Chaotic Phenomena and Oscillations in Dynamical Behaviour of Financial System via Fractional Calculus

Abstract

In this research work, we formulate the phenomena of the financial system in the fractional framework to describe the complex nature of finance. The basic definitions and ideas of the Caputo-Fabrizio fractional operator are listed. We introduce a novel numerical technique for the dynamical behaviour of our fractional model. The oscillatory and chaotic behaviour of the model is studied with the variation of various input parameters on the model. We have shown that there exists strong oscillatory and chaotic behaviour in the system; moreover, we highlighted the impact of fractional-order parameter on the model. The conditions for the local complex behaviour of the system are analyzed with the impact of certain parameters on the macroeconomic. Our findings suggest that the values of initial conditions, fractional order, and input parameters can lessen the chaos of the proposed financial system.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The financial system plays a vital role in the development of a country and is involved in almost all the sectors of a society. Novel methods and techniques are needed to conceptualize and illustrate the financial system. Furthermore, more accurate, precise, and reliable information is required for financial forecasting. The intricate phenomena of the financial system cannot be easily understandable and are out of the range of a common person which badly affects markets, investment banks, stock exchanges, insurance companies, and institutions. It has been acknowledged that finance is an influential area of research to provide information for academic research and public policy. Different questions about the financial system arise in the mind of individuals who are not fully cleared: What is a financial system? What is the importance of the financial system? Why this statement should be relevant and true? What are the main factors of a system, and how determine its quality? What policy implications it might have? What are the advantages for a common person? How to evaluate financial systems and what is their quality significance? Is there some reasonable empirical and theoretical basis for the assumption of a system? and How policy-makers can improve a financial system for the development of a society?

It is eminent that mathematical models play a dominant role to visualize the oscillatory and chaotic behaviour of a system [1–4]. Now a day, financial models are a prominent problem for researchers due to its nature and application in micro and macroeconomics [5, 6]. Numerous researchers formulated and investigated several financial models based on different assumptions to visualize its dynamics [7–12]. Periodic and chaotic behaviour naturally occurs in these nonlinear models of economics. Some mathematical systems are very complicated, and their findings cannot be used for predictions or suggestions. In [6, 7], Chen formulated a financial model in a fractional framework where the authors investigated the chaotic motion, periodic motion, oscillatory behaviour, fixed point, and identified period doubling. Gao and Ma examined the chaotic behaviour and worked out on the Hopf bifurcation of a finance model [13]. The global attractor and the bifurcation phenomena of a financial system were studied by Jun-Hai and Yu-Shu in their research [7]. Moreover, the topological horseshoe and Hope bifurcation for a financial system were discussed by Ma and Wang in [14]. In [15], the authors introduced a financial model with a time delay to show the influence of delay on its chaotic behaviour. They determined that the time delay in an approximate case can suppress and enhance the cause of chaos.

Now a day, the intricate dynamics of the financial system has become a center of attention for researchers. Researchers use scientific experiments and mathematical theory for the analysis of these dynamical systems. Humans’ mind constructs a logical and reasonable model for financial phenomena which are then investigated through mathematical techniques to identify the outcomes of the proposed model. Therefore, it is significant to study financial system through mathematical tools to reduce the issues of economic and financial systems. In this research work, our main focus is to construct a mathematical model for a financial system to visualize its nature through analytic and numerical skills. We choose to explore the dynamical behaviour of the system with the fluctuation of state variables and input parameters in order to emphasise the influence of these input values on the system’s output. We also prefer to check the nature of price exponent, investment demand, and rate of interest in different scenarios.

This work includes a robust study of the financial system to conceptualize the complex phenomena of price exponent, investment demand, and the rate of interest. The article is organized in the following manner: we formulate the natural laws of finance in the form of the Caputo-Fabrizio (CF) fractional structure in the second section. In the third section, we represent the most important results and theorem of the proposed fractional operator for the analysis of our financial system. A novel technique is derived for the numerical analysis and graphical representation of our financial system in section four. We visualize the chaotic and oscillatory behaviour of the system; moreover, we visualized and studied the effect of fractional order on the solution pathway of our model. Finally, concluding remarks are illustrated in the fifth section of the article.

2. Evaluation of Fractional Financial Model

In the formulation of the model, we assume three state variables that are x the rate of interest, y the demand for investment, and z the index of the price. In financial phenomena, the state variable x is influenced by the surplus between savings and investments, and by the goods’ prices. The rate of changes of the second state variable y is proportional to the investment and is also in inversion proportional to the rate of interest and cost of investment. The rate of change of the third state variable z is influenced by the inflation rate and is controlled by the contradiction between supply and demand of the commercial market. In our formulation, we indicate the saving amount by “a” while the cost per investment is denoted by “b” and the elasticity of demand of the commercial markets is denoted by “c.” Then, the financial system in terms of ordinary differential equations with the above assumptions is given by$$\begin{array}{c}\text{(1)}& \frac{\text{d}x}{\text{d}t}=y-ax+z,\frac{\text{d}y}{\text{d}t}=1-x^2-by,\\ \frac{\text{d}z}{\text{d}t}=-cz-x,\end{array}$$where $a\ge 0,b\ge 0$ and $c\ge 0$. Any one of a, b, and c can be chosen as a control parameter for the system while the other two parameters will be fixed.

It is reported that the results and outcomes of fractional models are realistic, reliable, accurate, and precise rather than integer models [16–19]. Fractional calculus has been effectively utilized in different areas of science, engineering, economics, and technology [20–25]. Therefore, motivated by the above accurate results, we opt to present the above financial model in the fractional framework to obtain more realistic results. Then, the above financial model (1) in the fractional framework can be represented as follows:$$\begin{array}{c}\text{(2)}& \frac{CF}{0}{D}_t^{\vartheta }x=y-ax+,\frac{CF}{0}{D}_t^{\vartheta }y=1-x^2-by,\\ \frac{CF}{0}{D}_t^{\vartheta }z=-cz-x,\end{array}$$where the fractional derivative is denoted by $CF/0D_t^{\vartheta }$ and $\vartheta$ indicates the order of the operator. In the next section, the basic concepts and theorems of Caputo-Fabrizio fractional operator are presented in detail for the analysis of our financial model.

3. Theory of Fractional Calculus

In this section of the article, the most important concept and definitions will be illustrated for the analysis of our fractional financial model. Following are the basic results of the fractional Caputo-Fabrizio operator.

Definition 1.

Assume a function h such that $\kappa \in H^{1}a,b$, in which a smaller than b, then the fractional derivative in CF [26] form is defined as$$\begin{array}{}\text{(3)}& D_t^{\vartheta }\kappa t=\frac{U\vartheta }{1-\vartheta }{\displaystyle {\int }_a^t{\kappa }^{\prime }z}\exp \frac{t-z}{1-\vartheta }-\vartheta \text{d}z,\end{array}$$where $\vartheta$ represents the order of the fractional derivative, and $U\tau$ indicates the normality factor with $U0=U1=1$ [26], and moreover, $\vartheta \in \mathrm{0,1}$. If $h\notin H^{1}a,b$, then the fractional derivative is converted into the following form:$$\begin{array}{}\text{(4)}& D_t^{\vartheta }\kappa t=\frac{\vartheta U\vartheta }{1-\vartheta }{\displaystyle {\int }_a^t\kappa t-\kappa z}\exp \frac{t-z}{1-\vartheta }-\vartheta \text{d}z.\end{array}$$

Remark 1.

Consider that $\vartheta =1/1+\alpha \in \mathrm{0,1}$ and $\alpha =1-\vartheta /\vartheta \in 0,\infty$, then the above (4) can be written in the following form:$$\begin{array}{c}\text{(5)}& D_t^{\vartheta }\kappa t=\frac{M\alpha }{\alpha }{\displaystyle {\int }_a^t{\kappa }^{\prime }z{e}^{-t-z/\alpha }\text{d}z,}M0=M\infty =1.\end{array}$$

Furthermore,$$\begin{array}{}\text{(6)}& \underset{\alpha ⟶0}{\text{lim}}\frac{1}{\alpha }\exp -\frac{t-z}{\alpha }=\delta z-t.\end{array}$$

In Figures 1–5, the fractional integral is also important for the analysis of a dynamic system. The definition of CF fractional integral is presented in [27] in detail which is defined as follows.

[figure(s) omitted; refer to PDF]

Definition 2.

Assume a function $h$, then its fractional integral is defined as$$\begin{array}{}\text{(7)}& I_t^{\vartheta }ht=\frac{21-\vartheta }{2-\vartheta U\vartheta }ht+\frac{2\vartheta }{2-\vartheta U\vartheta }{\displaystyle {\int }_o^{t}hu\text{d}u},\hspace{1em}t\ge 0,\end{array}$$where the order of the integral is given by $\vartheta$ with the condition that $0<\vartheta <1$.

Remark 2.

Here, we further investigate the Definition 2 and get the following$$\begin{array}{}\text{(8)}& \frac{21-\vartheta }{2-\vartheta U\vartheta }+\frac{2\vartheta }{2-\vartheta U\vartheta }=1.\end{array}$$

It provides $U\vartheta =2/2-\vartheta$, $0<\vartheta <1$. Recently, a novel fractional operator of order $\vartheta$ was developed in [27] by using equation (8) which can be written as follows:$$\begin{array}{}\text{(9)}& D_t^{\vartheta }ht=\frac{1}{1-\vartheta }{\displaystyle {\int }_0^t{h}^{\prime }x}\exp \vartheta \frac{t-x}{1-\vartheta }\text{d}x,0<\vartheta <1.\end{array}$$

4. Numerical Approach for Financial System

Here, we opt to introduce a novel numerical approach to the dynamical behaviour of our fractional financial system (2). Our main objective is (Figures 6–10) to highlight the dynamical behaviour of the system with the variation of different input parameters numerically. Numerous numerical techniques have been developed for the Caputo-Fabrizio fractional models [28–30]. For our fractional financial system (2), we will introduce a novel numerical technique for the graphical representation of the system. The proposed technique is given as follows: from the first equation of the financial system, we have$$\begin{array}{}\text{(10)}& x_{1}t-x_{1}0=\frac{1-\vartheta }{U\vartheta }{\mathrm{ℒ}}_{1}t,x_1+\frac{\vartheta }{U\vartheta }{\displaystyle {\int }_0^t{\mathrm{ℒ}}_1\zeta ,x_1\text{d}\zeta }.\end{array}$$

[figure(s) omitted; refer to PDF]

For further steps, the time is chosen as $t=t_{m+1},m=\mathrm{0,1},\dots$ and obtains the following equation:$$\begin{array}{}\text{(11)}& x_1{t}_{m+1}-x_{1}0=\frac{1-\vartheta }{U\vartheta }{\mathrm{ℒ}}_1{t}_m,x_1{t}_m+\frac{\vartheta }{U\vartheta }{\displaystyle {\int }_0^{t_{m+1}}{\mathrm{ℒ}}_{1}t,x_1\text{d}t}.\text{(12)}& x_1{t}_m-x_{1}0=\frac{1-\vartheta }{U\vartheta }{\mathrm{ℒ}}_1{t}_{m-1},x_1{t}_{m-1}+\frac{\vartheta }{U\vartheta }{\displaystyle {\int }_0^{t_m}{\mathrm{ℒ}}_{1}t,x_1\text{d}t}.\end{array}$$

For the above, we get the following equation:$$\begin{array}{}\text{(13)}& x_{1_{m+1}}-x_{1_m}=\frac{1-\vartheta }{U\vartheta }{\mathrm{ℒ}}_1{t}_m,x_{1_m}-{\mathrm{ℒ}}_1{t}_{m-1},x_{1_{m-1}}+\frac{\vartheta }{U\vartheta }{\displaystyle {\int }_m^{t_{m+1}}{\mathrm{ℒ}}_{1}t,x_1\text{d}t}.\end{array}$$

The following approximation is obtained through interpolation polynomial for the given function ${\mathrm{ℒ}}_{1}t,x_1$ in $t_k,t_{k+1}$$$\begin{array}{}\text{(14)}& {\mathrm{\mathscr{H}}}_{k}t\cong \frac{{\mathrm{ℒ}}_1{t}_k,x_k}{h}t-t_{k-1}-\frac{{\mathrm{ℒ}}_1{t}_{k-1},x_{k-1}}{h}t-t_k,\end{array}$$where $h=t_m-t_{m-1}$ is representing the time and the above ${\mathrm{\mathscr{H}}}_{k}t$ is utilized to calculate the integral$$\begin{array}{c}\text{(15)}& {\displaystyle {\int }_m^{t_{m+1}}{\mathrm{ℒ}}_{1}t,x_1\text{d}t}={\displaystyle {\int }_m^{t_{m+1}}\frac{{\mathrm{ℒ}}_1{t}_m,x_{1_m}}{h}t-t_{m-1}-\frac{{\mathrm{ℒ}}_1{t}_{m-1},x_{1_{m-1}}}{h}t-t_m}\text{d}t,=\frac{3h}{2}{\mathrm{ℒ}}_1{t}_m,x_{1_m}-\frac{h}{2}{\mathrm{ℒ}}_1{t}_{m-1},x_{1_{m-1}}.\end{array}$$

In the next step, we put (15) in (13) and get the following equation:$$\begin{array}{}\text{(16)}& x_{1_{m+1}}=x_{1_m}+\frac{1-\vartheta }{U\vartheta }+\frac{3\vartheta h}{2U\vartheta }{\mathrm{ℒ}}_1{t}_m,x_{1_m}-\frac{1-\vartheta }{U\vartheta }+\frac{\vartheta h}{2U\vartheta }{\mathrm{ℒ}}_1{t}_{m-1},x_{1_{m-1}},\end{array}$$which is the proposed numerical technique for the first equation of our fractional financial model (2). Taking the same steps, we can obtain the following for the remaining equations:$$\begin{array}{}\text{(17)}& x_{2_{m+1}}=x_{2_m}+\frac{1-\vartheta }{U\vartheta }+\frac{3\vartheta h}{2U\vartheta }{\mathrm{ℒ}}_2{t}_m,x_{2_m}-\frac{1-\vartheta }{U\vartheta }+\frac{\vartheta h}{2U\vartheta }{\mathrm{ℒ}}_2{t}_{m-1},x_{2_{m-1}},\text{(18)}& x_{3_{m+1}}=x_{3_m}+\frac{1-\vartheta }{U\vartheta }+\frac{3\vartheta h}{2U\vartheta }{\mathrm{ℒ}}_3{t}_m,x_{3_m}-\frac{1-\vartheta }{U\vartheta }+\frac{\vartheta h}{2U\vartheta }{\mathrm{ℒ}}_3{t}_{m-1},x_{3_{m-1}}.\end{array}$$

In numerical simulations, we use the above numerical technique to highlight the dynamic and chaotic behaviour of the proposed fractional financial system. The theory of chaos is concerned with occurrences that have unexpected outcomes. To be more specific, it is the name given to the irregular and unexpected temporal development of many nonlinear and complicated linear systems. It addresses the characteristics of the transition from stability to instability or from order to chaos. The contemporary economy revolves upon finance. The financial system’s security and stability are essential for stable economic and social progress. Due to deterministic instability, financial chaos such as extreme turbulence in the financial market and the financial crisis occurred during the functioning of the financial system, having a significant detrimental influence on social stability and economic growth. Analysis of the dynamical and chaotic behaviour is important in the sense to observe the most sensitive input factor and also to determine the control input parameter for the chaotic behaviour of the system. Through our simulations, we will also interrogate the chaotic behaviour and oscillation in our systems that are the point of interest for the researcher and policy maker. These findings are important in the sense to point out the most critical parameters of the system.

In Figures 1–3, we illustrated the fractional dynamics of the financial system with variation in fractional order $\vartheta$. In Figure 1, we choose the fractional order $\vartheta =0.35$ and then select $\vartheta =0.72$ in Figure 2. In the third simulation presented in Figure 3, the solution pathways and chaotic phenomena of the system are presented with $a=3,b=0.1,c=1.0$, and $\vartheta =0.5$. In these simulations, we showed the oscillatory behaviour and phase portrait of the system with the variation of $\vartheta$. It has been noticed in these scenarios that the input parameter $\vartheta$ significantly affects the system and can be used as a control parameter for the oscillatory behaviour. In our simulations, we noticed that there was chaos in the system; therefore, we tried through different simulations to find the main reason for the occurrence of these phenomena. We examined the financial system with different initial conditions and different values of the input parameter to analyze the fluctuation of system (2).

The phase portrait of the hypothesized financial system has been illustrated in the upcoming simulations to investigate the most sensitive parameter of the system. In Figure 4, the phase portrait of the system is presented by taking the input parameter values $a=0.3,b=0.1$, and $c=3$ while in Figure 5, we change the parameters to $a=1,b=0.1$, and $c=1$. It has been noticed that there is a strong chaotic phenomena in the system with these values of input parameters. In Figure 6, we change the parameter $b$ from 0.1 to 0.001 while $a=0.3$ and $c=3$. The effect of the input parameter $b$ has been observed in this simulation. In seventh simulation illustrated in Figure 7, the value of $a$ increased from 0.3 to 3 while the value of $c$ is decreased from 3 to 0.5. To identify the influence of $c$, in Figure 8, we increased the parameter from 0.5 to 2.0 with fixed values of other parameters. In Figures 9 and 10, we mainly changed the input parameter a and c to study their influence on the system. In second last simulation visualized in Figure 9, the values of a and c are decreased to 0.3 and 0.3 while the values of these parameters are increased in the last simulation highlighted in Figure 10. We examined that the input parameter a and c are significantly responsible for the chaotic phenomena of the system. On the basis of our analysis, we predicted that a, c, and $\vartheta$ may be used as control parameter to reduce the chaos of the system.

We discovered that the system exhibits substantial chaotic and oscillatory activity and that these phenomena are tightly linked. It has been noted that fractional system discoveries are more promising than integer system findings. Our findings imply that adjusting the beginning circumstances, fractional order, and input parameters can reduce the financial system’s chaos. Therefore, we recommend to make the suitable situation for the control of these parameters and values because desirable outcomes can be obtained by controlling these values.

5. Conclusion

The concept of finance is broad and important for all sectors influencing humans’ life; furthermore, it is indirectly related to individuals, societies, cities, and countries. Its investigation is of great importance for the development of a society; therefore, it is an area of interest for researchers. In this article, we formulated the phenomena of the financial system in the framework of fractional CF derivative. We listed the fundamental ideas and theorems of the Caputo-Fabrizio operator for the analysis of our financial system. We derived a numerical technique for the dynamical and chaotic behaviour of our proposed system and showed oscillation and chaos in the system through a numerical scheme. Then, we highlighted the influence of fractional order and other input parameters on the behaviour of the system. Our findings suggest that the chaotic behaviour of the system is strongly associated with the initial conditions and values of the input parameter. It has been observed that the chaos of the system is closely related to oscillations. We suggested that the input parameters can be used to control the chaos of the system.

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