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

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. The Conformable Derivative ESDDFD Discrete Model Construction Fundamentals

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

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

**Definition**

**1.**

## 4. Numerical Experiments

#### 4.1. Three-Dimensional Systems Comparison

#### 4.2. Five-Dimensional Systems Comparison: Varying ${\alpha}_{5},k$, and $p$

#### 4.2.1. Varying α_{5} with Fixed k = 2 and p = 1 and α_{5} ∈ [0.232, 0.328]

_{5}∈ [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.

#### 4.2.2. Varying p with Fixed $k=2,{\alpha}_{5}=0.3,$ and $p\in \left[1,2\right]$

#### 4.2.3. Varying k with Fixed $p=1$ and ${\alpha}_{5}=0.3,$ with k ∈ [1.5, 2.5]

#### 4.2.4. With Fixed k = 2, p = 1 and ${\alpha}_{5}=0.24$

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Phase portraits (

**a**) CEFD model (20) (

**b**) MCEFD model (16) (

**c**) Model 17 (

**d**) Model 18 (

**e**) model (19).

**Figure 2.**CEFD model (21) profiles of (

**a**) $x-y-z,$ (

**b**) $x-u-z,$ (

**c**) $x-w-z,$ at $h=0.002,k=2,p=1,{\alpha}_{5}=0.232$.

**Figure 3.**Phase portraits (

**a**) $x-y-z,$ (

**b**) $x-u-z,$ (

**c**) $x-z-w$, at $h=0.002,k=2,p=1,{\alpha}_{5}=0.232$ for models (22) through (25).

**Figure 4.**CEFD model (21); bifurcation of (

**a**) $u$ (

**b**) $x$ (

**c**) $z$ versus ${\alpha}_{5}$ for $h=0.002$.

**Figure 5.**MCEFD Model (22); (

**a**) $u\mathrm{vs}.{\alpha}_{5},$ (

**b**) $x\mathrm{vs}.{\alpha}_{5},$ (

**c**) $z\mathrm{vs}.{\alpha}_{5}$, at $k=2,p=1,{\alpha}_{5}\in \left[0.232,0.328\right]$.

**Figure 6.**ESDDFD model (23); (

**a**) $u\mathrm{vs}.{\alpha}_{5},$ (

**b**) $x\mathrm{vs}.{\alpha}_{5},$ (

**c**) $z\mathrm{vs}.{\alpha}_{5}$, at $k=2,p=1,{\alpha}_{5}\in \left[0.232,0.328\right]$.

**Figure 7.**ESDDFD model (24); (

**a**) $u\mathrm{vs}.{\alpha}_{5},$ (

**b**) $x\mathrm{vs}.{\alpha}_{5},$ (

**c**) $z\mathrm{vs}.{\alpha}_{5}$, at $k=2,p=1,{\alpha}_{5}\in \left[0.232,0.328\right]$.

**Figure 8.**ESDDFD model (25); (

**a**) $u\mathrm{vs}.{\alpha}_{5},$ (

**b**) $x\mathrm{vs}.{\alpha}_{5},$ (

**c**) $z\mathrm{vs}.{\alpha}_{5}$, at $k=2,p=1,{\alpha}_{5}\in \left[0.232,0.328\right]$.

**Figure 9.**Phase portraits (

**a**) $x-y-z,$ (

**b**) $x-u-z,$ (

**c**) $x-z-w,$ at $h=0.1,k=2,p=1,{\alpha}_{5}=0.232$ for models (22) through (25).

**Figure 10.**Phase portraits (

**a**) $x-y-z,$ (

**b**) $x-u-z,$ (

**c**) $x-z-w$, at $h=1.0,k=2,p=1,{\alpha}_{5}=0.232$ for models (22) through (25). $h=1.0,{\alpha}_{5}=0.232$ for (23) through (25).

**Figure 11.**CEFD model (21); (

**a**) $u\mathrm{vs}.p,$ (

**b**) $x\mathrm{vs}.p,$ (

**c**) $z\mathrm{vs}.p$, at $k=2,{\alpha}_{5}=0.3,p\in \left[1,2\right]$.

**Figure 12.**MCEFD model (22); (

**a**) $u\mathrm{vs}.p,$ (

**b**) $x\mathrm{vs}.p,$ (

**c**) $z\mathrm{vs}.p$, at $k=2,{\alpha}_{5}=0.3,p\in \left[1,2\right]$.

**Figure 13.**ESDDFD1 model (23); (

**a**) $u\mathrm{vs}.p,$ (

**b**) $x\mathrm{vs}.p,$ (

**c**) $z\mathrm{vs}.p$, at $k=2,{\alpha}_{5}=0.3,p\in \left[1,2\right]$.

**Figure 14.**ESDDFD2 model (24); (

**a**) $u\mathrm{vs}.p,$ (

**b**) $x\mathrm{vs}.p,$ (

**c**) $z\mathrm{vs}.p$, at $k=2,{\alpha}_{5}=0.3,p\in \left[1,2\right]$.

**Figure 15.**ESDDFD2 model (25); (

**a**) $u\mathrm{vs}.p,$ (

**b**) $x\mathrm{vs}.p,$ (

**c**) $z\mathrm{vs}.p$, at $k=2,{\alpha}_{5}=0.3,p\in \left[1,2\right]$.

**Figure 16.**Phase portraits (

**a**) $x-y-z,$ (

**b**) $x-u-z,$ (

**c**) $x-z-w,$ at $h=0.002,k=2,p=1.94,{\alpha}_{5}=0.3$ for models (22) through (25).

**Figure 17.**Model (21) phase portraits (

**a**) $x-y-z,$ (

**b**) $x-z-u,$ and (

**c**) $x-z-w$ at $k=2,p=1.94,{\alpha}_{5}=0.3$.

**Figure 18.**CEFD model (21); (

**a**) $u\mathrm{vs}.k,$(

**b**) $x\mathrm{vs}.k,$ (

**c**) $z\mathrm{vs}.k,$ at $p=1,{\alpha}_{5}=0.3,k\in \left[1.5,2.5\right]$.

**Figure 19.**MCEFD model (22); (

**a**) $u\mathrm{vs}.k,$(

**b**) $x\mathrm{vs}.k,$ (

**c**) $z\mathrm{vs}.k,$ at $p=1,{\alpha}_{5}=0.3,k\in \left[1.5,2.5\right]$.

**Figure 20.**ESDDFD1 model (23); (

**a**) $u\mathrm{vs}.k,$(

**b**) $x\mathrm{vs}.k,$ (

**c**) $z\mathrm{vs}.k,$ at $p=1,{\alpha}_{5}=0.3,k\in \left[1.5,2.5\right]$.

**Figure 21.**ESDDFD2 model (24); (

**a**) $u\mathrm{vs}.k,$(

**b**) $x\mathrm{vs}.k,$ (

**c**) $z\mathrm{vs}.k,$ at $p=1,{\alpha}_{5}=0.3,k\in \left[1.5,2.5\right]$.

**Figure 22.**ESDDFD2 model (25); (

**a**) $u\mathrm{vs}.k,$(

**b**) $x\mathrm{vs}.k,$ (

**c**) $z\mathrm{vs}.k,$ at $p=1,{\alpha}_{5}=0.3,k\in \left[1.5,2.5\right]$.

**Figure 23.**Phase portraits (

**a**) $x-y-z,$ (

**b**) $x-u-z,$ (

**c**) $x-z-w,$ at $h=0.1,k=2.45,p=1,{\alpha}_{5}=0.3$ for models (22) through (25).

**Figure 24.**Model (21) phase portraits; (

**a**)$x-y-z,$ (

**b**) $x-z-u,$and (

**c**)$x-z-w$ at $k=2.45,p=1,{\alpha}_{5}=0.3$.

**Figure 25.**Phase portraits $\left(a\right)y-z-u,\left(b\right)x-\mathrm{y}-\mathrm{w}$, at $h=0.002,k=2,p=1,{\alpha}_{5}=0.24$ for model (21) CEFD.

**Figure 26.**Phase portraits (

**a**) $y-z-u,$ (

**b**) $x-y-w,$at $h=0.002,k=2,p=1,{\alpha}_{5}=0.24$ for models (22) through (25).

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**MDPI and ACS Style**

Clemence-Mkhope, D.P.; Gibson, G.A.
Taming Hyperchaos with Exact Spectral Derivative Discretization Finite Difference Discretization of a Conformable Fractional Derivative Financial System with Market Confidence and Ethics Risk. *Math. Comput. Appl.* **2022**, *27*, 4.
https://doi.org/10.3390/mca27010004

**AMA Style**

Clemence-Mkhope DP, Gibson GA.
Taming Hyperchaos with Exact Spectral Derivative Discretization Finite Difference Discretization of a Conformable Fractional Derivative Financial System with Market Confidence and Ethics Risk. *Mathematical and Computational Applications*. 2022; 27(1):4.
https://doi.org/10.3390/mca27010004

**Chicago/Turabian Style**

Clemence-Mkhope, Dominic P., and Gregory A. Gibson.
2022. "Taming Hyperchaos with Exact Spectral Derivative Discretization Finite Difference Discretization of a Conformable Fractional Derivative Financial System with Market Confidence and Ethics Risk" *Mathematical and Computational Applications* 27, no. 1: 4.
https://doi.org/10.3390/mca27010004