$. Figure 6a–c represent the bifurcation diagrams for $\alpha =0.86$, $\alpha =0.9$, and $\alpha =0.95$, accordingly. The three diagrams as a whole represent Neimark–Sacker bifurcations, with ${\epsilon }_1$ varying among 0 and $0.5$. The plot of the largest Lyapunov exponent in Figure 6 validates the bifurcation diagram’s shape, as shown in Figure 5. Figure 7 plots the outcome of $C_0$ complexity [43] and $SE$ [44]. As can be seen, the fluctuation in $\sigma$ and $\kappa$ correlated with the complexity of this model. Furthermore, in order to obtain a somewhat high structural complexity, we need to know the fractional order that was chosen. Therefore, to obtain a relatively high structural complexity, one should be careful when choosing the values of $\sigma$, $\kappa$, and fractional orders in system (16). These results are consistent with those of earlier analyses. The choice of constant fractional orders in modelling and influencing the dynamics of the 4D discrete economic system (16) is crucial, since this highlights the system’s complexity and diversity.

4. The 4D Fractional Discrete Economic System with Variable Fractional Order

In order to highlight the importance of fractional processes that depend on time in describing system behavior and complexity, numerical simulations were perfumed to investigate the interplay between parameters $\sigma$ and variable fractional order $\alpha \left(v\right)$. We will investigate the computation of the largest Lyapunov exponents (LLE), bifurcation charts, phase space orbits, 0-1 test, $C_0$ complexity, $SE$, and the evolution states of the system. Therefore, the previously defined Caputo-like difference operator ${}^C\Delta _t^{\alpha }$ is converted into its fractional variable-order version ${}^C\Delta _t^{\alpha \left(v\right)}$. The variable-order fractional 4D discrete economic system is effectively formulated as

$$\left\{\begin{array}{c}{}^C{\Delta }_t^{\alpha \left(v\right)}{y}_1\left(v\right)=y_3\left(\mathcal{U}\right)+y_1\left(\mathcal{U}\right)(y_2\left(\mathcal{U}\right)-\delta )+y_4\left(\mathcal{U}\right),\hfill \\ {}^C{\Delta }_t^{\alpha \left(v\right)}{y}_2\left(v\right)=1-\gamma y_2\left(\mathcal{U}\right)-y_1^2\left(\mathcal{U}\right),\hfill \\ {}^C{\Delta }_t^{\alpha \left(v\right)}{y}_3\left(v\right)=-y_1\left(\mathcal{U}\right)-\mu y_3\left(\mathcal{U}\right),\hfill \\ {}^C{\Delta }_t^{\alpha \left(v\right)}{y}_4\left(v\right)=-\sigma y_1\left(\mathcal{U}\right)y_2\left(\mathcal{U}\right)-\kappa y_4\left(\mathcal{U}\right).\hfill \end{array}\right.$$

(18)

where $\mathcal{U}=v-1+\alpha \left(v\right)$, $\alpha \left(v\right)$ is the fractional variable-order such that $\alpha \left(v\right)\in [0,1]$. The numerical formula for the system’s solution, (18), is required in order to discuss how the variable fractional order influences the dynamical behavior of the 4D discrete economic system. The numerical scheme in Theorem 1 can be extended to variable-order systems by treating the fractional order as a function of the discrete time index. In this case, the summation weights and gamma function terms become time-dependent, leading to the numerical formulation presented as follows [45]:

$$\left\{\begin{array}{cc}\hfill y_1\left(\mathfrak{e}\right)& {\displaystyle =y_1\left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right(j-1\left)\right)}}\sum _{j=1}^{\mathfrak{e}}{\displaystyle \frac{\Gamma (\mathfrak{e}-j+\alpha (j-1\left)\right)}{\Gamma (\mathfrak{e}-j+1)}}\{y_3(j-1)+y_1(j-1)(y_2(j-1)-\delta )+y_4(j-1)\},}\hfill \\ \hfill y_2\left(\mathfrak{e}\right)& {\displaystyle =y_2\left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right(j-1\left)\right)}}\sum _{j=1}^{\mathfrak{e}}{\displaystyle \frac{\Gamma (\mathfrak{e}-j+\alpha (j-1\left)\right)}{\Gamma (\mathfrak{e}-j+1)}}\left\{1-\gamma y_2(j-1)-y_1^2(j-1)\right\},}\hfill \\ \hfill y_3\left(\mathfrak{e}\right)& {\displaystyle =y_3\left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right(j-1\left)\right)}}\sum _{j=1}^{\mathfrak{e}}{\displaystyle \frac{\Gamma (\mathfrak{e}-j+\alpha (j-1\left)\right)}{\Gamma (\mathfrak{e}-j+1)}}\left\{-y_1(j-1)-\mu y_3(j-1)\right\},}\hfill \\ \hfill y_4\left(\mathfrak{e}\right)& {\displaystyle =y_4\left(0\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right(j-1\left)\right)}}\sum _{j=1}^{\mathfrak{e}}{\displaystyle \frac{\Gamma (\mathfrak{e}-j+\alpha (j-1\left)\right)}{\Gamma (\mathfrak{e}-j+1)}}\left\{-\sigma y_1(j-1)y_2(j-1)-\kappa y_4(j-1)\right\}.}\hfill \end{array}\right.$$

(19)

The intention of this part is to establish that, under variable fractional orders, the 4D discrete economic system displays rich dynamics. Examining the system’s dynamic features, particularly the impact of the bifurcation parameter and fractional variable-order functions, was our primary objective. For example, with starting values $y_1\left(0\right)=1.5, y_2\left(0\right)=1,y_3\left(0\right)=0.5,y_4\left(0\right)=0.5$ (I.C) and assuming $\delta =0.09,\gamma =1.5,\mu =1.5,\sigma =0.5$ and $\kappa =0.1$, we investigated the following three variable fractional orders:

$$\alpha \left(v\right)=0.94+0.05\sin \left(\frac{v}{5}\right),\alpha \left(v\right)=0.97+0.03\cos \left(\frac{v}{2}\right),\alpha \left(v\right)=0.9+0.1\sin \left(\frac{v}{2}\right).$$

Firstly, the evolution of states $y_1$, $y_2$, $y_3$, and $y_4$ was independent of time for the variable fractional $\alpha \left(v\right)=0.97+0.03\cos \left(\frac{v}{2}\right)$ order, as displayed in Figure 8. Furthermore, the analysis shows that the 4D discrete economic with variable fractional order has Brownian-like dynamics and chaotic attractors, as shown in Figure 9 for $\sigma =0.5$. Such figures are noteworthy because they show that even a slight alteration in the bifurcation parameter $\sigma$ may impact the behavior. Notably, we observed more chaotic behavior for the variable fractional function $\alpha \left(v\right)=0.97+0.03\cos \left(\frac{v}{2}\right)$ than for the other two functions, as indicated by the evolution of states in Figure 8. Further, these chaotic attractors have a different structure from those found in the context of a constant fractional order. Taking $\sigma$ as the bifurcation parameter, we examine the dynamics of the variable fractional system (18) and create bifurcation diagrams to comprehend system behavior. Specifically, $\delta =0.09,\gamma =1.5,\mu =1.5,$ $\kappa =0.1$ and I.C were set. For three variable fractional-order functions, bifurcation charts versus $\sigma$ were calculated: $\alpha \left(v\right)=0.94+0.05\sin \left(\frac{v}{5}\right),\alpha \left(v\right)=0.97+0.03\cos \left(\frac{v}{2}\right),\alpha \left(v\right)=0.9+0.1\sin \left(\frac{v}{2}\right)$, as shown in Figure 10. Notably, $\sigma$ ranges from 0 to $0.5$ in Figure 10a, while $\sigma$ ranges from 0 to $0.4$ in Figure 10b,c. We can see that the system exhibits a periodic condition at first. Through Neimark–Sacker bifurcation, it eventually descends into chaos. Next, in relation to the largest Lyapunov exponents $LLE$ in Figure 11, the system is similar to the bifurcation pattern of the variable fractional system (18) seen in Figure 10. It is well known that when a system’s $LLE$ is positive, it behaves chaotically, reflecting sensitivity to complicated dynamics and starting conditions. On the other hand, the system is in a periodic state if $LLE$ is negative. Thus, in all three situations, the time-varying nature of variable fractional orders results in more severe transitory behaviors than their constant counterparts. In Figure 12, we apply the previously used numerical techniques in addition to other analytical tools like $SE$ and $C_0$ complexity. These additional tools offer a more thorough evaluation of the chaotic structure of the system (18). Adding variable fractional-order dynamics to the 4D discrete economic system greatly enhances its behavior. Using various fractional operators (Caputo-left difference via constant order versus Caputo-left difference via variable order) produces different dynamic behaviors, as evidenced by this comparison of the two scenarios. The economic system with variable fractional order is a novel addition to studies on chaotic discrete systems with variable fractional-order dynamics.

5. Stabilization of the 4D Fractional Discrete Economic System

In this section, the capacity to stabilize chaotic economic systems (16), (18) is given special consideration. In order to move the chaotic systems states asymptotically towards zero, we refer to the control with the addition of adaptively updated terms. It should be noted that, in the context of this economic model, driving the state variables toward zero does not imply a shutdown of economic activity. Instead, the zero state corresponds to a normalized equilibrium point representing a steady balance among investment, demand, inflation feedback, and fiscal response. Hence, the control schemes are designed to eliminate chaotic fluctuations and restore this equilibrium condition.

5.1. Constant Fractional-Order Case

Here, we present the proposed constant fractional 4D discrete economic system’s control method (16). The presented control strategy dynamically modifies the system parameters according to the current state in order to reduce the chaotic behavior in the system. Employing the characteristics of constant fractional-order dynamics, the control rule is made to be both responsive and adaptive to modifications in the behavior of the system. In order to stabilize the constant fractional 4D discrete economic system, the next theorem presents a set of control laws.

Theorem 4.

The constant fractional 4D discrete economic system’s control method (16) can be asymptotically stabilized to the zero equilibrium under the proposed control law

$$\left\{\begin{array}{c}{\mathcal{C}}_1\left(\mathcal{W}\right)=-y_1\left(\mathcal{W}\right)y_2\left(\mathcal{W}\right)-y_1\left(\mathcal{W}\right),\hfill \\ {\mathcal{C}}_2\left(\mathcal{W}\right)=y_1^2\left(\mathcal{W}\right)-y_2\left(\mathcal{W}\right)-1,\hfill \\ {\mathcal{C}}_3\left(\mathcal{W}\right)=-y_3\left(\mathcal{W}\right),\hfill \\ {\mathcal{C}}_4\left(\mathcal{W}\right)=-y_4\left(\mathcal{W}\right)+\sigma y_1\left(\mathcal{W}\right)y_2\left(\mathcal{W}\right).\hfill \end{array}\right.$$

(20)

Proof. 

The following is a description of the controller system:

$$\left\{\begin{array}{c}{}^C{\Delta }_t^{\alpha }{y}_1\left(v\right)=y_3\left(\mathcal{W}\right)+y_1\left(\mathcal{W}\right)(y_2\left(\mathcal{W}\right)-\delta )+y_4\left(\mathcal{W}\right)+{\mathcal{C}}_1\left(\mathcal{W}\right),\hfill \\ {}^C{\Delta }_t^{\alpha }{y}_2\left(v\right)=1-\gamma y_2\left(\mathcal{W}\right)-y_1^2\left(\mathcal{W}\right)+{\mathcal{C}}_2\left(\mathcal{W}\right),\hfill \\ {}^C{\Delta }_t^{\alpha }{y}_3\left(v\right)=-y_1\left(\mathcal{W}\right)-\mu y_3\left(\mathcal{W}\right)+{\mathcal{C}}_3\left(\mathcal{W}\right),\hfill \\ {}^C{\Delta }_t^{\alpha }{y}_4\left(v\right)=-\sigma y_1\left(\mathcal{W}\right)y_2\left(\mathcal{W}\right)-\kappa y_4\left(\mathcal{W}\right)+{\mathcal{C}}_4\left(\mathcal{W}\right).\hfill \end{array}\right.$$

(21)

Thus, the below system was obtained by replacing the control term ${\mathcal{C}}_i\left(\mathcal{W}\right),i=1,2,3,4$ in (21), as explained in (20):

$$\left\{\begin{array}{c}{}^C{\Delta }_t^{\alpha }{y}_1\left(v\right)=y_3\left(\mathcal{W}\right)-\delta y_1\left(\mathcal{W}\right)+y_4\left(\mathcal{W}\right)-y_1\left(\mathcal{W}\right),\hfill \\ {}^C{\Delta }_t^{\alpha }{y}_2\left(v\right)=-\gamma y_2\left(\mathcal{W}\right)-y_2\left(\mathcal{W}\right),\hfill \\ {}^C{\Delta }_t^{\alpha }{y}_3\left(v\right)=-y_1\left(\mathcal{W}\right)-\mu y_3\left(\mathcal{W}\right)-y_3\left(\mathcal{W}\right),\hfill \\ {}^C{\Delta }_t^{\alpha }{y}_4\left(v\right)=-\kappa y_4\left(\mathcal{W}\right)-y_4\left(\mathcal{W}\right).\hfill \end{array}\right.$$

(22)

This is expressed in the following manner:

$${}^C{\Delta }_t^{\alpha }\left(\begin{array}{c}{y}_1\left(v\right)\\ y_2\left(v\right)\\ y_3\left(v\right)\\ y_4\left(v\right)\end{array}\right)=\mathfrak{Q}\left(\begin{array}{c}{y}_1\left(v\right)\\ y_2\left(v\right)\\ y_3\left(v\right)\\ y_4\left(v\right)\end{array}\right),$$

(23)

where

$$\mathfrak{Q}=\left(\begin{array}{cccc}-(1+\delta )& 0& 1& 1\\ 0& -(1+\gamma )& 0& 0\\ -1& 0& -(1+\mu )& 0\\ 0& 0& 0& -(1+\kappa )\end{array}\right).$$

(24)

It is clear that the eigenvalues of $\mathfrak{Q}$ fulfill

$$|\arg {\lambda }_j|>{\displaystyle \frac{\alpha \pi }{2}}\mathrm{and}\left|{\lambda }_j\right|<{\left(2\cos \left({\displaystyle \frac{|\arg {\lambda }_j|-\pi }{2-\alpha }}\right)\right)}^{\alpha },j=1,2,3,4.$$

(25)

The zero solution of system (22) is asymptotically stable, as per Theorem 2. As a result, the proposed control law fully stabilizes the system (16). □

Through numerical simulations, we evaluate the outcomes of Theorem 4. Figure 13 shows the time series of the 4D discrete economic system (16) with controlled constant fractional order. The system’s states asymptotically approach zero, as the figure makes clear, verifying its successful stabilization.

5.2. Variable Fractional-Order Case

In the following part, by making use of the stabilization strategy for variable-order fractional systems, our goal is to develop a strong control rule that propels all states of the fractional 4D discrete economic system (18) with variable fractional order in the direction of asymptotic stability at the origin.

Accordingly, the controlled system is as follows:

$$\left\{\begin{array}{c}{}^C{\Delta }_t^{\alpha \left(v\right)}{y}_1\left(v\right)=y_3\left(\mathcal{U}\right)+y_1\left(\mathcal{U}\right)(y_2\left(\mathcal{U}\right)-\delta )+y_4\left(\mathcal{U}\right)+{\mathcal{T}}_1\left(\mathcal{U}\right),\hfill \\ {}^C{\Delta }_t^{\alpha \left(v\right)}{y}_2\left(v\right)=1-\gamma y_2\left(\mathcal{U}\right)-y_1^2\left(\mathcal{U}\right)+{\mathcal{T}}_2\left(\mathcal{U}\right),\hfill \\ {}^C{\Delta }_t^{\alpha \left(v\right)}{y}_3\left(v\right)=-y_1\left(\mathcal{U}\right)-\mu y_3\left(\mathcal{U}\right)+{\mathcal{T}}_3\left(\mathcal{U}\right),\hfill \\ {}^C{\Delta }_t^{\alpha \left(v\right)}{y}_4\left(v\right)=-\sigma y_1\left(\mathcal{U}\right)y_2\left(\mathcal{U}\right)-\kappa y_4\left(\mathcal{U}\right)+{\mathcal{T}}_4\left(\mathcal{U}\right).\hfill \end{array}\right.$$

(26)

Theorem 5.

The variable fractional 4D discrete economic system’s control method (18) can be asymptotically stabilized to the zero equilibrium under the proposed control law

$$\left\{\begin{array}{c}{\mathcal{T}}_1\left(\mathcal{U}\right)=-y_1\left(\mathcal{U}\right)y_2\left(\mathcal{U}\right)-y_1\left(\mathcal{U}\right),\hfill \\ {\mathcal{T}}_2\left(\mathcal{U}\right)=y_1^2\left(\mathcal{U}\right)-y_2\left(\mathcal{U}\right)-1,\hfill \\ {\mathcal{T}}_3\left(\mathcal{U}\right)=-y_3\left(\mathcal{U}\right),\hfill \\ {\mathcal{T}}_4\left(\mathcal{U}\right)=-y_4\left(\mathcal{U}\right)+\sigma y_1\left(\mathcal{U}\right)y_2\left(\mathcal{U}\right).\hfill \end{array}\right.$$

(27)

Proof. 

Replacing (27) the control term ${\mathcal{T}}_i\left(\mathcal{U}\right),i=1,2,3,4$ in (26):

$$\left\{\begin{array}{c}{}^C{\Delta }_t^{\alpha \left(v\right)}{y}_1\left(v\right)=y_3\left(\mathcal{U}\right)-\delta y_1\left(\mathcal{U}\right)+y_4\left(\mathcal{U}\right)-y_1\left(\mathcal{U}\right),\hfill \\ {}^C{\Delta }_t^{\alpha \left(v\right)}{y}_2\left(v\right)=-\gamma y_2\left(\mathcal{U}\right)-y_2\left(\mathcal{U}\right),\hfill \\ {}^C{\Delta }_t^{\alpha \left(v\right)}{y}_3\left(v\right)=-y_1\left(\mathcal{U}\right)-\mu y_3\left(\mathcal{U}\right)-y_3\left(\mathcal{U}\right),\hfill \\ {}^C{\Delta }_t^{\alpha \left(v\right)}{y}_4\left(v\right)=-\kappa y_4\left(\mathcal{U}\right)-y_4\left(\mathcal{U}\right).\hfill \end{array}\right.$$

(28)

□

The values of ${\lambda }_1=0.91$, ${\lambda }_2={\lambda }_3=-0.5$, ${\lambda }_4=0.9$, and $\alpha =0.9$ are provided. Applying MATLAB, we find that System (28)’s eigenvalues fulfill Theorem (3)’s necessities, validating the stability of the variable fractional 4D discrete economic system (18) in Figure 14.

6. Conclusions

In this article, the evolution of the fractional four-dimensional (4D) discrete economic system with constant- and variable-order derivatives’ chaotic and stable states is presented. Beyond conventional integer-order systems, these two kinds of fractional orders are broadening the investigation of system dynamics. Different numerical techniques were used to verify that the suggested fractional variants of the 4D discrete economic system are chaotic. These consisted of 0-1 test plots, Lyapunov exponents, time states, phase space orbit, bifurcation diagrams, spectral entropy, and $C_0$ complexity. The research validates the existence of chaotic attractors in the system across many fractional-order scenarios by employing these methods for comprehensive investigation. The 4D fractional discrete economic system’s states with constant- and variable-order derivatives were also guided to the origin within an established period of time using two distinct types of control techniques. In future research, we plan to apply the variable fractional order in the framework of other economic models in order to explore this subject further.