Chaos in a Financial System with Fractional Order and Its Control via Sliding Mode

In this paper, the dynamical behaviors and chaos control of a fractional-order ﬁnancial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.13 for the incommensurate order case. Also, period-doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional-order ﬁnancial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional-order systems. Numerical simulations are carried out to verify the analytical results.


Introduction
Investigating chaos in dynamical systems is one of the most interesting topics which has been carried out extensively in different scientific fields such as medicine [1], biology [2], mathematics [3], and many others. In the literature, several dynamical systems presenting chaotic behaviors have been proposed such as the Lorenz system [4], the Chen system [5], the Lü system [6], and the Newton-Leipnik system [7]. During the last decades, many researchers have taken a great interest in the subject of chaotic systems' control. In [8], the authors used the sliding mode controller to eliminate chaos in a new uncertain chaotic dynamical system (Liu system). In [9], a robust adaptive sliding mode controller is used to remove chaos in a novel class of chaotic systems. Chaos control in the Lorenz, Chen, and Lü systems using the backstepping technique is performed in [10]. Chaos control in the Newton-Leipnik system is carried out using linear feedback controllers in [11].
Fractional calculus involving fractional order derivatives, i.e., derivatives of noninteger order, has a history of over 300 years [12]. During those years, this theory was considered a purely mathematical concept. Recently, fractional order derivatives have been widely applied to several systems in many areas of research to better understand these systems [13][14][15][16][17][18][19]. Indeed, fractional order derivatives, possessing memory, can describe more accurately different nonlinear phenomena than integer order derivatives [20,21]. Very recently, some interesting developments and results in the theory and applications of fractional calculus have been obtained in the literature. For example, see [22], in which authors used Lie symmetry analysis to obtain an exact solution of the conformable heat equation. In [23], useful properties of the Lie group method with the invariance subspace method are combined to obtain a large family of exact solutions for the fractional Black-Scholes equation. Also, in [24], the necessary optimality conditions of Euler-Lagrange type of variational problems in which variational functional depends on the Atangana-Baleanu derivative are proved. Finally, the Hydon method to determine discrete symmetries for a differential equation is employed to construct discrete symmetries for a family of ordinary, partial, and fractional differential equations in [25]. In fractional order systems, it was found that the systems with derivation orders q 1 , q 2 , and q 3 generate chaotic behaviors when q � q 1 + q 2 + q 3 < 3 [13][14][15][16][17][18][19]; in other words, chaos continues to exist in these systems for derivation orders less than 3.
Since the discovery by Strotz et al. [26] of chaos in an economical model, various financial and economical models have been proposed in the literature to better understand this complex dynamic of these systems. Among others, we have the forced Van der Pol model [27], the IS-LM model (Investment Saving-Liquidity Money) [28], the new hyperchaotic finance model [29], and many others [30][31][32][33]. In 2001, Ma and Chen [34] proposed a very interesting model to represent the dynamics of financial systems. e analysis of this model revealed interesting dynamics and also an extreme sensitivity to the initial conditions of the variables and parameters of the system. Financial variables such as the exchange rate, gross domestic product, interest rate, and production, to name a few, have a long memory [35,36], i.e., all future fluctuations of these financial variables are influenced by past and present fluctuations. us, fractional order derivatives, possessing memory effect, can describe more accurately the dynamics of financial systems than integer order derivatives. In 2008, Chen [14] studied the generalization of the Ma and Chen system [34], i.e., considered this system with fractional order. Two routes to chaos were found in this fractional order system, namely, the route to chaos via intermittency and period doubling. e lowest derivation order found for which the system generates chaotic behavior was 2.55 in the commensurate fractional-order case and 2.35 in the incommensurate fractional-order case.
Also, to better understand the dynamics of financial systems, a new model was presented by Liao et al. [37] in 2020. Compared to the model proposed by Ma and Chen [34], this new model takes into account the fact that the price index is affected by investment demand. e study of this model was carried out numerically by the authors. It was found that the interaction between three factors in this financial system results in a complex behavior of the system. Complex dynamics behaviors such as period doubling and chaos were found in this system [37]. Chaotic behavior is undesirable in financial systems because it makes predictions in finance and economics impossible and, therefore, constitutes a risk for investments. en, its control in the presence of uncertainties related to the parameters of the system and external disturbances turned out to be necessary.
Chaos control in financial systems has been widely studied in the literature. Several techniques have been designed for the control of financial chaotic systems [38,39].
One of these techniques is the sliding mode control. Indeed, as a mathematical model cannot represent a physical situation perfectly, it is necessary to consider the uncertainties linked to the values of the system parameters and also any external disturbances to which a model may be subjected. erefore, it is better to design a robust controller, i.e., insensitive to uncertainties and external disturbances. e sliding mode control is a powerful technique to robustly control uncertain dynamical systems subject to uncertainties and external disturbances [40][41][42]. e design of the sliding mode controller has been widely discussed in the literature [43][44][45][46]. ese different criteria make the sliding mode control the right technique for controlling financial systems with uncertain dynamics. For example, in [38], a fractionalorder sliding mode controller was designed to eliminate the chaotic behavior in an economical system in the presence of model uncertainties and external disturbances.
Motivated by the above discussions, in this paper, chaos in the financial system presented by Liao et al. [37] with fractional order and robust control of this chaotic behavior are investigated.
e study of chaos in this system with fractional order is carried out for the commensurate and the incommensurate fractional order to find the minimum effective dimension, i.e., the lowest sum of derivation orders from which chaos arise in the system using analytical methods and numerical simulations. Finally, a sliding mode control law is designed to control the chaos in this fractionalorder financial system with or without uncertainties and external disturbances. Numerical simulations are carried out to show that the controller can suppress chaos in the system and also can stabilize and maintain the system states on the sliding surface. e remaining part of this paper is structured as follows. In Section 2, some definitions and analytical condition for the existence of chaos in fractional order systems are given. e fractional-order financial system is presented in Section 3.
e dynamics study of this fractional-order financial system is carried out in Section 4. In Section 5, a simple but robust fractional-order sliding mode controller is designed to globally and asymptotically stabilize the system. Numerical simulations are performed in Section 6 to verify the analytical results obtained. is work ends with a conclusion in Section 7.

Definitions and Lemma
Fractional calculus is a generalization of ordinary calculus. Some definitions of fractional derivatives are given in [12]. e most commonly used definitions in the literature are the Grunwald-Letnikov, the Riemann-Liouville, and the Caputo definitions [12].
During this work, we will only use the Caputo fractional derivative because unlike the Riemann-Liouville definition of the fractional derivative which involves initial conditions of the fractional order, this fractional derivative involves initial conditions which take the same form as the case of the integer order, which has a physical interpretation.
e Caputo (C) fractional derivative of order q is given by with a and t which are numbers representing the limits of the operator C a D q t . e symbol Γ(·) is the gamma function.
Definition 1 (see [47][48][49]). A saddle equilibrium point is a fixed point at which the equivalent linearized model has at least one eigenvalue in the stable region (eigenvalue with the negative real part) and one in the unstable region (eigenvalue with the positive real part).
Definition 2 (see [47][48][49][50]). In a three-dimensional (3D) dynamical system, a saddle fixed point is called a saddle equilibrium point of index 1 if one of its eigenvalues is unstable (positive) and the other two are stable (negative), whereas a saddle fixed point is called a saddle equilibrium point of index 2 if two of its eigenvalues are unstable and the other is stable.
To analyze the stability of a fractional-order system fixed points, the following lemma can be used [47,48,51].
Lemma 1 (see [47]). For an incommensurate fractionalorder system, i.e., a fractional order system in which derivation orders are not the same, an equilibrium point E * of the system is asymptotically stable if the following condition is satisfied for all roots λ of the following equation: in which J is the Jacobian matrix of the system evaluated at the equilibrium point E * and M represents the Least Common Condition (2) can be rewritten as follows: So, an equilibrium point E will be asymptotically stable if its roots λ i satisfy condition (4). e term (π/2M) − min i |arg(λ i )| is called the Instability Measure for equilibrium points in Fractional Order Systems (IMFOS). is measure is a necessary [47], but not a sufficient condition for the presence of chaos in a fractional order system [52][53][54].

The Chaotic Financial System with Fractional Order
Recently, as reported in the literature [37], the financial model takes into account the interaction between the interest rate x, the investment demand y, and the price index z. e system is described as follows: in which the parameters e, k, c, m, l, ρ, and δ are constants. e authors numerically investigated the chaotic behaviors in this system. In [37], system (5) exhibits chaotic behavior when , and initial conditions (1.2, 1.5, 1.6) are considered (see Figure 1).
In this paper, we consider system (5) with the fractional order. Standard derivatives are replaced by fractional order derivatives as follows: where q i ∈ (0, 1) and (6) is called a commensurate fractional-order system; otherwise, it is called an incommensurate order system [47]. e Jacobian matrix of system (6) evaluated at one of its equilibrium points E * � (x * , y * , z * ) is given by When the values of the system parameters are chosen as above, the equilibrium points of system (6) can be calculated by solving the equations D q 1 x � 0, D q 2 y � 0, and D q 3 z � 0. e system has four equilibrium points which are given by   Table 1.
Taking into account Definition 2 and from Table 1, it can be seen that fixed points E 1 and E 2 are saddle equilibrium points of index 1 and the others are saddle equilibrium points of index 2.
Using the Adams-Bashforth-Moulton predictorcorrector method proposed by Diethelm et al. [55], the numerical solution of system (6) can be written as follows: Complexity  4 Complexity

Dynamics Analysis of the Financial System with Fractional Order
In this section, the numerical method proposed by Diethelm et al. [55] and presented in the previous section is used to solve numerically system (6) in the commensurate and incommensurate fractional-order cases. e parameters' values defined in Section 3 and initial conditions (x 0 , y 0 , z 0 ) � (1.2, 1.5, 1.6) will be considered in this part. Using the well-known tools for studying dynamical systems such as phase diagrams, time series, bifurcation diagram, and largest Lyapunov exponent, the dynamics of the financial system with the fractional order will be investigated.

Dynamics for Commensurate Fractional-Order System.
Here, we consider system (6) when q 1 � q 2 � q 3 � q (commensurate order). System (6) does not exhibit chaotic behavior if it satisfies the inequality q < (2/π)min i |arg(λ i )| [47,48,51] with λ i being the eigenvalue of the Jacobian matrix of system (6) evaluated at one of its fixed points. For the equilibrium points E 3 and E 4 , we have min i |arg(λ i )| � 1.2819; so, Figure 2 shows that the largest Lyapunov exponent of system (6) with the commensurate fractional order is positive only if q > 0.82. erefore, the system does not show chaotic behavior when q < 0.82. It is found using numerical simulations that the system exhibits a chaotic attractor for q ≥ 0.83. For q � 0.83, phase diagrams are shown in Figure 3(a) for the x − y phase plane and in Figure 3(b) for the x − z phase plane. Figures 3(c) and 3(d) show the time series of state variables x and y, respectively, for q � 0.83. As it can be seen, the system exhibits chaotic behavior, and this is confirmed by a positive Lyapunov exponent for q � 0.83. erefore, the minimum effective dimension of system (6) for the commensurate fractional order is 0.83 × 3 � 2.49.
us, simulation results show that chaos exists in this fractional-order financial system with the derivation order less than 3. For numerical simulations, the step size h � 0.01 is used.

Dynamics for Incommensurate Fractional-Order System.
When one of the system's derivation orders has a different value from the other two, we get an incommensurate fractional-order system [47].
By solving equation (17), the IMFOS of the system is In this case, IMFOS > 0 but system (6) does not exhibit chaotic behavior (λ max < 0). is shows that the condition IMFOS ≥ 0 is a necessary condition for chaos to exist and not the sufficient one.
e system's IMFOS is In this case, the system exhibits a chaotic behavior as it can be seen in Figure 6, where we can observe the chaotic attractor of the system. erefore, the lowest value for which q 2 in this case generates chaotic behavior is 0.87, where λ max > 0. Case 3: fix q 1 � q 2 � 1, and let us take different values of q 3 .
When the value of q 3 increases from 0.85 to 0.90, the route to chaos via period doubling is found. e bifurcation diagram and largest Lyapunov exponent when the derivation order q 3 varies on the closed interval [0.85, 0.90] are plotted in Figure 8. Clearly, from the bifurcation diagram, the period-doubling route to chaos can be seen. Figures 9(a)-9(d) show that the system has period-1, period-2, period-4, and chaotic attractors for q 3 � 0.85, 0.87, 0.878, and 0.89, respectively. Note that the step size used for the numerical simulations in this section is h � 0.01.
From these three cases, we deduce that the minimum effective dimension of system (6) in the incommensurate fractional-order case is 2.13.
In the next section, a simple but robust fractional-order sliding mode control law will be designed to control chaos in system (6).

The Sliding Mode Controller Design
According to the sliding mode control theory, to design a sliding mode controller, we have two steps: (i) Build a sliding surface, which represents the desired dynamics of the system such as stability (ii) Develop a control law in a way that the system states are brought towards the sliding surface in a finite time and are maintained in a neighborhood of the sliding surface when time evolves [38] e sliding mode control law is itself composed of two parts; the first, which is continuous, is called the equivalent control law, and the second is discontinuous. e equivalent control law describes the behavior of the system to be controlled when its trajectories are on the sliding surface. e discontinuous reaching law ensures the convergence of all the system states towards the sliding surface.
To control chaos in fractional-order financial system (6), the controller u(t) is added at the level of the second state equation as follows: Motivated by literature [38,43], as a choice for the sliding surface, we choose the following fractional-order sliding surface: where η is an arbitrary positive constant. In the sliding mode, the invariance conditions of the surface must be satisfied and are defined as follows (i.e., the sliding surface and its derivative must satisfy the relations which appear in the following equation): From equations (26) and (27), we get D q 2 y(t) � − ky 2 (t) + lx 2 (t) + ηy(t).
From system (25) and equation (28), we obtain the equivalent control law as follows: Regarding the discontinuous reaching law, it is chosen as follows: in which and G r is the gain of the controller. Finally, the total control law has the following form: Theorem 1. System (6) with control law (32), i.e., system (25) is globally and asymptotically stable if the controller gain G r < 0.
Proof. For this, let us choose the Lyapunov quadratic function as follows: and its derivative gives erefore, we have found a Lyapunov function which satisfies the conditions of Lyapunov theorem, i.e., V > 0 and V ′ < 0.
Proof. For this, let us choose Lyapunov quadratic function (33); thus, we have us, the proof is achieved. fractional-order case, q 1 � q 2 � q 3 � q � 0.83 is chosen and for which the system is chaotic (see Figure 3). For the incommensurate fractionalorder case, q 1 � 1, q 2 � 0.88, and q 3 � 1 are chosen. For this choice, the largest Lyapunov exponent is positive (see Figure 4(b)). e corresponding phase diagrams and time series are shown in Figure 10. Note that the controller can be activated in the system at any time. In this part, it is activated at t � 20 sec.
Case 1: commensurate order without uncertainty and an external disturbance. As it was shown above, system (25) without the controller u(t) is chaotic for q 1 � q 2 � q 3 � q � 0.83. Now, applying controller (32) to the system, simulation results can be seen in Figure 11. From this figure, it is observed that control law (32) can effectively asymptotically stabilize the state variables of system (25) (see Figures 11(a)-11(c)). Besides, the time series of the sliding surface σ(t) is plotted in Figure 11(d). From this figure, it can be observed that the controller stabilizes the trajectories of the system on the sliding surface and maintains them on this surface when time evolves. Case 2: commensurate order with uncertainty and an external disturbance. In this part, we disturb the fractional-order financial system by an uncertainty defined by Δg(x, y, z) � 0.2 sin( ���������� x 2 + y 2 + z 2 ) and an external disturbance p(t) � 0.5 sin(2t), where |Δg(x, y, z)| ≤ μ 1 � 0.2 and |p(t)| ≤ μ 2 � 0.5. e time series of system (25) state variables (see Figures 12(a)-12(c)) and the time series of sliding surface (26) (see Figure 12(d)) in the presence of control law (32) can be seen through Figure 12. From this figure, we can see that the controller stabilizes the system in the presence of uncertainty and external disturbance.   Figure 11: e time series of the controlled commensurate fractional-order system state variables and corresponding time series of the sliding surface without uncertainty and external disturbance (the control input is activated at t � 20 sec).

Complexity
Case 3: incommensurate order with uncertainty and an external disturbance.
In this case , we disturb financial system (25) with the incommensurate fractional order by the same uncertainty and external disturbance as case 2.
e time series of system (25) state variables and the time series of the sliding surface in the presence of control law (32) are shown in Figure 13. From this figure, it can be observed that the state variables of the system are stabilized.
In each figure, the time series of the sliding surface is plotted to show the ability of the controller to bring back all the system states onto the sliding surface and to maintain them on this surface when time evolves. Simulation results show that controller (32) is able to stabilize systems (25) and (35) in commensurate and incommensurate fractional-order cases.
erefore, the designed control law can suppress chaos in this fractional-order financial system in the presence or not of uncertainty and external disturbance.

Conclusions
In this paper, the dynamics of a financial system with the fractional order as well as the robust chaos control in this system are studied analytically, and numerical simulations are performed to confirm the analytical results. e existence of chaos in this study is validated by a positive Lyapunov exponent and by an analytical condition existing in the  literature. e fractional order system exhibits rich dynamics behaviors such as periodic and chaotic behaviors. A perioddoubling route to chaos is found in this system. Numerical simulations revealed that chaos exists in this fractional order system for derivation orders less than 3. e lowest derivation order found to have chaos in the commensurate fractional-order case is 2.49 and 2.13 for the incommensurate fractional-order case. Regarding the robust control of chaos in the system, by using Lyapunov's stability theorem, a simple but robust fractional-order sliding mode control law has been designed to stabilize the chaotic trajectories of the fractional-order financial system in the presence or not of uncertainty and external disturbance. It should be noted that the controller has been applied only at the investment demand state equation to fully control the system. Numerical simulations show that this controller is effective and can control the financial system with commensurate and incommensurate fractional orders. In [14], Chen studied the fractional order version of the financial system proposed by Ma and Chen [34]. e lowest derivation order obtained in [14] for chaos to exist is 2.55 in the commensurate order case and 2.35 in the incommensurate order case. Compared to the fractional order version of Liao et al. [37] proposed in this paper, the emergence of chaos is enhanced in both cases. It is well known that time delay can affect the behavior of dynamical systems. For future works, chaotic dynamics analysis of this fractional-order financial system with time delay can be considered.

Data Availability
No data were used to support the findings of this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.