# A New Fractional Discrete Memristive Map with Variable Order and Hidden Dynamics

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## Abstract

**:**

## 1. Introduction

## 2. The Fractional Discrete Memristive-Based Duffing Map

#### 2.1. Preliminaries

#### 2.2. Model Description

#### 2.3. Analysis of Equilibrium Point

## 3. Dynamical Analysis

#### 3.1. Coexisting Multiple Hidden Attractors

#### 3.2. The Invariance Property

#### 3.3. Effect of the Fractional Variable Order $({\nu}_{1},{\nu}_{2},{\nu}_{3})$

**Remark 1.**

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Bifurcation diagrams and LEs versus ${\alpha}_{1}$ of the FMD (4) for variable orders ${\nu}_{0}=0.99$, ${\nu}_{1}=0.98$ and ${\nu}_{2}=0.97$, and system parameters ${\alpha}_{2}=0.244$, $\gamma =-2$: (

**a**) Bifurcation diagram, (

**b**) corresponding LEs.

**Figure 2.**Hidden attractors of the FMD (4) with short memory for ${\alpha}_{2}=0.244$, $\gamma =-2$, and IC $(1,-0.2,0.1)$, and variable order ${\nu}_{0}=0.99$, ${\nu}_{1}=0.98$ and ${\nu}_{2}=0.97$ for different ${\alpha}_{1}$ values: (

**a**) ${\alpha}_{1}=1.7$, (

**b**) ${\alpha}_{1}=1.8$, (

**c**) ${\alpha}_{1}=1.83$, (

**d**) ${\alpha}_{1}=1.9$.

**Figure 3.**Bifurcation diagram and Lyapunov exponents varying with ${z}_{0}$. (

**a**) Bifurcation diagram, (

**b**) corresponding Lyapunov exponents.

**Figure 4.**Hidden attractor coexistence of the FMD (4) with respect to initial condition; (

**a**) for ${\alpha}_{1}=1.7$; (

**b**) for ${\alpha}_{1}=1.8$.

**Figure 6.**(

**a**) Bifurcation diagram versus fractional order ${\nu}_{2}$ of the proposed system, (

**b**) corresponding $0-1$ test.

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**MDPI and ACS Style**

Almatroud, O.A.; Khennaoui, A.-A.; Ouannas, A.; Alshammari, S.; Albosaily, S.
A New Fractional Discrete Memristive Map with Variable Order and Hidden Dynamics. *Fractal Fract.* **2024**, *8*, 322.
https://doi.org/10.3390/fractalfract8060322

**AMA Style**

Almatroud OA, Khennaoui A-A, Ouannas A, Alshammari S, Albosaily S.
A New Fractional Discrete Memristive Map with Variable Order and Hidden Dynamics. *Fractal and Fractional*. 2024; 8(6):322.
https://doi.org/10.3390/fractalfract8060322

**Chicago/Turabian Style**

Almatroud, Othman Abdullah, Amina-Aicha Khennaoui, Adel Ouannas, Saleh Alshammari, and Sahar Albosaily.
2024. "A New Fractional Discrete Memristive Map with Variable Order and Hidden Dynamics" *Fractal and Fractional* 8, no. 6: 322.
https://doi.org/10.3390/fractalfract8060322