1.
Introduction
Chaos is widely applied to various fields of natural and social sciences, as well as forming new subjects or new research fields [1]. There are many applications of chaotic systems in finance and management such as business cycles, firm growth, investment, and chaotic behavior in foreign capital investment [2]. Until now, econophysics has been appointed as an alternative scientific methodology to understand the very complex dynamics of the economic and financial system [3]. In the economic field, financial risk is one of the most important issues that concerns the State and all economic entities [4]. In recent years, the chaotic state of the economy has received intensive attention and engineering applications have been made to understand the complex behavior of financial markets [5].
Control methods for nonlinear dynamical systems are important topics in chaos theory [6,7,8]. Much research attention has been given to system modelling and control in engineering and science [9,10,11]. Chaos phenomenon developed by creating irregular phenomena can be desirable for many types of control applications [12]. Chaotic systems have been studied using various techniques like passive control [13] adaptive control [14], fuzzy control [15,16], active control [17,18,19], sliding mode control [20,21,22], etc.
Sliding mode control (SMC) is attractive for nonlinear systems because of its insensitivity to parametric and nonparametric uncertainties [23,24,25]. SMC is a well-known method for chaotic system synchronization [26,27,28]. However, SMC invariance is not guaranteed in the attainment phase. Integral sliding mode control (ISMC) eliminates the gain phase in such a way that invariance is achieved in the entire system response. Hence, ISMC is a popular method for chaotic system control and synchronization [29,30].
There are several properties that identify the behavior of chaotic systems. Sensitivity to initial conditions is the most common trait of chaotic systems [31]. Any nonlinear dynamical system can have the ability to exhibit chaotic behavior if it is at least a 3-dimensional system for an autonomous system, and a 2-dimensional system for a non-autonomous system [32]. One way to identify a chaotic system is to use the method of Lyapunov exponents. A 3-D dynamical system is chaotic if it has a positive Lyapunov exponent. Also, a system is said to be dissipative if the sum of all Lyapunov exponents of the system is negative.
In the chaos literature, there is a significant interest in the modelling of financial chaotic systems. Gao and Ma [33] investigated the behavior of chaos and the existence of Hopf bifurcation in a financial chaotic system. Vaidyanathan et al. [34] proposed a new financial chaotic system and presented its circuit design. Tacha et al. [35] introduced a new financial chaotic system with dissaving and detailed its circuit simulation.
In this work, we propose a new finance chaotic system and study its dynamic behavior with help of bifurcation diagrams and Lyapunov exponents. We compare the new finance chaotic system with the three finance chaotic systems [33,34,35] and show that the proposed chaotic system has a great value of Maximum Lyapunov Exponent (MLE) and Kaplan-Yorke Dimension than the three finance chaotic systems [33,34,35].
We show that the proposed chaotic financial system has three unstable equilibrium points. We discuss its properties such as rotational symmetry, multi-stability, etc. A nonlinear chaotic system is called multistable if it has coexistent chaotic attractors for the same parameter values but different initial states [36]. The existence of this new financial chaos system is expected to predict chaos in the financial sector with variables of interest rates, investment demand, price exponents, household savings, investment costs, and the elasticity of demand for the commercial market. Chaotic systems have applications in many engineering fields such as cryptosystems [37,38] and secure communication devices [39,40].
This work is organized as follows. In Section 2, we present a dynamic analysis of the new chaotic system. In Section 3, we obtain new results for complete synchronization of the new chaotic system via adaptive integral sliding mode control. In Section 4, we draw the main conclusions.
2.
A new finance chaotic system
Gao & Ma [33] reported 3-D dynamics with a financial risk model provided by
In the 3-D financial system (1), x1 represents the interest rate, x2 represents the level of investment demand and x3 represents the price exponential. Also, the constant a stands for the household savings rate and b represents the cost of investment.
We assume a and b are positive. Gao and Ma [33] showed that the finance system (1) is chaotic when a=6 and b=0.1.
Using MATLAB, the Lyapunov exponent (LE) values of the Gao-Ma finance system (1) are obtained for (a,b)=(6,0.1) and X(0)=(0.5,3,−0.4) as follows:
This calculation shows that the Gao-Ma financial system (1) is chaotic with the maximal Lyapunov exponent (MLE) value as LE1=0.0909.
Also, the Kaplan-Yorke dimension of the Gao-Ma financial chaotic system (1) is calculated as follows:
Vaidyanathan et al. [34] proposed a new financial system by replacing the quadratic nonlinearity x21 with the quartic nonlinearity x41 in the second differential equation of the Gao-Ma financial chaotic system (1).
Vaidyanathan et al. [34] reported 3-D dynamics with a financial risk model provided by
Vaidyanathan et al. [34] showed that the finance system (4) is chaotic when a=7.5 and b=0.1.
Using MATLAB, the Lyapunov exponent (LE) values of the Vaidyanathan finance system (4) are obtained for (a,b)=(7.5,0.1) and X(0)=(0.5,3,−0.4) as follows:
This calculation shows that the Vaidyanathan financial system (3) is chaotic with the maximal Lyapunov exponent (MLE) value as LE1=0.1285.
Also, the Kaplan-Yorke dimension of the Vaidyanathan financial chaotic system (3) is calculated as follows:
Tacha et al. [35] proposed a new financial system with dissaving by replacing the term −ax1 with ax1 in the first differential equation of the Gao-Ma financial chaotic system (1) and also by replacing the quadratic nonlinearity x21 with the absolute function nonlinearity |x1| in the second differential equation of the Gao-Ma financial chaotic system (1).
Tacha et al. [35] reported 3-D dynamics with a financial risk model with dissaving provided by
Tacha et al. [34] showed that the finance system (7) is chaotic when a=1 and b=0.2.
Using MATLAB, the Lyapunov exponent (LE) values of the Tacha finance system (7) are obtained for (a,b)=(1,0.2) and X(0)=(0.5,3,−0.4) as follows:
This calculation shows that the Tacha financial system (7) is chaotic with the maximal Lyapunov exponent (MLE) value as LE1=0.0673.
Also, the Kaplan-Yorke dimension of the Tacha financial chaotic system (7) is calculated as follows:
In this work, we propose a new financial chaos system by retaining the quadratic nonlinearity x21 but adding the absolute function nonlinearity |x1| in the second differential equation of the Gao-Ma chaotic system (1).
We propose a new 3-D dynamics with a financial risk model provided by
We shall show that the new finance system (10) is chaotic when a=1 and b=0.04.
Using MATLAB, the Lyapunov exponent (LE) values of the new finance system (10) are obtained for (a,b)=(1,0.04) and X(0)=(0.5,3,−0.4) as follows:
This calculation shows that the new financial system (10) is chaotic with the maximal Lyapunov exponent (MLE) value as LE1=0.1862.
Also, the Kaplan-Yorke dimension of the new financial chaotic system (10) is calculated as follows:
Table 1 shows a comparison of the Gao-Ma finance system [33], Vaidyanathan finance system [34], Tacha finance system [35] and the new finance chaotic system (10) proposed in this research work. From Table 1, it is very clear that the new finance chaotic system (10) is more chaotic than the Gao-Ma finance system [33], Vaidyanathan finance system [34] and Tacha finance system [35].
We obtain the equilibrium point of the new finance chaotic system (10) by solving the following equation for the chaotic case of the parameters (a,b)=(1,0.04):
From Eq (13c), it is clear that
Using (14), we can simplify the Eqs (13a) and (13b) as follows:
We have two cases to consider: (A) x1=0 and (B) x1≠0.
In Case (A), we suppose that x1=0. Since x3=−x1, it is immediate that x3=0. From Eq (15b), we get 1−0.04x2=0 or x2=25. Thus, E0=(0,25,0) is a balance point of the new finance system (10).
In Case (B), we suppose that x1≠0. From Eq (15a), we get x2=2.
Thus, Eq (15b) can be simplified as follows:
Solving Eq (16), we get two real roots x1=±0.5816653820.
Since x3=−x1, we deduce that x3=∓0.5816653820.
Thus, in case (B), we have two balance points given as follows: E1=(0.5816,2,−0.5816) and E2=(−0.5816,2,0.5816) of the new finanace chaotic system (10).
The Jacobian matrix of the new financial chaos system (10) at E0=(0,25,0) is written as follows:
The spectral value of W0 is determined using MATLAB as follows
This establishes that E0 is a saddle point and an unstable balance point for the new finance chaotic system (10).
The Jacobian matrix of the new financial chaos system (10) at E1=(0.5816,2,−0.5816) is shown as follows:
The spectral value of W1 was determined using MATLAB as follows
This indicates that E1 is a saddle focus and an unstable equilibrium point for a chaotic financial system (10).
The Jacobian matrix of the new financial chaos system (10) at E2=(−0.5816,2,0.5816) is determined
The spectral value of W2 was determined using MATLAB as follows
This establishes that E1 is a saddle-focus and an unstable balance point for the new finance chaotic system (10).
The phase plot of the new financial chaotic system (10) for X(0)=(0.5,3,−0.4) and (a,b)=(1,0.04) in the (x1,x2)− plane is shown in Figure 1. Plane phase orbit in Figure 1 depicts the high complexity of the new financial chaotic system (10) in the (x1,x2)− plane.
The phase plot of the new financial chaotic system (10) for X(0)=(0.5,3,−0.4) and (a,b)=(1,0.04) in the (x2,x3)− plane is shown in Figure 2. Plane phase orbit in Figure 2 depicts the high complexity of the new financial chaotic system (10) in the (x2,x3)− plane.
The phase plot of the new financial chaotic system (10) for X(0)=(0.5,3,−0.4) and (a,b)=(1,0.04) in the (x1,x3)− plane is shown in Figure 3. Plane phase orbit in Figure 3 depicts the high complexity of the new financial chaotic system (10) in the (x1,x3)− plane.
Figure 4 shows a Lyapunov exponents diagram for the new financial chaotic system (10) when a takes values in the interval [0,18] and b=0.04. Figure 5 shows a bifurcation diagram of the new financial system (10) showing a graph of X3max versus a for b=0.04. These results show that the new finance system (10) has a chaotic attractor for a∈[0,15] and a periodic orbit for a>15.
Figure 6 shows a Lyapunov exponents diagram for the new financial chaotic system (10) when b takes values in the interval [0,0.8] and a=1. Figure 7 shows a bifurcation diagram of the new financial system (10) showing a graph of X3max versus b for a=1. These results show that the new finance system (10) has a chaotic attractor for b∈[0,0.25] and a periodic orbit for b>0.25.
Figure 8 shows the multistability of the financial chaotic system (10) where the blue color trajectory starts with the initial state X0=(0.5,3,−0.4), and the red color trajectory starts with the initial state Y0=(−1,3,1), and the parameter values are kept fixed as (a,b)=(1,0.04).
3.
Complete synchronization of the new finance chaotic system via integral sliding mode control
In this section, we achieve global chaos synchronization of the new finance chaotic system (10) with itself via feedback controllers designed with integral sliding mode control (ISMC).
The drive system is taken as the new finance chaotic system:
The response system is taken as the new finance chaotic system with sliding mode controls:
We denote the states of the systems (23) and (24) as X=(x1,x2,x3) and Y=(y1,y2,y3) respectively. In (24), v1,v2,v3 are the integral sliding mode controls (ISMC) to be determined.
First, we define the synchronization error between the finance systems (23) and (24) as:
It is easy to see that
In the IMSC design, we associate an integral sliding surface with every synchronization error variable as follows:
Differentiating the surface equations given in (27), we get:
We suppose that μ1,μ2,μ3 are positive constants, and consider the integral sliding mode controls as follows:
In Eq (29), αi,ki,(i=1,2,3) are positive constants.
Substituting (29) into the error Eq (26), we obtain:
Next, we shall establish the main synchronization result of this section.
Theorem 1. The integral sliding mode control law (29) renders the drive financial system (23) and response system (24) synchronized globally for all values of initial states X(0),Y(0)∈R3, where it is assumed that μi,αi,ki,(i=1,2,3) are positive constants.
Proof. We start the proof by taking the Lyapunov function defined by
It is easy to see that V is a strictly positive definite function defined on R3.
Next, we find that
A simple calculation gives
Clearly, ˙V is a negative definite function since αi,ki>0 for i=1,2,3.
Using Lyapunov stability theory, we deduce that Si→0,(i=1,2,3) as t→∞.
Hence, it follows that εi→0,(i=1.2.3) as t→∞. This completes the proof.
For MATLAB simulations, the constants are taken as in the chaos case, viz. a=1 and b=0.04.
We assume the sliding constants as αi=0.2,ki=8 and μi=8 for i=1,2,3.
The initial state of the drive system (23) is taken as X(0)=(1.4,2.9,8.3) and the initial state of the response system (24) as Y(0)=(2.7,10.4,1.5).
Figure 9 shows the complete synchronization between the drive system (23) and response system (24) as the synchronization errors converge to zero in 1.5 sec. The fast convergence of the synchronization error shows the efficiency of the integral sliding mode control (ISMC) effect. Chaos synchronization has been carried out by using integral sliding mode control to synchronize the new financial chaotic system, taken as the master-slave systems. ISMC is a very useful technique for nonlinear control systems for both parametric and nonparametric uncertainties.
4.
Conclusions
In this paper, we proposed a new 3-D financial chaotic system and detailed about the novelty of the new financial system by comparing it with three financial chaotic systems in the literature, viz. Gao-Ma system [33], Vaidyanathan system [34] and Tacha system [35]. We showed that the proposed chaotic system is more chaotic than the three financial chaotic systems [33,34,35] by considering the maximal Lyapunov exponent (MLE) and the Kaplan-Yorke dimension of systems. We carried out a bifurcation analysis of the proposed financial system using bifurcation diagrams and Lyapunov exponents. Using integral sliding mode control and Lyapunov stability theory, we derived new results for global chaos synchronization for a pair of new financial chaotic systems taken as the drive-response systems. We illustrated all the main results of this study using MATLAB simulation plots.
Acknowledgments
We would like to thank Universitas Padjadjaran, which has provided the Universitas Padjadjaran Internal Research Grant, Fiscal Year 2022, the "ACADEMIC LEADERSHIP GRANT (ALG)" program under Prof. Dr. Sukono, MM., M.Si., with contract Number: 2203/UN6.3.1/PT.00/2022.
Conflict of interest
We declare that we have no conflict of interest.