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Theory and Modern Applications

Modeling, discretization, and hyperchaos detection of conformable derivative approach to a financial system with market confidence and ethics risk


We propose a new chaotic financial system by considering ethics involvement in a four-dimensional financial system with market confidence. We present a five-dimensional conformable derivative financial system by introducing conformable fractional calculus to the integer-order system. We propose a discretization scheme to calculate numerical solutions of conformable derivative systems. We illustrate the scheme by testing hyperchaos for the system.

1 Introduction

A hyperchaotic system is typically defined as a chaotic system with at least two positive Lyapunov exponents [1,2,3]. Considering one null exponent along the flow and one negative exponent to ensure the boundedness of the solution, the dimension for a hyperchaotic fractional-order system is at least four. Many types of fractional-order hyperchaotic systems have been investigated, such as hyperchaos in fractional-order Rössler systems [4], fractional hyperchaotic systems with flux controlled memristors [5], fractional-order hyperchaotic systems realized in circuits [6,7,8], fractional-order Lorenz hyperchaotic systems [9], and fractional-order hyperchaotic cellular neural networks [10]. Huang and Li [11] proposed an interesting nonlinear financial system depicting the relationship among interest rates, investments, prices, and savings. Chen [12] presented a fractional form of a nonlinear financial system. Wang, Huang and Shen [13] established an uncertain fractional-order form of the financial system. Mircea et al. [14] set up a delayed form of a financial system. Xin, Chen, and Ma [15] proposed a discrete form of a financial system. Yu, Cai, and Li [16] extended the financial system with the average profit margin. Xin, Li, and Zhang [17, 18] introduced investment incentive and market confidence to a nonlinear financial system to set up novel four-dimensional financial systems. Most of these are fractional-order hyperchaotic systems [19, 20]. In this paper, we construct a 5D fractional-order hyperchaotic financial system.

Although Riemann–Liouville, Caputo, and Grunwald–Letnikov fractional derivatives [21,22,23,24,25,26,27,28] are widely used in physics, mathematics, medicine, economics, and engineering, these derivative definitions lack some of the agreed properties of the classical differential operator, such as the chain rule. The conformable derivative can be regarded as a natural extension of the classical differential operator, which satisfies most important properties, such as the chain rule [29,30,31]. Researchers have recently applied conformable derivatives to many scientific fields [32,33,34,35,36,37,38,39,40,41,42]. In this paper, we introduce the conformable derivative for the financial system with market confidence and ethics risk.

Researchers [43,44,45,46,47,48] have paid much attention to the impact of confidence on the financial system since the financial crisis began in 2008. Financial crises are sometimes associated with a vicious circle in which confidences and economic indicators interact. For example, lower consumer confidence can lead to weaker consumption expectation, which will bring a sharp decrease in the market demand, stagnation in orders, and a marked decrease in sales. This can hold down investor confidence and trigger a decrease in employment and wages, which also lead to lower consumer confidence. Alternatively, we can boost the demands of consumers and investors by improving individual incomes, confidences, and expectations. Certainly, investor confidence can be rescued by lowering interest rates. Consumers’ balance sheets can result in a price index, which will affect the confidences of both investors and consumers [18]. Thus, it is necessary to study financial systems with market confidence.

As for mainstream finance theory, financial participants’ profit drive and ethics-denunciation can rub together, so the profit motive leads them to crimp on ethics and disregard a broader social impact of their actions. Unrestrained demand for profit turns financial participants into psychopaths. Although the profit-driven segment can earn abnormal returns through the violation of ethics in the short run, these profit-generating opportunities do not persist in the long run for a socially responsible investment movement [49], because it violates the inherent requirement of financial efficiency, entailing a high risk that the financial market falls into instability. Analogous to Rasmussen’s logic [50], the drive for profit is indeed a double-edged sword. As has long been recognized, it is an inevitable result of flourishing financial markets as a positively useful means of encouraging investment. Conversely, it has some of the other effects of extreme profit-driven behavior, which leads people to sympathize more fully and readily with selfish profit maximization, as opposed to altruistic social responsibility, and this distortion in our sympathies, in turn, undermines both investment ethics and public welfare. Ethics may be easy for many subjects to understand, but implementing it in financial markets requires faith, dedication, determination, and regulation. Profit-driven behavior in a market economy is not naturally reasonable, but it must be restrained. Therefore, it is meaningful for us to consider ethics when we analyze a financial system.

After we propose a conformable derivative financial system, we need a suitable scheme to obtain its solutions. Though there are several methods to solve a conformable derivative system [41, 51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72], these are too complex for many people. Inspired by the discretization process for the Caputo derivative [73, 74], we propose a simple discretization process for a conformable derivative. As shown in Sect. 3, our results agree with Mohammadnezhad’s conformable Euler’s method [75]. Using our proposed discretization scheme, we detect the hyperchaotic attractor of a five-dimensional fractional-order financial system.

The remainder of this paper is organized as follows. In Sect. 2, we present a conformable derivative hyperchaotic financial system with market confidence and ethics risk. In Sect. 3, we provide a conceptual overview of conformable calculus and propose a conformable discretization process, which coincides with Mohammadnezhad’s conformable Euler’s method [75]. In Sect. 4, we detect the hyperchaotic attractor from the proposed financial system. Some concluding remarks in Sect. 5 close the paper.

2 A conformable derivative hyperchaotic financial system with market confidence and ethics risk

Based on the dynamical mechanism of financial systems, as shown in Fig. 1, Huang and Li [11] proposed an interesting nonlinear financial system, as follows:

$$ \textstyle\begin{cases} \dot{x} = z+(y-a)x,\\ \dot{y}= 1-by-x^{2},\\ \dot{z}= -x-cz, \end{cases} $$

where \(x, y, z, a, b, c\) are the interest rate, investment demand, price index, saving amount, cost per investment, and demand elasticity of commercial markets, respectively, and \(a, b, c\geq 0\).

Figure 1
figure 1

Schematic diagram of a simple financial system (1)

Xin and Zhang [18] took into account the market confidence in system (1), updated the dynamical mechanism of financial systems with the market confidence, as shown in Fig. 2, and obtained the following financial system:

$$ \textstyle\begin{cases} \dot{x} = z+(y-a)x+m_{1}w,\\ \dot{y}= 1-by-x^{2}+m_{2}w,\\ \dot{z}= -x-cz+m_{3}w,\\ \dot{w}=-xyz, \end{cases} $$

where \(x, y, z, a, b, c \) are defined as in system (1), w indicates the market confidence, and \(m_{1}, m_{2}, m_{3} \) are the impact factors.

Figure 2
figure 2

Schematic diagram of financial system (2) with market confidence

Ethics risk is often portrayed as a threat under asymmetric information. Uncertain or incomplete contracts cause the responsible stakeholders to not bear all of the consequences for maximizing their own utility while harming other stakeholders through improper behavior such as lying, cheating, and breaking the terms of a contract. However, there are no clear guidelines for governments to deal with ethical dilemmas, whether in law or often in religion. Thus, ethical risk may occur when stakeholders must choose among alternatives, for example, when significant value conflicts exit among differing interests, real alternatives with justifiable equality and mutual benefit, significant consequences on them. Thus we can update the dynamical mechanism of financial system (2) by accounting for ethical risk, as shown in Fig. 3.

Figure 3
figure 3

A schematic diagram of financial system (3) with market confidence and ethics risk

Ethics risk can negatively affects market confidence. They will offset to some extent the impact of market confidence on the interest rate, investment demand, and price index. In addition, the higher the market confidence, the less the motivation to harm other stakeholders, and the lower the ethics risk. Moreover, ethical risks become increasingly lower with continuous improvement of regulations. Market confidence is influenced by many factors, including ethics, law, and religion, so we will consider the impact of ethics risk on market confidence. Thus we can obtain the following financial system accounting for both market confidence and ethics risk:

$$ \textstyle\begin{cases} \dot{x} = z+(y-a)x+k(w-pu),\\ \dot{y}= 1-by-x^{2}+k(w-pu),\\ \dot{z}= -x-cz+k(w-pu),\\ \dot{w}=-dxyz,\\ \dot{u}=k(w-pu), \end{cases} $$

where \(x, y, z, w, a, b, c \) mean the same as in system (2), u denotes the ethics risk, and \(k, p, d \) are the impact factors.

Fractional-order economic systems [18, 76,77,78,79,80] can generalize their integer-order forms [17, 81, 82]. As a natural extension of the integer-order differential operator, the conformable fractional operator is a suitable tool to generalize integer-order forms, so we will introduce conformable fractional derivatives to financial system (3) as follows:

$$ \textstyle\begin{cases} T_{\alpha _{1}}x = z+(y-a)x+k(w-pu),\\ T_{\alpha _{2}}y= 1-by-x^{2}+k(w-pu), \\ T_{\alpha _{3}}z= -x-cz+k(w-pu),\\ T_{\alpha _{4}}w= -dxyz,\\ T _{\alpha _{5}}u= k(w-pu), \end{cases} $$

where \(\alpha =(\alpha _{1},\alpha _{2},\alpha _{3},\alpha _{4},\alpha _{5})\) is subject to \(\alpha _{1},\alpha _{2},\alpha _{3},\alpha _{4}, \alpha _{5}\in (0,1)\).

Remark 1

When \(\alpha =(1,1,1,1,1)\), system (4) degenerates to system (3).

3 Discretization of conformable derivative systems

3.1 Preliminaries

Definition 1

(See [30])

For a function \(f:[t_{0}, \infty )\rightarrow \mathbb{R}\), its left conformable derivative starting from \(t_{0}\) of order \(\alpha \in (0,1)\) is defined by

$$ T_{\alpha }^{t_{0}}f(t)=\lim_{\varepsilon \rightarrow 0} \frac{f(t+ \varepsilon (t-t_{0})^{1-\alpha })-f(t)}{\varepsilon }, $$

where the function f is called α-differentiable.

Definition 2

(See [30])

For a function \(f:[t_{0}, \infty )\rightarrow \mathbb{R}\), its left conformable integral starting from \(t_{0}\) of order \(\alpha \in (0,1)\) is defined as

$$ I_{\alpha }^{t_{0}}f(t)= \int ^{t}_{t_{0}}(s-t_{0})^{\alpha -1}f(s) \,ds, $$

where the integral is the usual Riemann improper integral.

Lemma 1

(See [30])

Suppose the derivative order \(\alpha \in (0,1)\) and that functions f are α-differentiable at a point \(t_{0}>0\). Then the left conformable derivative satisfies

$$ T_{\alpha }^{t_{0}}f(t)=(t-t_{0})^{1-\alpha } \frac{df(t)}{dt}. $$

Lemma 2

(Conformable Euler’s method [75])

Consider the following conformable derivative system:

$$ T_{\alpha }x(t)=f \bigl(x(t) \bigr),\quad 0\leq t\leq T, x(0)=x_{0}, $$

where \(h=\frac{T}{N}=t_{n+1}-t_{n}\), \(t_{n} = nh, n=0,1,\ldots,N\). If h is small enough and \(T_{\alpha }x(t), T_{2\alpha }x(t)\in C^{0}[a,b]\), then we obtain the following discretization of Eq. (8)

$$ x_{n+1}\approx x_{n} + \frac{h^{\alpha }}{\alpha }f(x_{n}). $$

3.2 Discretization process

Introducing partially piecewise constant arguments to Eq. (8), Kartal and Gurcan [59] obtained the form

$$ T_{\alpha }x(t)=f \biggl(x(t),x \biggl( \biggl[ \frac{t}{h} \biggr]h \biggr) \biggr),\quad 0\leq t \leq T, x(0)=x_{0}, $$

where \(h=\frac{T}{N}\), that is, \(t\in [nh,(n+1)h), n=0,1,2,\ldots, \frac{T}{h}\). Then Kartal and Gurcan [59] proposed the discretization process of system (10), but it was unsuitable for the following form obtained by introducing completely piecewise constant arguments to Eq. (8):

$$ T_{\alpha }x(t)=f \biggl(x \biggl( \biggl[\frac{t}{h} \biggr]h \biggr) \biggr),\quad 0\leq t \leq T, x(0)=x_{0}. $$

Theorem 1

(Conformable discretization by piecewise constant approximation)

According to system (11), we obtain the following discretization of Eq. (8):

$$ x_{n+1}= x_{n} + \frac{h^{\alpha }}{\alpha }f(x_{n}), $$

where \(x_{n}\) denotes \(x_{n}(nh)\).


Using Lemma 1, we rewrite Eq. (11) as

$$ (t-nh)^{1-\alpha }\frac{dx(t)}{dt}=f \bigl(x(nh) \bigr),\quad 0\leq t\leq T, x(0)=x_{0}, $$

which leads to

$$ \frac{dx(t)}{dt}=(t-nh)^{\alpha -1}f \bigl(x(nh) \bigr),\quad 0 \leq t\leq T, x(0)=x_{0}. $$

Drawing on the step method [59, 73], we detail the steps of the discretization process:

(i) Let \(n=0\). Then \(t\in [0,h)\), and we rewrite Eq. (13) as

$$ \frac{dx(t)}{dt}=(t-0)^{\alpha -1}f(x_{0}),\quad t\in [0,h), $$

and the solution of Eq. (14) is

$$\begin{aligned} x_{1}(t) &=x_{0} + \int _{0}^{t} \bigl((s-0)^{\alpha -1}f(x_{0}) \bigr)\,ds \\ &=x_{0} +f(x_{0}) \int _{0}^{t} s^{\alpha -1}\,ds \\ &=x_{0} +\frac{t _{\alpha }}{\alpha }f(x_{0})). \end{aligned}$$

(ii) Let \(n=1\). Then \(t\in [h,2h)\), and we rewrite Eq. (13) as

$$ \frac{dx(t)}{dt}=(t-h)^{\alpha -1}f \bigl(x_{1}(h) \bigr),\quad t\in [h,2h), $$

and the solution of Eq. (15) is

$$\begin{aligned} x_{2}(t) &=x_{1}(h) + \int _{h}^{t} \bigl((s-h)^{\alpha -1}f \bigl(x_{1}(h) \bigr) \bigr)\,ds \\ &=x_{1}(h) +f \bigl(x_{1}(h) \bigr) \int _{h}^{t} (s-h)^{\alpha -1}\,ds \\ &=x_{1}(h) +\frac{(t-h)^{\alpha }}{\alpha }f \bigl(x_{1}(h) \bigr). \end{aligned}$$

(iii) By repeating the previous process, we obtain the following solution of Eq. (13):

$$ x_{n+1}(t)=x_{n}(nh)+\frac{(t-nh)^{\alpha }}{\alpha }f \bigl(x_{n}(nh) \bigr), \quad t\in \bigl[nh,(n+1)h\bigr). $$

Let \(t\rightarrow (n+1)h\). We deduce the following discretization:

$$ x_{n+1} \bigl((n+1)h \bigr)=x_{n}(nh)+\frac{h^{\alpha }}{\alpha }f \bigl(x_{n}(nh) \bigr),\quad t\in \bigl[nh,(n+1)h\bigr), $$

that is,

$$ x_{n+1}=x_{n}+\frac{h^{\alpha }}{\alpha }f(x_{n}), $$

and the theorem is proved. □

Remark 2

The conformable discretization by piecewise constant approximation well coincides with the conformable Euler method [75].

4 Hyperchaos detection

Using Theorem 1, we rewrite system (4) using piecewise constant approximation as follows:

$$ \textstyle\begin{cases} x_{n+1} = x_{n} + \frac{h^{\alpha _{1}}}{\alpha _{1}} (z+(y-a)x+k(w-pu)), \\ y_{n+1}= y_{n} + \frac{h^{\alpha _{2}}}{\alpha _{2}} (1-by-x^{2}+k(w-pu)), \\ z_{n+1}= z_{n} + \frac{h^{\alpha _{3}}}{\alpha _{3}} (-x-cz+k(w-pu)), \\ w_{n+1}= w_{n} + \frac{h^{\alpha _{4}}}{\alpha _{4}} (-dxyz),\\ u _{n+1}= u_{n} + \frac{h^{\alpha _{5}}}{\alpha _{5}} k(w-pu). \end{cases} $$

In this section, we implement hyperchaos detection by varying the parameters related to ethics risk, such as \(\alpha _{5}\), the confidence factor k, and the risk factor p. To detect hyperchaos in system (16) using conformable discretization process, we fix the following parameters and initial point values: \(h=0.002, a=0.8, b=0.6, c=1, d=2, \alpha _{1}=0.3, \alpha _{2}=0.5, \alpha _{3}=0.6, \alpha _{4}=0.24\), \(x_{0}=0.4, y_{0}=0.6, z_{0}=0.8, w_{0}=0.3, u _{0}=0.4\).

4.1 Varying \(\alpha _{5}\) with fixed \(k=2\) and \(p=1\)

Using the iterative algorithm, we obtain a bifurcation diagram of system (16) when we vary \(\alpha _{5}\) and fix \(k=2\) and \(p=1\), as shown in Fig. 4(a). In Fig. 4(a), we can always find two positive Lyapunov exponents corresponding to any \(\alpha _{5}\), which is well validated by Figs. 4(b)–(d). Thus, we can say that system (16) is hyperchaotic with \(\alpha _{5}\in [0.232, 0.328]\). We can fix \(\alpha _{5}=0.24\) and pick up a set of Lyapunov exponents \((\lambda _{1}, \lambda _{2}, \lambda _{3}, \lambda _{4}, \lambda _{5})=(0.2120, 0.1308, -0.0491, -0.2048, -0.5201)\). Obviously, there are two positive Lyapunov exponents \(\lambda _{1}, \lambda _{2}\) and three negative Lyapunov exponents \(\lambda _{3}, \lambda _{4}, \lambda _{5}\) when \(\alpha _{5}=0.24\), that is, there is a hyperchaotic attractor, as shown in Figs. 4(e)–(f).

Figure 4
figure 4

Complex behavior of system (16) with \(\alpha _{5}\)

4.2 Varying p with fixed \(k=2\) and \(\alpha _{5}=0.3\)

Here we employ an iterative algorithm to produce a bifurcation diagram of system (16) when we vary p and fix \(k=2\) and \(\alpha _{5} =0.3\), as shown in Fig. 5(a). In the figure, we can always find two positive Lyapunov exponents corresponding to any p, which is well confirmed by Figs. 5(b)–(d). Thus, we can say that system (16) is hyperchaotic with \(p\in [1, 2]\). We can set \(p=1\) and obtain a set of Lyapunov exponents \((\lambda _{1}, \lambda _{2}, \lambda _{3}, \lambda _{4}, \lambda _{5})=(0.2182, 0.1468, -0.0653, -0.4007, -1.9060)\). There are also two positive Lyapunov exponents \(\lambda _{1}, \lambda _{2}\) and three negative Lyapunov exponents \(\lambda _{3}, \lambda _{4}, \lambda _{5}\) when \(p=1\), that is, there exists hyperchaos in system (16), as shown in Figs. 5(e)–(f).

Figure 5
figure 5

Complex behavior of system (16) with p

4.3 Varying k with fixed \(p=1\) and \(\alpha _{5}=0.3\)

By means of the iterative algorithm, we draw a bifurcation diagram of system (16) when we vary k and fix \(p=1\) and \(\alpha _{5} =0.3\), as shown in Fig. 6(a). In Fig. 6(a), we can always obtain two positive Lyapunov exponents corresponding to any k, which is well verified by Figs. 6(b)–(d). Thus, we can say that system (16) is hyperchaotic with \(k\in [1.5, 2.5]\). We also can let \(k=1.5\) and obtain a set of Lyapunov exponents \((\lambda _{1}, \lambda _{2}, \lambda _{3}, \lambda _{4}, \lambda _{5})=(0.1918, 0.1049, -0.0902, -0.3301, -1.5310)\). There are also two positive Lyapunov exponents \(\lambda _{1}, \lambda _{2}\) and three negative Lyapunov exponents \(\lambda _{3}, \lambda _{4}, \lambda _{5}\) when \(k=1.5\), that is, a hyperchaotic attractor occurs in system (16), as shown in Figs. 6(e)–(f).

Figure 6
figure 6

Complex behavior of system (16) with k

5 Conclusion

As we know, a real financial system is very complicated, and it is impossible to accurately depict it. We try to introduce the ethics risk and conformable derivatives to an existing financial system with market confidence to set up a five-dimensional conformable derivative hyperchaotic financial system. To obtain the system’s numerical solutions, we employ a piecewise constant approximation to design a new discretization process whose result is the same to that of the conformable Euler method. Finally, we use the proposed method to detect the hyperchaos in the proposed financial system.

The following extensions are of interest for future research.

(i) Compared with traditional fractional-order derivatives, such as Riemann–Liouville and Caputo, the conformable derivative retains some most critical properties. A useful one is a relation between the conformable derivative and the classical integer-order derivative in the definition of the composite function. So, we can regard the conformable derivative financial system as a natural extension of its ODE form. However, the conformable derivative is a local derivative and has no some remarkable properties, such as memory and nonlocality. Further development is to consider the memory for which a traditional fractional-order financial system with market confidence and ethics risk will be interesting.

(ii) In the real world, our government makes some decisions based on particular parameters of a real system. So it is essential for us to study the parameter estimation [83,84,85] of the conformable derivative financial system with market confidence and ethics.

(iii) Though we draw some dynamical system model and implement some evolutionary analysis by conformable derivative, it will be more interesting to verify the proposed model by experimental economics approach [86, 87].

(iv) The proposed discretization process for conformable derivative systems is easy to be implemented and can be used in many scientific fields, such as engineering, physics, economics, environment, ecology, and materials science.


  1. Liu, Y., Li, J., Wei, Z., Moroz, I.: Bifurcation analysis and integrability in the segmented disc dynamo with mechanical friction. Adv. Differ. Equ. 2018, 210 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Wei, Z., Moroz, I., Sprott, J.C., et al.: Hidden hyperchaos and electronic circuit application in a 5D self-exciting homopolar disc dynamo. Chaos 27(3), 033101 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Wei, Z., Rajagopal, K., Zhang, W., et al.: Synchronisation, electronic circuit implementation, and fractional-order analysis of 5D ordinary differential equations with hidden hyperchaotic attractors. Pramana 90(4), 50 (2018)

    Article  Google Scholar 

  4. Li, C., Chen, G.: Chaos and hyperchaos in the fractional-order Rossler equations. Physica A 341, 55–61 (2004)

    Article  MathSciNet  Google Scholar 

  5. Rajagopal, K., Karthikeyan, A., Duraisamy, P.: Hyperchaotic chameleon: fractional-order FPGA implementation. Complexity 2017, 8979408 (2017)

    MathSciNet  MATH  Google Scholar 

  6. El-Sayed, A., et al.: Dynamical behaviors, circuit realization, chaos control, and synchronization of a new fractional-order hyperchaotic system. Appl. Math. Model. 40(5–6), 3516–3534 (2016)

    Article  MathSciNet  Google Scholar 

  7. El-Sayed, A., Elsonbaty, A., Elsadany, A., Matouk, A.: Dynamical analysis and circuit simulation of a new fractional-order hyperchaotic system and its discretization. Int. J. Bifurc. Chaos 26(13), 1650222 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mou, J., Sun, K.: Characteristic analysis of fractional-order 4D hyperchaotic memristive circuit. Math. Probl. Eng. 2017, 2313768 (2017)

    Article  MathSciNet  Google Scholar 

  9. Wang, Y., He, S., Wang, H., et al.: Bifurcations and synchronization of the fractional-order simplified Lorenz hyperchaotic system. J. Appl. Anal. Comput. 5(2), 210–219 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Huang, X., Zhao, Z., Wang, Z., et al.: Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing 94(3), 13–21 (2012)

    Article  Google Scholar 

  11. Huang, D., Li, H.: Theory and Method of the Nonlinear Economics. Sichuan University Press, Chengdu (1993)

    Google Scholar 

  12. Chen, W.: Nonlinear dynamics and chaos in a fractional-order financial system. Chaos Solitons Fractals 36, 1305–1314 (2008)

    Article  Google Scholar 

  13. Wang, Z., Huang, X., Shen, H.: Control of an uncertain fractional-order economic system via adaptive sliding mode. Neurocomputing 83, 83–88 (2012)

    Article  Google Scholar 

  14. Mircea, G., Neamtu, M., Bundau, O., Opris, D.: Uncertain and stochastic financial models with multiple delays. Int. J. Bifurc. Chaos 22, 1250131 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Xin, B., Chen, T., Ma, J.: Neimark–Sacker bifurcation in a discrete-time financial system. Discrete Dyn. Nat. Soc. 2010, 405639 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yu, H., Cai, G., Li, Y.: Dynamic analysis and control of a new hyperchaotic finance system. Chaos Solitons Fractals 45, 1048–1057 (2012)

    Article  MATH  Google Scholar 

  17. Xin, B., Li, Y.: 0–1 test for chaos in a fractional-order financial system with investment incentive. Abstr. Appl. Anal. 2013, 876298 (2013)

    MathSciNet  Google Scholar 

  18. Xin, B., Zhang, J.: Finite-time stabilizing a fractional-order chaotic financial system with market confidence. Nonlinear Dyn. 79(2), 1399–1409 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang, L., Sun, K., He, S., et al.: Solution and dynamics of a fractional-order 5-D hyperchaotic system with four wings. Eur. Phys. J. Plus 132(1), 31 (2017)

    Article  Google Scholar 

  20. Wang, S., Wu, R.: Dynamic analysis of a 5D fractional-order hyperchaotic system. Int. J. Control. Autom. Syst. 15(3), 1003–1010 (2017)

    Article  Google Scholar 

  21. Zheng, R., Jiang, X.: Spectral methods for the time-fractional Navier–Stokes equation. Appl. Math. Lett. 91, 194–200 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Xu, H., Jiang, X.: Creep constitutive models for viscoelastic materials based on fractional derivatives. Comput. Math. Appl. 73(6), 1377–1384 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fan, W., Qi, H.: An efficient finite element method for the two-dimensional nonlinear time–space fractional Schrodinger equation on an irregular convex domain. Appl. Math. Lett. 86, 103–110 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, X., Qi, H., Jiang, X.: Numerical analysis for electroosmotic flow of fractional Maxwell fluids. Appl. Math. Lett. 78, 1–8 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gao, X., Chen, D., Yan, D., et al.: Dynamic evolution characteristics of a fractional-order hydropower station system. Mod. Phys. Lett. B 32(2), 1750363 (2018)

    Article  Google Scholar 

  26. Wang, F., Chen, D., Zhang, X., Wu, Y.: Finite-time stability of a class of nonlinear fractional-order system with the discrete time delay. Int. J. Syst. Sci. 48, 984–993 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wu, G., Baleanu, D., Huang, L.: Novel Mittag-Leffler stability of linear fractional delay difference equations with impulse. Appl. Math. Lett. 82, 71–78 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wu, G., Baleanu, D., Luo, W.: Analysis of fractional non-linear diffusion behaviors based on Adomian polynomials. Therm. Sci. 21(2), 813–817 (2017)

    Article  Google Scholar 

  29. Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Abdeljawad, T., Al-Mdallal, Q., Jarad, F.: Fractional logistic models in the frame of fractional operators generated by conformable derivatives. Chaos Solitons Fractals 119, 94–101 (2019)

    Article  MathSciNet  Google Scholar 

  32. Acan, O., Firat, O., Keskin, Y.: Conformable variational iteration method, conformable fractional reduced differential transform method and conformable homotopy analysis method for non-linear fractional partial differential equations. Waves Random Complex Media 8, 1–19 (2018)

    Article  Google Scholar 

  33. Attia, R., Lu, D., Khater, M.: Chaos and relativistic energy-momentum of the nonlinear time fractional Duffing equation. Math. Comput. Appl. 24(1), 10 (2019)

    Google Scholar 

  34. Bohner, M., Hatipoglu, V.: Dynamic cobweb models with conformable fractional derivatives. Nonlinear Anal. Hybrid Syst. 32, 157–167 (2019)

    Article  MathSciNet  Google Scholar 

  35. Tarasov, V.: No nonlocality. No fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 62, 157–163 (2018)

    Article  MathSciNet  Google Scholar 

  36. Rosales, J., Godínez, F., Banda, V.: Analysis of the Drude model in view of the conformable derivative. Optik 178, 1010–1015 (2019)

    Article  Google Scholar 

  37. Akbulut, A., Melike, K.: Auxiliary equation method for time-fractional differential equations with conformable derivative. Comput. Math. Appl. 75(3), 876–882 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  38. Martinez, L., Rosales, J., Carreno, C.: Electrical circuits described by fractional conformable derivative. Int. J. Circuit Theory Appl. 46(5), 1091–1100 (2018)

    Article  Google Scholar 

  39. Rezazadeh, H., Khodadad, F., Manafian, J.: New structure for exact solutions of nonlinear time fractional Sharma–Tasso–Olver equation via conformable fractional derivative. Appl. Appl. Math. 12(1), 13–21 (2017)

    MathSciNet  MATH  Google Scholar 

  40. Korkmaz, A.: Explicit exact solutions to some one-dimensional conformable time fractional equations. Waves Random Complex Media 29(1), 124–137 (2019)

    Article  MathSciNet  Google Scholar 

  41. Perez, J., Gomez-Aguilar, J., Baleanu, D., Tchier, F.: Chaotic attractors with fractional conformable derivatives in the Liouville–Caputo sense and its dynamical behaviors. Entropy 20(5), 384 (2018)

    Article  MathSciNet  Google Scholar 

  42. He, S., Banerjee, S., Yan, B.: Chaos and symbol complexity in a conformable fractional-order memcapacitor system. Complexity 2018, 4140762 (2018)

    MATH  Google Scholar 

  43. Lu, Y., Yang, L., Liu, L.: Volatility spillovers between crude oil and agricultural commodity markets since the financial crisis. Sustainability 11, 396 (2019)

    Article  Google Scholar 

  44. Erfani, G., Vasigh, B.: The impact of the global financial crisis on profitability of the banking industry: a comparative analysis. Economies 6, 66 (2018)

    Article  Google Scholar 

  45. Dinoer, H., Yuksel, S., Senel, S.: Analyzing the global risks for the financial crisis after the great depression using comparative hybrid hesitant fuzzy decision-making models: policy recommendations for sustainable economic growth. Sustainability 10, 3126 (2018)

    Article  Google Scholar 

  46. Li, R., Liu, W., Liu, Y., Tsai, S.B.: IPO underpricing after the 2008 financial crisis: a study of the Chinese stock markets. Sustainability 10, 2844 (2018)

    Article  Google Scholar 

  47. Cavdar, S.C., Aydin, A.D.: An empirical analysis for the prediction of a financial crisis in Turkey through the use of forecast error measures. J. Risk Financial Manag. 8, 337–354 (2015)

    Article  Google Scholar 

  48. Zhao, H., Zhao, H., Guo, S., Li, F., Hu, Y.: The impact of financial crisis on electricity demand: a case study of North China. Energies 9, 250 (2016)

    Article  Google Scholar 

  49. Derwall, J., Koedijk, K., Ter Horst, J.: A tale of values-driven and profit-seeking social investors. J. Bank. Finance 35(8), 2137–2147 (2011)

    Article  Google Scholar 

  50. Rasmussen, D.: Adam Smith on what is wrong with economic inequality. Am. Polit. Sci. Rev. 110(2), 342–352 (2016)

    Article  Google Scholar 

  51. Eslami, M., Rezazadeh, H.: The first integral method for Wu–Zhang system with conformable time-fractional derivative. Calcolo 53(3), 475–485 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  52. Ilie, M., Biazar, J., Ayati, Z.: The first integral method for solving some conformable fractional differential equations. Opt. Quantum Electron. 50(2), 55 (2018)

    Article  Google Scholar 

  53. Hosseini, K., Bekir, A., Ansari, R.: New exact solutions of the conformable time-fractional Cahn–Allen and Cahn–Hilliard equations using the modified Kudryashov method. Optik 132, 203–209 (2017)

    Article  Google Scholar 

  54. Unal, E., Gokdogan, A.: Solution of conformable fractional ordinary differential equations via differential transform method. Optik 128, 264–273 (2017)

    Article  Google Scholar 

  55. Kumar, D., Seadawy, A., Joardar, A.: Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology. Chin. J. Phys. 56(1), 75–85 (2018)

    Article  Google Scholar 

  56. Srivastava, H., Gunerhan, H.: Analytical and approximate solutions of fractional-order susceptible-infected-recovered epidemic model of childhood disease. Math. Methods Appl. Sci. 42(3), 935–941 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  57. Kaplan, M.: Applications of two reliable methods for solving a nonlinear conformable time-fractional equation. Opt. Quantum Electron. 49(9), 312 (2017)

    Article  Google Scholar 

  58. Yavuz, M., Ozdemir, N.: A different approach to the European option pricing model with new fractional operator. Math. Model. Nat. Phenom. 13(1), 12 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  59. Kartal, S., Gurcan, F.: Discretization of conformable fractional differential equations by a piecewise constant approximation. Int. J. Comput. Math. 25, 1–2 (2018)

    Article  Google Scholar 

  60. Iyiola, O., Tasbozan, O., Kurt, A., Cenesiz, Y.: On the analytical solutions of the system of conformable time-fractional Robertson equations with 1-D diffusion. Chaos Solitons Fractals 94, 1–7 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  61. Ruan, J., Sun, K., Mou, J., He, S., Zhang, L.: Fractional-order simplest memristor-based chaotic circuit with new derivative. Eur. Phys. J. Plus 133(1), 3 (2018)

    Article  Google Scholar 

  62. He, S., Sun, K., Mei, X., Yan, B., Xu, S.: Numerical analysis of a fractional-order chaotic system based on conformable fractional-order derivative. Eur. Phys. J. Plus 132(1), 36 (2017)

    Article  Google Scholar 

  63. Yokus, A.: Comparison of Caputo and conformable derivatives for time-fractional Korteweg–de Vries equation via the finite difference method. Int. J. Mod. Phys. B 32(29), 1850365 (2018)

    Article  MathSciNet  Google Scholar 

  64. Rezazadeh, H., Ziabarya, B.: Sub-equation method for the conformable fractional generalized Kuramoto–Sivashinsky equation. Comput. Res. Prog. App. Sci. Eng. 2(3), 106–109 (2016)

    Google Scholar 

  65. Zhong, W., Wang, L.: Basic theory of initial value problems of conformable fractional differential equations. Adv. Differ. Equ. 1, 321 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  66. Tayyan, B., Sakka, A.: Lie symmetry analysis of some conformable fractional partial differential equations. Arab. J. Math. 2018, 1–12 (2018)

    Google Scholar 

  67. Yaslan, H.: Numerical solution of the conformable space-time fractional wave equation. Chin. J. Phys. 56(6), 2916–2925 (2018)

    Article  MathSciNet  Google Scholar 

  68. Kurt, A., Cenesiz, Y., Tasbozan, O.: On the solution of Burgers’ equation with the new fractional derivative. Open Phys. 13, 355–360 (2015)

    Article  Google Scholar 

  69. Khalil, R., Abu-Shaab, H.: Solution of some conformable fractional differential equations. Int. J. Pure Appl. Math. 103(4), 667–673 (2015)

    Article  Google Scholar 

  70. Unal, E., Gokdogan, A., Celik, E.: Solutions of sequential conformable fractional differential equations around an ordinary point and conformable fractional Hermite differential equation (2015). Preprint. arXiv:1503.05407

  71. Liu, S., Wang, H., Li, X., Li, H.: The extremal iteration solution to a coupled system of nonlinear conformable fractional differential equations. J. Nonlinear Sci. Appl. 10, 5082–5089 (2017)

    Article  MathSciNet  Google Scholar 

  72. Cenesiz, Y., Kurt, A.: The solutions of time and space conformable fractional heat equations with conformable Fourier transform. Acta Univ. Sapientiae Math. 7(2), 130–140 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  73. El-Sayed, A., Salman, S.: On a discretization process of fractional-order Riccati differential equation. J. Fract. Calc. Appl. 4(2), 251–259 (2013)

    Google Scholar 

  74. Agarwal, R., El-Sayed, A., Salman, S.: Fractional-order Chua’s system: discretization, bifurcation and chaos. Adv. Differ. Equ. 1, 320 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  75. Mohammadnezhad, V., Eslami, M., Rezazadeh, H.: Stability analysis of linear conformable fractional differential equations system with time delays. Bol. Soc. Parana. Mat. 38(6), 159–171 (2020)

    Google Scholar 

  76. Xin, B., Chen, T., Liu, Y.: Synchronization of chaotic fractional-order WINDMI systems via linear state error feedback control. Math. Probl. Eng. 2010, 859685 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  77. Yavuz, M., Ozdemir, N.: European vanilla option pricing model of fractional-order without singular kernel. Fractal Fract. 2(1), 3 (2018)

    Article  MathSciNet  Google Scholar 

  78. Baskonus, H., Mekkaoui, T., Hammouch, Z., Bulut, H.: Active control of a chaotic fractional-order economic system. Entropy 17, 5771–5783 (2015)

    Article  Google Scholar 

  79. Ma, J., Ren, W.: Complexity and Hopf bifurcation analysis on a kind of fractional-order IS-LM macroeconomic system. Int. J. Bifurc. Chaos 26(11), 1650181 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  80. Huang, Y., Wang, D., Zhang, J., Guo, F.: Controlling and synchronizing a fractional-order chaotic system using stability theory of a time-varying fractional-order system. PLoS ONE 13(3), e0194112 (2018)

    Article  Google Scholar 

  81. Xin, B., Chen, T., Liu, Y.: Projective synchronization of chaotic fractional-order energy resources demand-supply systems via linear control. Commun. Nonlinear Sci. Numer. Simul. 16, 4479–4486 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  82. Almeida, R., Malinowska, A.B., Monteiro, M.T.T.: Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Methods Appl. Sci. 41(1), 336–352 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  83. Yuan, L., Yang, Q.: Parameter identification and synchronization of fractional-order chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 17(1), 305–316 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  84. Behinfaraz, R., Badamchizadeh, M., Ghiasi, A.R.: Parameter identification and synchronization of fractional-order chaotic systems. Appl. Math. Model. 40(7–8), 4468–4479 (2016)

    Article  MathSciNet  Google Scholar 

  85. Belkhatir, Z., Laleg-Kirati, T.M.: Parameters and fractional differentiation orders estimation for linear continuous-time non-commensurate fractional order systems. Syst. Control Lett. 115, 26–33 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  86. Pikulina, E., Renneboog, L., Tobler, P.: Overconfidence and investment: an experimental approach. J. Corp. Finance 43(4), 175–192 (2017)

    Article  Google Scholar 

  87. Deaves, R., Kluger, B., Miele, J.: An exploratory experimental analysis of path-dependent investment behaviors. J. Econ. Psychol. 43(4), 175–192 (2017)

    Google Scholar 

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The authors would like to express many thanks to referees for their time and efforts in reviewing this manuscript. Their helpful comments and constructive suggestions greatly improved this manuscript.

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This work is supported partly by Natural Science Foundation of Shandong Province (Grant No. ZR2016FM26), National Social Science Foundation of China (Grant No. 16FJY008), and National Natural Science Foundation of China (Grant No. 11801060).

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Xin, B., Peng, W., Kwon, Y. et al. Modeling, discretization, and hyperchaos detection of conformable derivative approach to a financial system with market confidence and ethics risk. Adv Differ Equ 2019, 138 (2019).

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