Taming hyperchaos with ESDDFD discretization of a conformable fractional derivative financial system with market confidence and ethics risk

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 𝛼 = 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.

demand, price index, market confidence, and ethics risk; the parameters a, b, 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.
Once proposed, and since analytic solutions do not exists, suitable numerical schemes to obtain solutions of the conformable derivative financial system. Though there are several methods to solve a conformable derivative system [41,, these are too complex for many people. Inspired by the discretization process for the Caputo derivative for Ricatti equations [73] and Chua systems [74], the conformable Euler's finite difference (CEFD) method [75] for the five-dimensional fractional-order financial system is proposed in [1]. 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 [1] is known to induce (see, e.g., [Garba et al] 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 (1.1) and the experiments of [Xin et al., 2019] are repeated with the new models. The remainder of this article is organized as follows. In Sect. 2, ESDDFD fundamentals, description of the model (1.1), and the CEFD model from [1] are presented. Section 3 presents construction of the denominator functions, (ℎ, ), 1 ≤ ≤ 5 , for the ESDDFD model (1.2) and compares sub-models of (1.2) with corresponding CEFD sub-models. In Sect. 4, experimental results are presented of hyperchaotic attractor detection from the proposed financial system using both methods. Some concluding remarks in Sect. 5 close the paper.

The conformable derivative hyperchaotic financial system and its CEFD model
The conformable fractional derivative financial system model (1.1) is based on a successive addition of various factors starting with the Huang and Li [11] nonlinear financial system model: modeling the interaction of interest rate, investment demand, and price index; the variables and parameters are the same as in (1.1). Model (2.1) was extended by Xin and Zhang [18] to account for market confidence: where m1, m2, m3 are the impact factors associated with market confidence; the remaining variables and parameters are the same as in (2.1). Model (1.1) is the fractionalization, predicated on the practice that fractional-order economic systems [18,[76][77][78][79][80] can generalize their integer-order forms [17,81,82], of the following extension of (2.2) in [1] to account for both market confidence and ethics risk: When α = (1, 1, 1, 1, 1), system (1.1) degenerates to system (2.3); in the absence of ethics risk, (2.3) reduces to (2.2); in the absence of market confidence, (2.2) reduces to (2.1). In these three cases, therefore, any discrete method developed for (1.1) must reduce to that of the respective three reduced systems. Chaotic behavior for both the CEFD and ESDDFD models will be investigated in Section 3 for (1.1) as well as the three reduced systems (2.1)-(2.3).
The following discrete model was obtained in [1] from the CEFD method and used to investigate hyperchaos of the system (1.1):

ESDDFD Discretization of 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, which has only positive solutions for any positive initial data. The exact discretization of (3.1), which has a solution identical to that of (3.1), is as follows: where the nonstandard denominators 1 (ℎ, ), 1 ≤ ≤ 5, are given by Since (1.1) reduces to (3.1), any valid discrete model for (1.1) must be reducible to one consistent with its exact discretization, that is, (3.2). By comparison, a reduction of the CEFD model (2.4) to the sub-system (3.1) yields the following discrete sub-system: which is positive only if the following condition is satisfied: To enable assessment of the effect of the non-local discretization of nonlinear terms, the following schemes are compared: The terms ( -a) , and x 2 are discretized non-locally as, respectively, ( +1 -a) and +1 , while discretization of the terms in Eqns. (3.4a) and in Eqn. (3.4c) as and ensures respective consistency with the terms of in Eqn. (3.4c) and in Eqn. (3.4a in the cases = 1 and = 1. By comparison, the scheme obtained by a reduction of the CEFD model (2.4) to its 3-dimensional sub-system (2.1) yields the following discrete sub-system: Since system (3.5) reduces to the − − sub-system of (3.3), 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 (1.2) are then obtained by discretizing ( − ) as ( − ) to ensure consistency with (3.2) and then discretizing non-locally as either While implicit, the schemes (????) can be explicitly solved for each = 1,2 in the order +1 , +1 , +1 , +1 , +1 to obtain the following:

Numerical Experiments
In this section, hyperchaos detection experiments are conducted parallel to those of [1] by varying the parameters related to ethics risk, such as α5, 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 While bifurcations can be seen in the CEFD model, they are absent from the results of the ESDDFD models.   .7) through (4.10) a graph of the five variables is given using the same step size and parameter values. These models produce identical graphs which differ significantly from the graphs for model (4.6). The Bifurcation tests for the ESDDFD model (3.4) are performed with the same parameters. The bifurcations diagrams for , and for models (4.6) through (4.10) are reproduced for ℎ = 0.002.

Discussion
A discrete model using the conformable Euler finite difference (CEFD) model, (