Abstract

A new four-dimensional hyperchaotic financial model is introduced. The novelties come from the fractional-order derivative and the use of the quadric function in modeling accurately the financial market. The existence and uniqueness of its solutions have been investigated to justify the physical adequacy of the model and the numerical scheme proposed in the resolution. We offer a numerical scheme of the new four-dimensional fractional hyperchaotic financial model. We have used the Caputo–Liouville fractional derivative. The problems addressed in this paper have much importance to approach the interest rate, the investment demand, the price exponent, and the average profit margin. The validation of the chaotic, hyperchaotic, and periodic behaviors of the proposed model, the bifurcation diagrams, the Lyapunov exponents, and the stability analysis has been analyzed in detail. The proposed numerical scheme for the hyperchaotic financial model is destined to help the agents decide in the financial market. The solutions of the 4D fractional hyperchaotic financial model have been analyzed, interpreted theoretically, and represented graphically in different contexts. The present paper is mathematical modeling and is a new tool in economics and finance. We also confirm, as announced in the literature, there exist hyperchaotic systems in the fractional context, which admit one positive Lyapunov exponent.

1. Introduction

Many behaviors of the dynamical systems are deterministic. The systems’ future behaviors follow the same evolutions and are explained by the initial conditions and the past of the systems. Chaos theory is one of the mathematical domain which studies these types of dynamical systems and has received many investigations [1, 2]. Lorenz [2] was the first author to propose the chaotic system in three-dimensional space, namely, chaotic attractor. Lorenz’s work can probably be considered as the beginning of this discipline. It is well known the chaotic systems are nonlinear dynamical systems and are sensitive to their initial conditions. That is, when the initial conditions of the considered system have small differences or changes, it becomes complicated to predict the behaviors of the system [1]. This field of mathematics is strongly in relation to the control theory. This reason explains the many investigations related to chaos control. In general, the controllers try to eliminate the chaotic behaviors using synchronization methods or other techniques. Many studies also focus on the stability analysis of the chaotic systems [1, 3]. Chaotic behaviors are observed in many real-world problems, in fluid flows, in weather and climate [2], in the stock market, in road traffics, and others. Chaos theory has many applications, too, in anthropology, computer science, economic [1], biology, physics [1], meteorology, and others. After Lorenz’s proposition, many other types of chaotic systems appear in the literature. Recent investigations focus on the chaotic and hyperchaotic systems in economics; we have the chaotic financial system with three-dimensional space (see in [4, 5]); we have the four-dimensional hyperchaotic financial model.

Chaotic and hyperchaotic systems have many applications in finance and economics. There exist many nonlinear dynamical systems in finance markets that use chaotic systems to predict the markets’ behavior. In [1], Xin et al. presented the 3D chaotic financial model and introduced a new four-dimensional fractional chaotic financial system. The numerical investigation to approximate the financial model’s solutions and the stability analysis have been proposed too in this paper. In [6], Chen et al. investigated the 4D hyperchaotic financial model using the Lyapunov direct method; the authors have found suitable control to stabilize the considered financial model. They also give the numerical simulation of their results. In [4], Gao and Ma investigated the complex dynamical behaviors of a finance system as the chaos and Hopf bifurcation. In [7], Yu et al. proposed a novel four-dimensional chaotic financial model based on the classical chaotic financial model with three dimension. The authors in [7] added the average profit margin to the classical financial model with three dimension to obtain a new chaotic model. They also presented the new model, the stability analysis, and provided the numerical simulation of their new model. In [8], Kumar et al. constructed a new finance model too, namely, the four-dimensional chaotic financial model and used the Lyapunov direct method to study the stability of the equilibrium points and also proposed the numerical simulations of the new model. For more investigations, like the works proposed by He in [9, 10], Yichen and others authors in [11, 12], Pham in [13, 14], and Shirkavand in [15], see also in [3, 5].

As previously observed, the stability and numerical simulations are the main interests in the three- and four-dimensional chaotic financial models. They are many methods for solving the fractional differential equations as the homotopy analysis, the homotopy perturbation, the domain decomposition, the numerical schemes as Adams–Bashforth numerical discretization, and others. In many contexts, the stability and the convergence of the previous cited methods are not trivial, and the effectiveness of the method can be discussed. For example, with homotopy methods, what is the good number of iterations to be considered under which we have good approximations for the solutions of the model. For the numerical schemes, including the numerical schemes of the fractional operators, the implementations of the methods are not easy, and the unconditional stability does not include the Lipschitz continuous of the functions constituting the model. Lipschitz continuous is known as indispensable for the existence of the solutions for a particular model. Many inconveniences exist in these previous cited methods. In this paper, we introduce a new hyperchaotic financial model; we propose a new procedure for getting the solutions of the four-dimensional hyperchaotic financial model by using implicit numerical discretization. Before investigating the solutions of the considered model, we propose qualitative properties. It is to justify the physical adequacy of the financial model. Our motivations are to introduce a new hyperchaotic financial model and introduce the fractional-order derivative [16–18] in modeling the financial equations. The fractional-order derivative well describes the memory effect. Thus, our second motivation is the use of the Caputo derivative into mathematical modeling. The main novelty in our study is the introduction of a numerical scheme to obtain the phases portraits of the considered system, which include the discretization of the Riemann–Liouville fractional integral and use the Lipschitz continuous conditions for the stability and the convergence of the numerical scheme. In other words, the existence of the model’s solutions is sufficient for the stability and convergence of the used numerical scheme. It is essential to mention in our numerical discretization that we do not use the discretization of the fractional derivative but we have used the discretization of the fractional integral, which is more useful and uses the analytical solutions discreetly. Our novelty in terms of the financial model introduces the quadric function to measure the variations of the interest rate. We manipulate very sensitive chaos systems. Thus, the introduction of this new function in the model will generate many changes in the stability of the equilibrium points and influence the chaos generated by the variation of the model’s parameters. The significant impact generated by the quadric function will be focused on this present paper. The bifurcation and the Lyapunov exponents in terms of the fractional derivative are also among the main novelties of this paper. This paper shows how to use the bifurcation and the Lyapunov exponents in the fractional context to analyze the chaos theory. In terms of the characterization of the chaos, we will provide the existence of two positive Lyapunov exponents that are sufficient for the hyperchaotic behaviors in the integer version, but in the context of fractional-order derivative, the numbers of positive Lyapunov exponents are not an adequate definition to characterize the hyperchaotic dynamics because there exist systems which are hyperchaotic with the bifurcation diagrams but admits one positive Lyapunov exponent. Therefore, fractional calculus must find theory on Lyapunov exponents to characterize chaotic behaviors and hyperchaotic behaviors. Our investigation will confirm the work on the Lyapunov exponent proposed by Danca in [19] who found hyperchaotic systems with one positive Lyapunov exponent. In the literature of the fractional calculus, the financial models, or chaotic systems are addressed in the papers [20–23]. For applications of the fractional derivatives in real-world problems, see in [17, 18, 24–31, 42–44].

In Section 2, we introduce the definitions and the tools necessary for our investigations. In Section 3, we present the 4D hyperchaotic financial model in the context of the fractional-order derivative and consider the memory effect. In Section 4, we focus on the qualitative properties of the proposed model. In Section 5, we describe the procedure of the solutions for the model. In Section 6, we represent the solutions graphically in different contexts. In Section 7, we analyze the characterizations of the chaos using the bifurcation diagrams and the Lyapunov exponents. In Section 8, we provide stability analysis in the context of fractional calculus. In Section 9, we give the future directions of research and final remarks.

2. Basic Definitions and Lemmas

This section is consecrated to the definitions of fractional derivatives and integrals. We will utilize them in our investigations. We work with fractional derivatives with singular kernels. In other words, we use the Caputo–Liouville fractional derivative, the Riemann–Liouville fractional derivative, and their associated generalizations.It is not hard to see when the order , we recover the classical integral. It proves the Liouville–Riemann integral is a generalization of the classical integral to the arbitrary noninteger order. Its associated derivative is called the Riemann–Liouville fractional derivative.

Definition 1. (see [35, 36]). The Liouville–Riemann fractional integral is described as the following form for the function :where the function represents the Gamma Euler function and with the order .

Definition 2. (see [35, 36]). The Liouville–Riemann fractional derivative of order is described as the following form for the function :where the function represents the Gamma Euler function.

Another fractional derivative proposed in the literature is called the Caputo–Liouville fractional derivative due to the inconvenience of the Riemann–Liouville derivative.

Definition 3. (see [35, 36]). The Caputo fractional derivative of order is described as the following form for the function :where the function represents the Gamma Euler function.

The generalization of two above fractional derivatives and integrals is represented in the following lines.

Definition 4. (see [28, 37–39]). The generalized Riemann–Liouville integral is described as the following form for the function :with the orders , satisfying the relation , , the function gamma is , and for all .

Definition 5. (see [28, 37–39]). The generalized Liouville–Riemann fractional derivative of order is described as the following form for the function :with the orders , satisfying the relation , , the function gamma is , and for all .

Definition 6. (see [26, 28, 40, 41]). The generalized Caputo fractional derivative of order is described as the following form for the function :with the orders , satisfying the relation , , the function gamma is , and for all .

We continue the rest of the paper with the Caputo–Liouville fractional derivatives. That is, we recall the Laplace transform of the Caputo–Liouville derivative; we have the relationwith the order respecting the condition .

3. Fractional Four-Dimensional Hyperchaotic Financial Model

In this section, we present the fractional model considered in our investigations. Yu, Cai, and Li proposed the integer-order version of the 4D hyperchaotic financial model in [7]. The fractional version of their model is described with the Caputo–Liouville derivative by the following equations:

We make the following assumptions related to the initial conditions:

In equations (8)–(11), the variable represents the interest rate, the variable represents the investment demand, the variable represents the price exponent, and the variable is an additional variable that represents the average profit margin. The parameter denotes the saving rate, the parameter represents the per investment cost, the parameter indicates the elasticity of demands, and the parameter is the average profit margin influence.

In our new modeling, we suppose, due to the external phenomena, the fluctuations of the interest rate into the financial market cannot be measured adequately by the quadratic function (in equation (9)) all times. Therefore, we consider more accurately quadric function , which can correct all the errors in the measurement of the real data. Thus, the new four-dimensional hyperchaotic financial model can be modeled as the following form:

The strange actuator is obtained with the following values:

Our motivations for the use of the fractional-order derivative are to extend the four-dimensional hyperchaotic financial model described by integer-order derivative to the fractional-order derivative. First, the fractional-order derivative takes into account the memory effect; that is, the past behaviors of the model explain the next behaviors of the model. The second is to adapt the answer given to Leibniz’s question to the financial model. Note that, it is proved in the literature that we can calculate the derivative of the function when is noninteger. In the literature, all the models, including the derivatives, are described by the integer-order derivatives; the question is now what will happen with these models when the order of the derivative is noninteger. It is also proved in the literature the fractional-order derivative is more realistic in modeling physical and economics models. For example, many diffusion processes as the subdiffusion, the ballistic diffusion process, the superdiffusion process, and the superdiffusion process which exist in real-world problems cannot be obtained with the integer derivative but with fractional-order derivatives. All these reasons have motivated us in this present works.

Note that the 3D financial chaotic model proposed in the literature is given by the following equation:

We can observe the average profit margin is added in the initial model equations (18)–(20) to obtain the hyperchaotic financial model. Note that the average profit margin captures more perfectly the behaviors in the market [7] because it depends on the interest rate and the investment demand. We will analyze in detail the impact of the interest rate and the investment demand in the average profit margin. It is not hard to observe the hyperchaotic financial model is another representation of the chaotic financial equation because the average profit margin impacts the interest rate. In addition, the price exponent and the average profit margin are independent indirectly. The first objective of this paper is to prove the physical adequacy of the fractional model by establishing the solution of the fractional model described by equations (13)–(16) exist and is unique. The Banach fixed theorem will be used.

4. Existence and Uniqueness of the Model

In this section, we prove the fractional equation defined by equations (13)–(16) has at least one solution. The technique of proof uses the Banach fixed theorem procedure. This section is important for proving the physical adequacy of the fractional differential equations. In mathematical views, it is not important to study a model when the solution does not exist. The previous reasons are the motivations of this section.

We consider the first differential equation (13), and we suppose the function defined by

The function needs to be Lipschitz continuous. We have the following procedure of demonstration:

Under the assumption, the state variable is bounded, that is, , and we get the condition of the Lipschitz continuous given bywith the Lipschitz constant expressed as the form . The second step of the application of the Banach fixed theorem consists of constructing Picard’s operator. It is clear that the solution of the fractional differential equation represented by equation (13) is given by the following expression:

Based on the form of the solution, we define the following Picard’s operator:

Before using this expression, it is important to prove the operator is well-bounded. We have the following procedure:From the assumption that the function is Lipschitz continuous, there exists such that and equation (26) becomeswith the condition . That is, the operator is well-bounded. In other words, it is well definite, and now we should prove the operator is also a contraction. We have the following reasoning:

Using the Lipschitz continuous condition established in equation (23), we have the following relationship:That is, the operator is a contraction when the following condition is held:

We conclude that the solution of the fractional differential equation represented in equation (13) exists after application of the Banach fixed point theorem. After existence, we will try to show the uniqueness of the solution. We first consider two different solutions expressed as and for our considered equation (13). We have in particular the relations given by

We evaluate the difference between equation (31) and equation (32), and we obtain the relation defined by

After the application of the Euclidean norm to equation (33), we obtain the equations

We have the following relation after calculations:from which we get that . By definition, Euclidean norm satisfies the condition . We conclude that after combining the previous relationships, the following result:

The unicity of the solution of equation (13) of our hyperchaotic model follows from equation (33). And then, we get the existence and the uniqueness of the solution of the first equation (13).

We repeat the previous reasoning by utilizing the second fractional differential equation (14), and we suppose a new function defined by

The function needs to be Lipschitz continuous. We have the following procedure of demonstration:

We have the Lipschitzian continuous condition given by the following condition:with the Lipschitz constant expressed as the form . The second step of the application of the Banach fixed theorem consists of constructing Picard’s operator. It is clear that the solution of the fractional differential equation represented by equation (14) is given by the following expression:

Based on the form of the solution, we define the following Picard’s operator:

Before using this expression, it is important to prove the operator is well-bounded. We have the following procedure:

Form the assumption the function is Lipschitz continuous, there exists such that , and equation (42) becomeswith the condition . That is, the operator is well-bounded. In other words, it is well definite, and now we should prove the operator is also a contraction. We have the following reasoning:

Using the Lipschitz continuous condition established in equation (38), we have the following relationship:

That is, the operator is a contraction when the following condition is held:

We conclude that the solution of the fractional differential equation represented in equation (14) exists after application of the Banach fixed point theorem. After existence, we will try to show the uniqueness of the solution. We first consider two different solutions expressed as and for our considered equation (14). We have in particular the relations given by

We evaluate the difference between equation (47) and equation (48), and we obtain the relation defined by

After the application of the Euclidean norm to equation (49), we have the equations

We obtain the following relation after calculations:from which we get that . By definition, Euclidean norm satisfies the condition . We conclude that, after combining the previous relationships, the following relationship:

The unicity of the solution of equation (14) of our hyperchaotic model follows from equation (52). And then, we get the existence and the uniqueness of the solution of the second equation in our model.

We continue with the third fractional differential equation (15), and we utilize the following function:

The function needs to be Lipschitz continuous. We have the following procedure of the proof:

We have the Lipschitzian continuous condition given by the following condition:with the Lipschitz constant expressed as the form . The second step of the application of the Banach theorem consists of constructing Picard’s operator. It is clear that the solution of the fractional differential equation represented by equation (15) is given by the following expression:

Based on the form of the solution, we define the following Picard’s operator:

Before using this expression, it is important to prove the operator is well-bounded. We have the following procedure:

From the assumption the function is Lipschitz continuous, there exists such that and equation (58) becomeswith the condition . That is, the operator is well-bounded. In other words, it is well definite, and now we should prove the operator is also a contraction. We have the following reasoning:

Using the Lipschitz continuous condition established in equation (55), we have the following relationship:That is, the operator is a contraction when the following condition is held:

We conclude that the solution of the fractional differential equation represented in equation (15) exists after application of the Banach fixed point theorem. Now, we will try to show the uniqueness of the solution. We first consider two different solutions expressed as and for our considered equation (15). We have in particular the relations given by

We evaluate the difference between equation (63) and equation (64), and we obtain the relation defined by

After the application of the Euclidean norm to equation (65), we obtain the following equations:

We obtain the following relation after calculations:from which we get that . By definition, Euclidean norm satisfies the condition . We conclude that after combining the previous relationships,

The unicity of the solution of equation (15) of our hyperchaotic model follows from equation (68). And then, we get the existence and the uniqueness of the solution of the third equation in our considered model.

We finish with the fourth fractional differential equation (16), and we use the following function:

The function needs to be Lipschitz continuous. We have the following procedure of the proof:

We have the Lipschitz continuous condition given by the following condition:with the Lipschitz constant expressed as the form . The second step of the application of the Banach fixed theorem consists of constructing Picard’s operator. It is clear that the solution of the fractional differential equation represented by equation (16) is given by the following expression: