Advertisement

Advances in Difference Equations

, 2019:138 | Cite as

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

  • Baogui XinEmail author
  • Wei Peng
  • Yekyung Kwon
  • Yanqin Liu
Open Access
Research
  • 125 Downloads

Abstract

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.

Keywords

Conformable calculus Fractional-order financial system Discretization process Hyperchaotic attractor Market confidence Ethics risk 

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

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

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

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:
$$ \textstyle\begin{cases} \dot{x} = z+(y-a)x+k(w-pu),\\ \dot{y}= 1-by-x^{2}+k(w-pu),\\ \dot{z}= -x-cz+k(w-pu),\\ \dot{w}=-dxyz,\\ \dot{u}=k(w-pu), \end{cases} $$
(3)
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:
$$ \textstyle\begin{cases} T_{\alpha _{1}}x = z+(y-a)x+k(w-pu),\\ T_{\alpha _{2}}y= 1-by-x^{2}+k(w-pu), \\ T_{\alpha _{3}}z= -x-cz+k(w-pu),\\ T_{\alpha _{4}}w= -dxyz,\\ T _{\alpha _{5}}u= k(w-pu), \end{cases} $$
(4)
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
$$ T_{\alpha }^{t_{0}}f(t)=\lim_{\varepsilon \rightarrow 0} \frac{f(t+ \varepsilon (t-t_{0})^{1-\alpha })-f(t)}{\varepsilon }, $$
(5)
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
$$ I_{\alpha }^{t_{0}}f(t)= \int ^{t}_{t_{0}}(s-t_{0})^{\alpha -1}f(s) \,ds, $$
(6)
where the integral is the usual Riemann improper integral.

Lemma 1

(See [30])

Suppose the derivative order\(\alpha \in (0,1)\)and that functionsfareα-differentiable at a point\(t_{0}>0\). Then the left conformable derivative satisfies
$$ T_{\alpha }^{t_{0}}f(t)=(t-t_{0})^{1-\alpha } \frac{df(t)}{dt}. $$
(7)

Lemma 2

(Conformable Euler’s method [75])

Consider the following conformable derivative system:
$$ T_{\alpha }x(t)=f \bigl(x(t) \bigr),\quad 0\leq t\leq T, x(0)=x_{0}, $$
(8)
where\(h=\frac{T}{N}=t_{n+1}-t_{n}\), \(t_{n} = nh, n=0,1,\ldots,N\). Ifhis 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)
$$ x_{n+1}\approx x_{n} + \frac{h^{\alpha }}{\alpha }f(x_{n}). $$
(9)

3.2 Discretization process

Introducing partially piecewise constant arguments to Eq. (8), Kartal and Gurcan [59] obtained the form
$$ T_{\alpha }x(t)=f \biggl(x(t),x \biggl( \biggl[ \frac{t}{h} \biggr]h \biggr) \biggr),\quad 0\leq t \leq T, x(0)=x_{0}, $$
(10)
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):
$$ T_{\alpha }x(t)=f \biggl(x \biggl( \biggl[\frac{t}{h} \biggr]h \biggr) \biggr),\quad 0\leq t \leq T, x(0)=x_{0}. $$
(11)

Theorem 1

(Conformable discretization by piecewise constant approximation)

According to system (11), we obtain the following discretization of Eq. (8):
$$ x_{n+1}= x_{n} + \frac{h^{\alpha }}{\alpha }f(x_{n}), $$
(12)
where\(x_{n}\)denotes\(x_{n}(nh)\).

Proof

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
$$ \frac{dx(t)}{dt}=(t-nh)^{\alpha -1}f \bigl(x(nh) \bigr),\quad 0 \leq t\leq T, x(0)=x_{0}. $$
(13)

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
$$ \frac{dx(t)}{dt}=(t-0)^{\alpha -1}f(x_{0}),\quad t\in [0,h), $$
(14)
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
$$ \frac{dx(t)}{dt}=(t-h)^{\alpha -1}f \bigl(x_{1}(h) \bigr),\quad t\in [h,2h), $$
(15)
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:
$$ \textstyle\begin{cases} x_{n+1} = x_{n} + \frac{h^{\alpha _{1}}}{\alpha _{1}} (z+(y-a)x+k(w-pu)), \\ y_{n+1}= y_{n} + \frac{h^{\alpha _{2}}}{\alpha _{2}} (1-by-x^{2}+k(w-pu)), \\ z_{n+1}= z_{n} + \frac{h^{\alpha _{3}}}{\alpha _{3}} (-x-cz+k(w-pu)), \\ w_{n+1}= w_{n} + \frac{h^{\alpha _{4}}}{\alpha _{4}} (-dxyz),\\ u _{n+1}= u_{n} + \frac{h^{\alpha _{5}}}{\alpha _{5}} k(w-pu). \end{cases} $$
(16)

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

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

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

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.

Notes

Acknowledgements

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.

Availability of data and materials

Not applicable.

Authors’ contributions

The authors have made the same contribution. All authors read and approved the final manuscript.

Funding

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).

Competing interests

The authors declare that they have no competing interests.

References

  1. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 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) CrossRefGoogle Scholar
  4. 4.
    Li, C., Chen, G.: Chaos and hyperchaos in the fractional-order Rossler equations. Physica A 341, 55–61 (2004) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Rajagopal, K., Karthikeyan, A., Duraisamy, P.: Hyperchaotic chameleon: fractional-order FPGA implementation. Complexity 2017, 8979408 (2017) MathSciNetzbMATHGoogle Scholar
  6. 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) MathSciNetCrossRefGoogle Scholar
  7. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Mou, J., Sun, K.: Characteristic analysis of fractional-order 4D hyperchaotic memristive circuit. Math. Probl. Eng. 2017, 2313768 (2017) MathSciNetCrossRefGoogle Scholar
  9. 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) MathSciNetzbMATHGoogle Scholar
  10. 10.
    Huang, X., Zhao, Z., Wang, Z., et al.: Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing 94(3), 13–21 (2012) CrossRefGoogle Scholar
  11. 11.
    Huang, D., Li, H.: Theory and Method of the Nonlinear Economics. Sichuan University Press, Chengdu (1993) Google Scholar
  12. 12.
    Chen, W.: Nonlinear dynamics and chaos in a fractional-order financial system. Chaos Solitons Fractals 36, 1305–1314 (2008) CrossRefGoogle Scholar
  13. 13.
    Wang, Z., Huang, X., Shen, H.: Control of an uncertain fractional-order economic system via adaptive sliding mode. Neurocomputing 83, 83–88 (2012) CrossRefGoogle Scholar
  14. 14.
    Mircea, G., Neamtu, M., Bundau, O., Opris, D.: Uncertain and stochastic financial models with multiple delays. Int. J. Bifurc. Chaos 22, 1250131 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Xin, B., Chen, T., Ma, J.: Neimark–Sacker bifurcation in a discrete-time financial system. Discrete Dyn. Nat. Soc. 2010, 405639 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Yu, H., Cai, G., Li, Y.: Dynamic analysis and control of a new hyperchaotic finance system. Chaos Solitons Fractals 45, 1048–1057 (2012) zbMATHCrossRefGoogle Scholar
  17. 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) MathSciNetGoogle Scholar
  18. 18.
    Xin, B., Zhang, J.: Finite-time stabilizing a fractional-order chaotic financial system with market confidence. Nonlinear Dyn. 79(2), 1399–1409 (2015) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 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) CrossRefGoogle Scholar
  20. 20.
    Wang, S., Wu, R.: Dynamic analysis of a 5D fractional-order hyperchaotic system. Int. J. Control. Autom. Syst. 15(3), 1003–1010 (2017) CrossRefGoogle Scholar
  21. 21.
    Zheng, R., Jiang, X.: Spectral methods for the time-fractional Navier–Stokes equation. Appl. Math. Lett. 91, 194–200 (2019) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Xu, H., Jiang, X.: Creep constitutive models for viscoelastic materials based on fractional derivatives. Comput. Math. Appl. 73(6), 1377–1384 (2017) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Yang, X., Qi, H., Jiang, X.: Numerical analysis for electroosmotic flow of fractional Maxwell fluids. Appl. Math. Lett. 78, 1–8 (2018) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 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) CrossRefGoogle Scholar
  26. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 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) CrossRefGoogle Scholar
  29. 29.
    Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  31. 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) MathSciNetCrossRefGoogle Scholar
  32. 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) CrossRefGoogle Scholar
  33. 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. 34.
    Bohner, M., Hatipoglu, V.: Dynamic cobweb models with conformable fractional derivatives. Nonlinear Anal. Hybrid Syst. 32, 157–167 (2019) MathSciNetCrossRefGoogle Scholar
  35. 35.
    Tarasov, V.: No nonlocality. No fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 62, 157–163 (2018) MathSciNetCrossRefGoogle Scholar
  36. 36.
    Rosales, J., Godínez, F., Banda, V.: Analysis of the Drude model in view of the conformable derivative. Optik 178, 1010–1015 (2019) CrossRefGoogle Scholar
  37. 37.
    Akbulut, A., Melike, K.: Auxiliary equation method for time-fractional differential equations with conformable derivative. Comput. Math. Appl. 75(3), 876–882 (2018) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Martinez, L., Rosales, J., Carreno, C.: Electrical circuits described by fractional conformable derivative. Int. J. Circuit Theory Appl. 46(5), 1091–1100 (2018) CrossRefGoogle Scholar
  39. 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) MathSciNetzbMATHGoogle Scholar
  40. 40.
    Korkmaz, A.: Explicit exact solutions to some one-dimensional conformable time fractional equations. Waves Random Complex Media 29(1), 124–137 (2019) MathSciNetCrossRefGoogle Scholar
  41. 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) MathSciNetCrossRefGoogle Scholar
  42. 42.
    He, S., Banerjee, S., Yan, B.: Chaos and symbol complexity in a conformable fractional-order memcapacitor system. Complexity 2018, 4140762 (2018) zbMATHGoogle Scholar
  43. 43.
    Lu, Y., Yang, L., Liu, L.: Volatility spillovers between crude oil and agricultural commodity markets since the financial crisis. Sustainability 11, 396 (2019) CrossRefGoogle Scholar
  44. 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) CrossRefGoogle Scholar
  45. 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) CrossRefGoogle Scholar
  46. 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) CrossRefGoogle Scholar
  47. 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) CrossRefGoogle Scholar
  48. 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) CrossRefGoogle Scholar
  49. 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) CrossRefGoogle Scholar
  50. 50.
    Rasmussen, D.: Adam Smith on what is wrong with economic inequality. Am. Polit. Sci. Rev. 110(2), 342–352 (2016) CrossRefGoogle Scholar
  51. 51.
    Eslami, M., Rezazadeh, H.: The first integral method for Wu–Zhang system with conformable time-fractional derivative. Calcolo 53(3), 475–485 (2016) MathSciNetzbMATHCrossRefGoogle Scholar
  52. 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) CrossRefGoogle Scholar
  53. 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) CrossRefGoogle Scholar
  54. 54.
    Unal, E., Gokdogan, A.: Solution of conformable fractional ordinary differential equations via differential transform method. Optik 128, 264–273 (2017) CrossRefGoogle Scholar
  55. 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) CrossRefGoogle Scholar
  56. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Kaplan, M.: Applications of two reliable methods for solving a nonlinear conformable time-fractional equation. Opt. Quantum Electron. 49(9), 312 (2017) CrossRefGoogle Scholar
  58. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Kartal, S., Gurcan, F.: Discretization of conformable fractional differential equations by a piecewise constant approximation. Int. J. Comput. Math. 25, 1–2 (2018) CrossRefGoogle Scholar
  60. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  61. 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) CrossRefGoogle Scholar
  62. 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) CrossRefGoogle Scholar
  63. 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) MathSciNetCrossRefGoogle Scholar
  64. 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. 65.
    Zhong, W., Wang, L.: Basic theory of initial value problems of conformable fractional differential equations. Adv. Differ. Equ. 1, 321 (2018) MathSciNetzbMATHCrossRefGoogle Scholar
  66. 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. 67.
    Yaslan, H.: Numerical solution of the conformable space-time fractional wave equation. Chin. J. Phys. 56(6), 2916–2925 (2018) MathSciNetCrossRefGoogle Scholar
  68. 68.
    Kurt, A., Cenesiz, Y., Tasbozan, O.: On the solution of Burgers’ equation with the new fractional derivative. Open Phys. 13, 355–360 (2015) CrossRefGoogle Scholar
  69. 69.
    Khalil, R., Abu-Shaab, H.: Solution of some conformable fractional differential equations. Int. J. Pure Appl. Math. 103(4), 667–673 (2015) CrossRefGoogle Scholar
  70. 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. 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) MathSciNetCrossRefGoogle Scholar
  72. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  73. 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. 74.
    Agarwal, R., El-Sayed, A., Salman, S.: Fractional-order Chua’s system: discretization, bifurcation and chaos. Adv. Differ. Equ. 1, 320 (2013) MathSciNetzbMATHCrossRefGoogle Scholar
  75. 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. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    Yavuz, M., Ozdemir, N.: European vanilla option pricing model of fractional-order without singular kernel. Fractal Fract. 2(1), 3 (2018) MathSciNetCrossRefGoogle Scholar
  78. 78.
    Baskonus, H., Mekkaoui, T., Hammouch, Z., Bulut, H.: Active control of a chaotic fractional-order economic system. Entropy 17, 5771–5783 (2015) CrossRefGoogle Scholar
  79. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  80. 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) CrossRefGoogle Scholar
  81. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  82. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Yuan, L., Yang, Q.: Parameter identification and synchronization of fractional-order chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 17(1), 305–316 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  84. 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) MathSciNetCrossRefGoogle Scholar
  85. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Pikulina, E., Renneboog, L., Tobler, P.: Overconfidence and investment: an experimental approach. J. Corp. Finance 43(4), 175–192 (2017) CrossRefGoogle Scholar
  87. 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

Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.College of Economics and ManagementShandong University of Science and TechnologyQingdaoChina
  2. 2.Division of Global Business AdministrationDongseo UniversityBusanKorea
  3. 3.College of Mathematical SciencesDezhou UniversityDezhouChina

Personalised recommendations