Taming Hyperchaos with Exact Spectral Derivative Discretization Finite Difference Discretization of a Conformable Fractional Derivative Financial System with Market Confidence and Ethics Risk

Abstract

Four discrete models, using the exact spectral derivative discretization finite difference (ESDDFD) method, are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) method and found to be hyperchaotic, and the CEFD method was recently shown to be valid only at fractional index $\alpha =1$, the source of the hyperchaos is in question. Through numerical experiments, illustration is presented that the hyperchaos previously detected is, in part, an artifact of the CEFD method, as it is absent from the ESDDFD models.

1. Introduction

Hyperchaotic systems [1,2]—typically defined as systems with at least two positive Lyapunov exponents [3,4,5]—of a fractional-order have been investigated in many contexts, such as systems of Rössler [6] or Lorenz [7] type, those with flux controlled memristors [8] or realized in circuits [9,10,11], those arising from cellular neural networks [12], and financial systems [13]. As recounted in [13], a nonlinear financial system depicting the relationship among interest rates, investments, prices, and savings was first introduced by Huang and Li [14]. It was extended to fractional-order in Chen [15], to uncertain fractional-order form in Wang et al. [16], to delayed form in Mircea et al. [17], and to discrete form in Xin et al. [18]. The average profit margin was added as a variable in Yu et al. [19], while investment incentive and market confidence were introduced in Xin et al. [20,21]. Xin and Zhang [21] updated the 3-dimensional Huang and Li [8] model to a 4-dimensional one by accounting for market confidence and [13] incorporated ethics risk to obtain a 5-dimensional system, which was then fractionalized to obtain the following fractional-order financial system considered in [13]:

$$\begin{array}{c}{T}_t^{{\alpha }_1}x=z+\left(y-a\right)x+k\left(w-pu\right)\\ T_t^{{\alpha }_2}y=1-by-x^2+k\left(w-pu\right)\\ T_t^{{\alpha }_3}z=-x-cz+k\left(w-pu\right)\\ T_t^{{\alpha }_4}w=-dxyz\\ T_t^{{\alpha }_5}u=k\left(w-pu\right)\end{array}$$

(1)

where $\alpha =\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5\right)$ is subject to ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5\in \left(0, 1\right)$, and $T_t^{{\alpha }_i},1\le i\le 5,$ denotes the conformable fractional derivative of order ${\alpha }_i$. The variables x, y, z, w, and u are the interest rate, investment demand, price index, market confidence, and ethics risk, respectively; the parameters a, b, and c are the saving amount, cost per investment, and demand elasticity of commercial markets, respectively, and a, b, c ≥ 0; k, p, d are impact factors associated with ethics risk.

Since analytic solutions do not exist, suitable numerical schemes to obtain solutions of the conformable derivative financial system are needed. Though there are several methods to solve a conformable derivative system [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47], these are too complex for many people. Inspired by the discretization process for the Caputo derivative for Ricatti equations [45] and Chua systems [46], the conformable Euler’s finite difference (CEFD) method [47] for the 5-dimensional fractional-order financial system is proposed in [13]. Numerical experiments with the resulting discrete model were conducted to detect a hyperchaotic attractor of the system. However, the standard Euler discretization of integer-order systems, such as studied in [13], is known to induce (see, e.g., [48,49]) numerical instabilities and spurious behavior where none exist in the continuous system. Moreover, the CEFD method has recently been shown [50] to be valid only for $\alpha =1$ and is, therefore, not a valid fractional method. Nonstandard finite difference (NSFD) models have extensively [48] been shown to eliminate induced chaos; the exact spectral derivative discretization finite difference (ESDDFD) methodology is a novel extension, developed in the context of advection–reaction–diffusion equations [51], of the NSFD method to non-integer derivatives [52].

It is, therefore, natural to ask whether some of the hyperchaotic behavior detected in the fractional financial system is an artifact of the method and whether ESDDFD models can be constructed to eliminate such induced hyperchaos. The purpose of the present study is to investigate this question—in particular, the effects of the discretization of the derivative and that of non-linear terms. To this end, the following four discrete models using the ESDDFD method are constructed for the system (1) and the bifurcation experiments of [13] are repeated with the new models.

$$\begin{array}{c}\frac{x_{k+1}-x_k}{{\varphi }_j\left(h,{\alpha }_1\right)}=F_i^x\left(x_k,y_k,z_k,u_k,w_k\right)\\ \frac{y_{k+1}-y_k}{{\varphi }_j\left(h,{\alpha }_2\right)}=F_i^y\left(x_k,y_k,z_k,u_k,w_k\right)\\ \frac{z_{k+1}-z_k}{{\varphi }_j\left(h,{\alpha }_3\right)}=-x_k-c{z}_k+k\left(w_k-p{u}_k\right)\\ \frac{u_{k+1}-u_k}{{\varphi }_j\left(h,{\alpha }_5\right)}=k\left(w_k-p{u}_k\right)\\ \frac{w_{k+1}-w_k}{{\varphi }_j\left(h,{\alpha }_4\right)}=F_i^w\left(x_k,y_k,z_k,z_k\right)\end{array}$$

(2)

$i=1, 2$ and $j=1, 2$, where:

$$\begin{array}{c}{F}_1^x\left(x_k,y_k,z_k,u_k,w_k\right)=z_k+\left(y_{k+1}-\mathrm{a}\right)x_k+k\left(w_k-p{u}_k\right)\\ F_1^y\left(x_k,y_k,z_k,u_k,w_k\right)=1-b{y}_k-x_k{x}_k+k\left(w_k-p{u}_k\right)\\ F_1^w\left(x_k,y_k,z_k,z_k\right)=-\frac{d}{2}{x}_k{y}_k\left(z_k+z_k\right)\\ F_2^x=F_1^x\left(x_k,y_{k+1},z_k,u_k,w_k\right)\\ F_2^y=F_1^y\left(x_k,y_{k+1},z_k,u_k,w_k\right)\\ F_2^w=F_1^w\left(x_k,y_{k+1},z_k,z_{k+1}\right)\end{array}$$

The remainder of this article is organized as follows. In Section 2, the ESDDFD fundamentals, a description of the model (1), and the CEFD model from [3] are presented. Section 3 presents the construction of the denominator functions, ${\varphi }_j\left(h,{\alpha }_m\right),1\le m\le 5$, for the ESDDFD model (2) and compares sub-models of (2) with corresponding CEFD sub-models. In Section 4, experimental results of hyperchaotic attractor detection from the proposed financial system using both methods are presented. Concluding remarks in Section 5 close the paper.

2. Preliminaries

2.1. The Conformable Derivative ESDDFD Discrete Model Construction Fundamentals

While the Riemann–Liouville, Caputo, Atangana–Baleanu, and Grünwald–Letnikov fractional derivatives [53,54,55,56,57,58,59,60] are widely used in various applications, their definitions lack the chain rule, a classical derivative property satisfied by the conformable fractional derivative (CFD) [61,62,63] and its various extensions (see e.g., [64]). A financial system with a market confidence and ethics risk model was recently [13] added to the many existing applications of the CFD in various scientific fields [22,65,66,67,68,69,70,71,72,73,74].

2.2. The Conformable Derivative Hyperchaotic Financial System and Its CEFD Model

The conformable fractional derivative financial system model (1) is based on a successive addition of various factors, starting with the Huang and Li [8] nonlinear financial system model:

$$\begin{array}{c}{x}^{\prime }=z+\left(y-a\right)x\\ y^{\prime }=1-by-x^2\\ z^{\prime }=-x-cz\end{array}$$

(3)

modeling the interaction of interest rate ($x$), investment demand $\left(y\right)$, and price index $\left(z\right)$; the variables and parameters are the same as in (1). Model (3) was extended, by Xin and Zhang [15], to account for market confidence:

$$\begin{array}{c}{x}^{\prime }=z+\left(y-a\right)x+m_{1}w\\ y^{\prime }=1-by-x^2+m_{2}w\\ z^{\prime }=-x-cz+m_{3}w\\ w^{\prime }=-dxyz\end{array}$$

(4)

where $m_1,m_2,m_3$ are the impact factors associated with market confidence $\left(w\right)$; the remaining variables and parameters are the same as in (3). Model (1) is the fractionalization, predicated on the practice that fractional-order economic systems [15,75,76,77,78,79] can generalize their integer-order forms [14,80,81], of the following extension of (4) in [13] to account for both market confidence and ethics risk $\left(u\right)$:

$$\begin{array}{c}{x}^{\prime }=z+\left(y-a\right)x+k\left(w-pu\right)\\ y^{\prime }=1-by-x^2+k\left(w-pu\right)\\ z^{\prime }=-x-cz+k\left(w-pu\right)\\ w^{\prime }=-dxyz\\ u^{\prime }=k\left(w-pu\right)\end{array}$$

(5)

When α = (1, 1, 1, 1, 1), system (1) degenerates to system (5); in the absence of ethics risk, (5) reduces to (4); in the absence of market confidence, (4) reduces to (3). In these three cases, therefore, any discrete method developed for (1) must reduce to that of the three respective reduced systems. Chaotic behavior for both the CEFD and ESDDFD models will be numerically investigated in Section 4 for (1) as well as the reduced fractional counterpart of system (3).

The following discrete model was obtained in [13] from the CEFD method and used to numerically investigate hyperchaos of the system (1):

$$\begin{array}{c}{x}_{k+1}=x_k+\frac{h^{{\alpha }_1}}{{\alpha }_1}\left(z_k+\left(y_k-a\right)x_k+k\left(w_k-p{u}_k\right)\right)\\ y_{k+1}=y_k+\frac{h^{{\alpha }_2}}{{\alpha }_2}\left(1-b{y}_k-x_k{x}_k+k\left(w_k-p{u}_k\right)\right)\\ z_{k+1}=z_k-\frac{h^{{\alpha }_3}}{{\alpha }_3}\left(x_k+c{z}_k-k\left(w_k-p{u}_k\right)\right)\\ u_{k+1}=u_k+\frac{h^{{\alpha }_5}}{{\alpha }_5}k\left(w_k-p{u}_k\right)\\ w_{k+1}=w_k-\frac{h^{{\alpha }_4}}{{\alpha }_4}d{x}_k{y}_k{z}_k\end{array}$$

(6)

3. ESDDFD Discretization of the Conformable Derivative System and Its Reductions

In the ESDDFD and NSFD discretization methodologies, the first step is to consider a linear sub-system whose exact or best scheme can be constructed. Such a sub-system, in this case, is the following:

$$\begin{array}{c}{T}_t^{{\alpha }_1}x=-ax,\\ T_t^{{\alpha }_2}y=-by,\\ T_t^{{\alpha }_3}z=-cz,\\ T_t^{{\alpha }_4}w=0,\\ T_t^{{\alpha }_5}u=-kpu,\end{array}$$

(7)

which has only positive solutions for any positive initial data. The exact discretization of (7), which has a solution identical to that of (7), is as follows:

$$\begin{array}{c}\frac{x_{k+1}-x_k}{{\varphi }_1\left(h,{\alpha }_1\right)}=-a{x}_k,\\ \frac{y_{k+1}-y_k}{{\varphi }_1\left(h,{\alpha }_2\right)}=-b{y}_k,\\ \frac{z_{k+1}-z_k}{{\varphi }_1\left(h,{\alpha }_3\right)}=-c{z}_k,\\ \frac{w_{k+1}-w_k}{{\varphi }_1\left(h,{\alpha }_4\right)}=0,\\ \frac{u_{k+1}-u_k}{{\varphi }_1\left(h,{\alpha }_5\right)}=-kp{u}_k,\end{array}$$

(8)

where the nonstandard denominators ${\varphi }_1\left(h,{\alpha }_i\right),1\le i\le 5,$are given by:

$$\begin{array}{c}{\varphi }_1\left(h,{\alpha }_i\right)=\frac{1}{Q_i}\left(1-e^{-\frac{Q_i}{{\alpha }_i}\left[{\left(t+h\right)}^{{\alpha }_i}-t^{{\alpha }_i}\right]}\right), \\ \mathrm{with} Q_1=a, Q_2=b, Q_3=c, Q_4=0, Q_5=kp.\end{array}$$

Since (1) reduces to (7), any valid discrete model for (1) must be reducible to one consistent with its exact discretization—that is, (8). By comparison, a reduction of the CEFD model (6) to the sub-system (7) yields the following discrete sub-system:

$$\begin{array}{c}{x}_{k+1}=x_k-\frac{h^{{\alpha }_1}}{{\alpha }_1}a{x}_k,\\ y_{k+1}=y_k-\frac{h^{{\alpha }_2}}{{\alpha }_2}b{y}_k,\\ z_{k+1}=z_k-\frac{h^{{\alpha }_3}}{{\alpha }_3}c{z}_k,\\ w_{k+1}=w_k+Q_4\frac{h^{{\alpha }_4}}{{\alpha }_4}{w}_k,\\ u_{k+1}=u_k-\frac{h^{{\alpha }_5}}{{\alpha }_5}kp{u}_k,\end{array}$$

(9)

which is positive only if the following condition is satisfied: $\left(1-\frac{h^{{\alpha }_i}}{{\alpha }_i}{Q}_i\right)\ge 0, 1\le i\le 5$, with the $Q_i$ as in (8); such conditional positivity is known to induce chaotic behavior. All of the sub-Equations (8) are of the form:

$$T_t^{\alpha }P=-\lambda P,$$

whose CEFD scheme is:

$$P_{k+1}=P_k-\frac{h^{\alpha }}{\alpha }\lambda P_k,$$

which has been conclusively shown in [50] to be valid only for $\alpha =1$.

It is shown in [50] that a modified CEFD (MCEFD) may be obtained from the following alternate CFD definition, which is equivalent to the fractional change of variables in the integer-valued derivative (see also [82]):

Definition 1.

Given a real-valued function on $\left[0,\infty \right)$, the conformable fractional derivative has the following alternative definition:

$${}_0{}^{C}T{}_t^{\alpha }\left[f\left(t\right)\right]\equiv \underset{h\to 0}{\lim }{}_0{}^{CFD}∆{}_t^{\alpha }\left[f\left(t\right)\right]=\alpha \underset{h\to 0}{\lim }\frac{f\left(t+h\right)-f\left(t\right)}{\left[{\left(t+h\right)}^{\alpha }-t^{\alpha }\right] },$$

where ${}_0{}^{C}T{}_t^{\alpha }\left[f\left(0\right)\right]$isunderstood to mean ${}_0{}^{C}T{}_t^{\alpha }\left[f\left(0\right)\right]=\underset{t\to 0^{+}}{\lim }{}_0{}^{C}T{}_t^{\alpha }\left[f\left(t\right)\right]$.

Therefore, the Euler scheme, resulting from the MCFED, is the same as that given in Equation (8), only with the denominator of:

$${\varphi }_1\left(h,{\alpha }_i\right)=\frac{1}{Q_i}\left(1-e^{-\frac{Q_i}{{\alpha }_i}\left[{\left(t+h\right)}^{{\alpha }_i}-t^{{\alpha }_i}\right]}\right)$$

replaced by:

$${\varphi }_2\left(h,{\alpha }_i\right)=\frac{1}{{\alpha }_i}\left[{\left(t+h\right)}^{{\alpha }_i}-t^{{\alpha }_i}\right], 1\le i\le 5,$$

which is equivalent to replacing $h^{{\alpha }_i}$ by ${\alpha }_i{\varphi }_2\left(h,{\alpha }_i\right)$ in the CEFD scheme (9).

To enable the assessment of the effect of the denominators ${\varphi }_j\left(h,{\alpha }_i\right), j=1,2$, the following schemes are compared:

$$\begin{array}{c}\frac{x_{k+1}-x_k}{{\varphi }_j\left(h,{\alpha }_1\right)}= z_{k }+\left(y_k-\mathrm{a}\right)x_k , \\ \frac{y_{k+1}-y_k}{{\varphi }_j\left(h,{\alpha }_2\right)}=1-b{y}_k-{\left(x_k\right)}^2,\\ \frac{z_{k+1}-z_k}{{\varphi }_j\left(h,{\alpha }_3\right)}=-x_k-c{z}_k, j=1,2.\end{array}$$

(10)

To enable the assessment of the effect of the non-local discretization of nonlinear terms, the following schemes are compared:

$$\begin{array}{c}\frac{x_{k+1}-x_k}{{\varphi }_j\left(h,{\alpha }_1\right)}=z_k+\left(y_{k+1}-\mathrm{a}\right)x_k,\\ \frac{y_{k+1}-y_k}{{\varphi }_j\left(h,{\alpha }_2\right)}=1-b{y}_k-x_{k+1}{x}_k,\\ \frac{z_{k+1}-z_k}{{\varphi }_j\left(h,{\alpha }_3\right)}=-x_k-c{z}_k, j=1, 2.\end{array}$$

(11)

The terms $\left(y-\mathrm{a}\right)x$, and $x^2$ are discretized non-locally as, respectively,$\left(y_{k+1}-\mathrm{a}\right)x_k$ and $x_{k+1}{x}_k$, while discretization of the terms $z$ (in the first Equation of (10)) and $x$ (in the third as $z_k$ and $x_k$) ensures respective consistency with the terms $cz$ in the third and $ax$ in the first Equation of (11) in the cases where $c=1$ and $a=1$.

By comparison, the scheme obtained through a reduction of the CEFD model (6) to its 3-dimensional sub-system (3) yields the following discrete sub-system:

$$\begin{array}{c}{x}_{k+1}=x_k+\frac{h^{{\alpha }_1}}{{\alpha }_1}\left(z_k+\left(y_k-\mathrm{a}\right)x_k\right)\\ y_{k+1}=y_k+\frac{h^{{\alpha }_2}}{{\alpha }_2}\left(1-b{y}_k-x_k{x}_k\right)\\ z_{k+1}=z_k+\frac{h^{{\alpha }_3}}{{\alpha }_3}\left(-x_k-c{z}_k\right).\end{array}$$

(12)

Since system (12) reduces to the $x-y-z$ sub-system of (9), which suffers from induced chaos, it is to be expected that it too suffers the same, which will be numerically investigated in the next section.

The ESDDFD models (2) are then obtained by discretizing $k\left(w-pu\right)$ as $k\left(w_k-p{u}_k\right)$ to ensure consistency with (8) and then discretizing $xyz$ non-locally as either $\frac{1}{2}{x}_k{y}_k\left(z_k+z_k\right)$ or $\frac{1}{2}{x}_k{y}_{k+1}\left(z_k+z_{k+1}\right)$, where the form $x_k{y}_{k+1}$ is used to match the $xy$ term in the $x$-equation.

$$\begin{array}{c}\frac{x_{k+1}-x_k}{{\varphi }_j\left(h,{\alpha }_1\right)}=z_k+\left(y_k-\mathrm{a}\right)x_k+k\left(w_k-p{u}_k\right)\\ \frac{y_{k+1}-y_k}{{\varphi }_j\left(h,{\alpha }_2\right)}=1-b{y}_k-{\left(x_k\right)}^2+k\left(w_k-p{u}_k\right)\\ \frac{z_{k+1}-z_k}{{\varphi }_j\left(h,{\alpha }_3\right)}=-x_k-c{z}_k+k\left(w_k-p{u}_k\right)\\ \frac{u_{k+1}-u_k}{{\varphi }_j\left(h,{\alpha }_5\right)}=k\left(w_k-p{u}_k\right)\\ \frac{w_{k+1}-w_k}{{\varphi }_j\left(h,{\alpha }_4\right)}=-\frac{d}{2}{x}_k{y}_k\left(z_k+z_k\right), j=1,2.\end{array}$$

(13)

and

$$\begin{array}{c}\frac{x_{k+1}-x_k}{{\varphi }_j\left(h,{\alpha }_1\right)}=z_k+\left(y_{k+1}-\mathrm{a}\right)x_k+k\left(w_k-p{u}_k\right)\\ \frac{y_{k+1}-y_k}{{\varphi }_j\left(h,{\alpha }_2\right)}=1-b{y}_k-x_{k+1}{x}_k+k\left(w_k-p{u}_k\right)\\ \frac{z_{k+1}-z_k}{{\varphi }_j\left(h,{\alpha }_3\right)}=-x_k-c{z}_k+k\left(w_k-p{u}_k\right)\\ \frac{u_{k+1}-u_k}{{\varphi }_j\left(h,{\alpha }_5\right)}=k\left(w_k-p{u}_k\right)\\ \frac{w_{k+1}-w_k}{{\varphi }_j\left(h,{\alpha }_4\right)}=-\frac{d}{2}{x}_k{y}_{k+1}\left(z_k+z_{k+1}\right), j=1,2.\end{array}$$

(14)

The schemes (13) are explicit and can be explicitly solved for each $j=1,2$, in the order $x_{k+1},y_{k+1},z_{k+1},u_{k+1},w_{k+1}$ to obtain the following:

$$\begin{array}{c}{x}_{k+1}=x_k+{\varphi }_j\left(h,{\alpha }_1\right)\left[z_k+\left(y_k-\mathrm{a}\right)x_k+k\left(w_k-p{u}_k\right)\right]\\ y_{k+1}=y_k+{\varphi }_j\left(h,{\alpha }_2\right)\left[1-b{y}_k-{\left(x_k\right)}^2+k\left(w_k-p{u}_k\right)\right]\\ z_{k+1}=z_k-{\varphi }_j\left(h,{\alpha }_3\right)\left[x_k+c{z}_k-k\left(w_k-p{u}_k\right)\right]\\ u_{k+1}=u_k+{\varphi }_j\left(h,{\alpha }_5\right)\left[k\left(w_k-p{u}_k\right)\right]\\ w_{k+1}=w_k-\frac{d}{2}{\varphi }_j\left(h,{\alpha }_4\right)x_k{y}_k\left(z_k+z_k\right), j=1,2.\end{array}$$

(15)

While implicit, the schemes (14) can be explicitly solved for each $j=1,2$ in the order $u_{k+1},z_{k+1},x_{k+1},y_{k+1},$ $w_{k+1}$ to obtain the following:

$$\begin{array}{c}{u}_{k+1}=u_k+{\varphi }_j\left(h,{\alpha }_5\right)\left[k\left(w_k-p{u}_k\right)\right]\\ z_{k+1}=z_k-{\varphi }_j\left(h,{\alpha }_3\right)\left[x_k+c{z}_k-k\left(w_k-p{u}_k\right)\right]\\ x_{k+1}=\frac{1}{\left[1+{\varphi }_j\left(h,{\alpha }_1\right)x_k{\varphi }_j\left(h,{\alpha }_2\right)x_k\right]}\left(x_k+{\varphi }_j\left(h,{\alpha }_1\right)x_k\left\{y_k+{\varphi }_j\left(h,{\alpha }_2\right)\left[1-b{y}_k+k\left(w_k-p{u}_k\right)\right]\right\}\right)\\ +\frac{1}{\left[1+\varphi \left(h,{\alpha }_1\right)x_k\varphi \left(h,{\alpha }_2\right)x_k\right]}{\varphi }_j\left(h,{\alpha }_1\right)\left[z_k-\mathrm{a}{x}_k+k\left(w_k-p{u}_k\right)\right]\\ \\ w_{k+1}=w_k-{\varphi }_j\left(h,{\alpha }_4\right)\frac{d}{2}{x}_k{y}_{k+1}\left(z_k+z_{k+1}\right)\end{array}$$

4. Numerical Experiments

In this section, hyperchaos detection experiments are conducted, parallel to those of [13], by varying the parameters related to ethics risk, such as ${\alpha }_5$, the confidence factor k, and the risk factor p, in the CEFD and ESDDFD models and their reductions. The following parameters and initial point values are fixed following [1]: $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. Three-Dimensional Systems Comparison

There were no experiments performed in [13] for this case. Simulations for both the ESDDFD model (11) and the CEFD model (12) are performed with the same parameters. The following models (16)–(19), obtained through the ESDDFD method,

$$\begin{array}{c}\frac{x_{k+1}-x_k}{\frac{1}{0.8}\left[1-e^{\frac{-0.8}{0.3}\left[{\left(t+h\right)}^{0.3}-t^{0.3}\right]}\right]}=z_k+\left(y_k-0.8\right)x_k,\\ \frac{y_{k+1}-y_k}{\frac{1}{0.6}\left[1-e^{\frac{-0.6}{0.5}\left[{\left(t+h\right)}^{0.5}-t^{0.5}\right]}\right]}=1-0.6y_k-{\left(x_k\right)}^2,\\ \frac{z_{k+1}-z_k}{\left[1-e^{\frac{-1}{0.6}\left[{\left(t+h\right)}^{0.6}-t^{0.6}\right]}\right]}=-x_k-z_k,\end{array}$$

(16)

$$\begin{array}{c}\frac{x_{k+1}-x_k}{\frac{1}{0.3}\left[{\left(t+h\right)}^{0.3}-t^{0.3}\right]}=z_k+\left(y_k-0.8\right)x_k,\\ \frac{y_{k+1}-y_k}{\frac{1}{0.5}\left[{\left(t+h\right)}^{0.5}-t^{0.5}\right]}=1-0.6y_k-{\left(x_k\right)}^2,\\ \frac{z_{k+1}-z_k}{\frac{1}{0.6}\left[{\left(t+h\right)}^{0.6}-t^{0.6}\right]}=-x_k-z_k,\end{array}$$

(17)

$$\begin{array}{c}\frac{x_{k+1}-x_k}{\frac{1}{0.8}\left[1-e^{\frac{-0.8}{0.3}\left[{\left(t+h\right)}^{0.3}-t^{0.3}\right]}\right]}=z_k+\left(y_{k+1}-0.8\right)x_k,\\ \frac{y_{k+1}-y_k}{\frac{1}{0.6}\left[1-e^{\frac{-0.6}{0.5}\left[{\left(t+h\right)}^{0.5}-t^{0.5}\right]}\right]}=1-0.6y_k-x_{k+1}{x}_k,\\ \frac{z_{k+1}-z_k}{\left[1-e^{\frac{-1}{0.6}\left[{\left(t+h\right)}^{0.6}-t^{0.6}\right]}\right]}=-x_k-z_k,\end{array}$$

(18)

$$\begin{array}{c}\frac{x_{k+1}-x_k}{\frac{1}{0.3}\left[{\left(t+h\right)}^{0.3}-t^{0.3}\right]}=z_k+\left(y_{k+1}-0.8\right)x_k,\\ \frac{y_{k+1}-y_k}{\frac{1}{0.5}\left[{\left(t+h\right)}^{0.5}-t^{0.5}\right]}=1-0.6y_k-x_{k+1}{x}_k,\\ \frac{z_{k+1}-z_k}{\frac{1}{0.6}\left[{\left(t+h\right)}^{0.6}-t^{0.6}\right]}=-x_k\end{array}$$

(19)

are compared to (20), obtained through the CEFD method:

$$\begin{array}{c}{x}_{k+1}-x_k+\frac{h^{0.3}}{0.3}\left(z_k+\left(y_k-0.8\right)x_k\right),\\ y_{k+1}=y_k+\frac{h^{0.5}}{0.5}\left(1-0.6y_k-x_k{x}_k\right),\\ z_{k+1}=z_k+\frac{h^{0.6}}{0.6}\left(x_k-z_k\right).\end{array}$$

(20)

While bifurcations can be seen in Figure 1a for the CEFD model, they are absent from the results of the ESDDFD models, Figure 1b–e.

4.2. Five-Dimensional Systems Comparison: Varying ${\alpha }_5, k$, and $p$

For this case, experiments performed in [13] are performed with the same parameters for models obtained through the ESDDFD method, for the various cases and values of $\left({\alpha }_5, k, p\right)$ used in [13]. Model (21) from the CEFD method,

$$\begin{array}{c}{x}_{k+1}=x_k+\frac{h^{0.3}}{0.3}\left(z_k+\left(y_k-0.8\right)x_k+k\left(w_k-p{u}_k\right)\right)\\ y_{k+1}=y_k+\frac{h^{0.5}}{05}\left(1-0.6y_k-x_k{x}_k+k\left(w_k-p{u}_k\right)\right)\\ z_{k+1}=z_k-\frac{h^{0.6}}{0.6}\left(x_k+z_k-k\left(w_k-p{u}_k\right)\right)\\ w_{k+1}=w_k-\frac{h^{0.24}}{0.24}2x_k{y}_k{z}_k\\ u_{k+1}=u_k+\frac{h^{{\alpha }_5}}{{\alpha }_5}k\left(w_k-p{u}_k\right)\end{array}$$

(21)

is compared to the following four models—respectively, MCEFD (22), ESDDFD1 (23), ESDDFD2 (24), ESDDFD3 (25)—obtained through the ESDDFD and NSFD methods:

$$\begin{array}{c}\frac{x_{k+1}-x_k}{\frac{1}{0.3}\left[{\left(t+h\right)}^{0.3}-t^{0.3}\right]}=z_k+\left(y_k-0.8\right)x_k+k\left(w_k-p{u}_k\right)\\ \frac{y_{k+1}-y_k}{\frac{1}{0.5}\left[{\left(t+h\right)}^{0.5}-t^{0.5}\right]}=1-0.6y_k-x_k{x}_k+k\left(w_k-p{u}_k\right)\\ \frac{z_{k+1}-z_k}{\frac{1}{0.6}\left[{\left(t+h\right)}^{0.6}-t^{0.6}\right]}=-x_k-z_k+k\left(w_k-p{u}_k\right)\\ \frac{w_{k+1}-w_k}{\frac{1}{0.24}\left[{\left(t+h\right)}^{0.24}-t^{0.24}\right]}=-x_k{y}_k\left(z_k+z_k\right)\\ \frac{u_{k+1}-u_k}{\frac{1}{{\alpha }_5}\left[{\left(t+h\right)}^{{\alpha }_5}-t^{{\alpha }_5}\right]}=k\left(w_k-p{u}_k\right)\end{array}$$

(22)

$$\begin{array}{c}\frac{x_{k+1}-x_k}{\frac{1}{0.8}\left[1-e^{\frac{-0.8}{0.3}\left[{\left(t+h\right)}^{0.3}-t^{0.3}\right]}\right]}=z_k+\left(y_k-0.8\right)x_k+k\left(w_k-p{u}_k\right)\\ \frac{y_{k+1}-y_k}{\frac{1}{0.6}\left[1-e^{\frac{-0.6}{0.5}\left[{\left(t+h\right)}^{0.5}-t^{0.5}\right]}\right]}=1-0.6y_k-x_k{x}_k+k\left(w_k-p{u}_k\right)\\ \frac{z_{k+1}-z_k}{\left[1-e^{\frac{-1}{0.6}\left[{\left(t+h\right)}^{0.6}-t^{0.6}\right]}\right]}=-x_k-z_k+k\left(w_k-p{u}_k\right)\\ \frac{w_{k+1}-w_k}{\left[1-e^{\frac{-1}{0.24}\left[{\left(t+h\right)}^{0.24}-t^{0.24}\right]}\right]}=-x_k{y}_k\left(z_k+z_k\right)\\ \frac{u_{k+1}-u_k}{\frac{1}{\mathrm{kp}}\left[1-e^{\frac{-kp}{{\alpha }_5}\left[{\left(t+h\right)}^{{\alpha }_5}-t^{{\alpha }_5}\right]}\right]}=k\left(w_k-p{u}_k\right)\end{array}$$

(23)

$$\begin{array}{c}\frac{x_{k+1}-x_k}{\frac{1}{0.3}\left[{\left(t+h\right)}^{0.3}-t^{0.3}\right]}=z_k+\left(y_{k+1}-0.8\right)x_k+k\left(w_k-p{u}_k\right)\\ \frac{y_{k+1}-y_k}{\frac{1}{0.5}\left[{\left(t+h\right)}^{0.5}-t^{0.5}\right]}=1-0.6y_k-x_{k+1}{x}_k+k\left(w_k-p{u}_k\right)\\ \frac{z_{k+1}-z_k}{\frac{1}{0.6}\left[{\left(t+h\right)}^{0.6}-t^{0.6}\right]}=-x_k-z_k+k\left(w_k-p{u}_k\right)\\ \frac{w_{k+1}-w_k}{\frac{1}{0.24}\left[{\left(t+h\right)}^{0.24}-t^{0.24}\right]}=-x_k{y}_{k+1}\left(z_k+z_{k+1}\right)\\ \frac{u_{k+1}-u_k}{\frac{1}{{\alpha }_5}\left[{\left(t+h\right)}^{{\alpha }_5}-t^{{\alpha }_5}\right]}=k\left(w_k-p{u}_k\right)\end{array}$$

(24)

$$\begin{array}{c}\frac{x_{k+1}-x_k}{\frac{1}{0.8}\left[1-e^{\frac{-0.8}{0.3}\left[{\left(t+h\right)}^{0.3}-t^{0.3}\right]}\right]}=z_k+\left(y_{k+1}-0.8\right)x_k+k\left(w_k-p{u}_k\right)\\ \frac{y_{k+1}-y_k}{\frac{1}{0.6}\left[1-e^{\frac{-0.6}{0.5}\left[{\left(t+h\right)}^{0.5}-t^{0.5}\right]}\right]}=1-0.6y_k-x_{k+1}{x}_k+k\left(w_k-p{u}_k\right)\\ \frac{z_{k+1}-z_k}{\left[1-e^{\frac{-1}{0.6}\left[{\left(t+h\right)}^{0.6}-t^{0.6}\right]}\right]}=-x_k-z_k+k\left(w_k-p{u}_k\right)\\ \frac{w_{k+1}-w_k}{\left[1-e^{\frac{-1}{0.24}\left[{\left(t+h\right)}^{0.24}-t^{0.24}\right]}\right]}=-x_k{y}_{k+1}\left(z_k+z_{k+1}\right)\\ \frac{u_{k+1}-u_k}{\frac{1}{\mathrm{kp}}\left[1-e^{\frac{-kp}{{\alpha }_5}\left[{\left(t+h\right)}^{{\alpha }_5}-t^{{\alpha }_5}\right]}\right]}=k\left(w_k-p{u}_k\right)\end{array}$$

(25)

4.2.1. Varying α₅ with Fixed k = 2 and p = 1 and α₅ ∈ [0.232, 0.328]

In this case, Ref. [13] concluded that system (6) is hyperchaotic with α₅ ∈ [0.232, 0.328]; fixing ${\alpha }_5=0.24$, a set of two positive Lyapunov exponents and three negative Lyapunov exponents were found. Profiles for $x, y, z, w$ and $u$, when ${\alpha }_5=0.232$ for model (21), are given below. Chaos can be clearly seen in Figure 2 which gives the phase portraits for the CEFD model. For each model (22) through (25). Figure 3 shows phase portraits using the same step size and parameter values. These models produce identical graphs, which differ significantly from the graphs for model (21). The bifurcation tests for the ESDDFD model are performed with the same parameters. The bifurcation diagrams for $x,z$and $u$ for model (21) are in Figure 4. These again show clear signs of chaos while the bifurcation diagrams for models (22) through (25), which are given in Figure 5, Figure 6, Figure 7 and Figure 8, do not.

For step sizes above 0.003, CEFD, (21), fails. MCEFD, (22) fails for step sizes above 0.573. The graphs in Figure 9 were produced using the same parameter values as before, except $h=0.1$. The graphs in Figure 10 were done with h = 1.0. These show the effect of larger step sizes on methods (23), (24), and (25). The ESDDFD methods preserve the end behavior at much larger step sizes than CEFD and MCEFD. Note the differences in the early behavior between the methods, especially when compared with h = 0.002.

4.2.2. Varying p with Fixed $k=2, {\alpha }_5=0.3,$ and $p\in \left[1, 2\right]$

In this case, Ref. [13] concluded that system 6 is hyperchaotic with p ∈ [1, 2]. Fixing p = 1, a set of two positive Lyapunov exponents and three negative Lyapunov exponents was determined. Bifurcation tests for the ESDDFD models are performed with the same parameters for the full discrete model (2). Figure 11 shows the bifurcation diagrams for $u,x$ and $z$ for the CEFD model (21). Figure 12, Figure 13, Figure 14 and Figure 15 show the bifurcation diagrams for the models (22) through (25). As in Section 4.2.1, the CEFD diagrams show evidence of chaos while the other models do not.

Setting $p=1.94$, phase portraits are given for models (22) through (25) in Figure 16. Figure 17 shows the phase portraits for model (21). There are clear signs of chaos in the phase portraits for model (21) and no chaos in those for the other models.

4.2.3. Varying k with Fixed $p=1$ and ${\alpha }_5=0.3,$ with k ∈ [1.5, 2.5]

In this case, Ref. [13] concluded that system (6) is hyperchaotic with k ∈ [1.5, 2.5]. Fixing k = 1.5, a set of two positive Lyapunov exponents and three negative Lyapunov exponents were determined. Bifurcation tests for the ESDDFD models are performed with the same parameters for the full discrete model (2). Figure 18 gives the bifurcation diagrams for CEFD, model (21). Figure 19, Figure 20, Figure 21 and Figure 22 give the bifurcation diagrams for $x,u$ and $z$, for models (22) through (25). Once again there is chaos evident in the CEFD diagrams but no chaos in the diagrams for the other models.

Setting $k=2.45$, phase portraits are given for models (22) through (25) in Figure 23. The phase portraits for CEFD, model (21), are given in Figure 24. Again, while the phase portraits for CEFD show chaos, it is lacking in the phase portraits for models (22) through (25).

4.2.4. With Fixed k = 2, p = 1 and ${\alpha }_5=0.24$

In this case, Ref. [13] concluded that system (6) has a hyperchaotic attractor in the $y-z-u$ and $x-y-w$ planes. Two phase portraits for model (21) are given in Figure 25 while the corresponding phase portraits for models (22) through (25) are given in Figure 26. While the results for model (21) show chaos, the results for models (22) through (25) do not.

5. Discussion

A discrete model using the conformable Euler finite difference (CEFD) model, (6), was constructed in [13] and used to detect hyperchaotic behavior of the system (1). In this paper, a discrete model (2) has been constructed for the system (1), and the parameters from [13] were used to study hyperchaos using bifurcation techniques. The discrete model (2) is constructed using the exact spectral derivative discretization finite difference (ESDDFD) method, a universal extension of the nonstandard finite difference method to fractional derivatives, which is designed to eliminate contrived chaos. Various cases are considered in parallel to those considered in [13] as well as for sub-systems relevant to the construction of the discrete model (2). While the proposed ESDDFD models produce similar results to each other, those results are significantly different from those obtained in [13] and exhibit no hyperchaotic behavior.

In view of the results obtained, it is reasonable to question the validity of the conclusions of hyperchaotic behavior previously reported for related models, which the authors intend to pursue in the future. While the conformable derivative is a local derivative and has neither memory nor nonlocality, it is a multiple of the Caputo FD [83], and therefore related to those with these properties. It will, therefore, be interesting to explore what, if any, properties of the conformable system are inherited by the Caputo and Riemann–Liouville FDs through these relationships. Further, as suggested in [13], studies incorporating real economic data with parameter estimation for the financial system with market confidence and ethics for all these derivatives are also necessary. Finally, as can be easily seen from Theorem 4.1 of [50], the discretization methods presented here for CFD systems are easy to implement and are equally applicable to all Caputo type derivatives, and hence, to Riemann–Liouville derivatives through their relationship; hence, they have potential to impact a wide range of fractional derivative applications.