# Design of a New Dimension-Changeable Hyperchaotic Model Based on Discrete Memristor

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*Symmetry*)

## Abstract

**:**

## 1. Introduction

## 2. Construction of Discrete Memristor Model for Improving Chaotic System

## 3. The Discrete Hyperchaotic Model with Variable Dimensions

#### 3.1. Chaotic Model Construction Method

#### 3.1.1. Cascade Operation

#### 3.1.2. Coupling

#### 3.2. The Coupling Model Based on Logistic Map and Sine Map (CLS)

## 4. Performances Analysis of CLS and MCLS

#### 4.1. Dynamic Performance in the Phase Space

#### 4.2. Evolution of the CLS and MCLS with the Variation of Parameters

#### 4.3. Global Dynamics Analysis Based on Lyapunov Exponents

#### 4.4. The Complexity Analysis

#### 4.4.1. The Permutation Entropy

#### 4.4.2. The Spectral Entropy Complexity

#### 4.5. The 0–1 Test

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Pinched hysteresis loops of the discrete memristor with different $\omega $ where $A=5$, ${q}_{0}=0.1$, $k=1$, ${R}_{1}=20$, ${R}_{2}=0.02$.

**Figure 3.**Phase diagram, temporal diagram, and frequency spectrum of 2D-CLS with initial values ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$ under the condition of parameter $(\mu ,a,b,\u03f5)=(3,0.5,\pi ,1.85)$. (

**a**) Phase diagram. (

**b**) Temporal diagram and frequency spectrum.

**Figure 4.**Phase diagram, temporal diagram, and frequency spectrum of 2D-CLS with initial values ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$ under the condition of parameter $(\mu ,a,b,\u03f5)=(3,0.3,\pi ,1.85)$. (

**a**) Phase diagram. (

**b**) Temporal diagram and frequency spectrum.

**Figure 5.**Phase diagram, temporal diagram, and frequency spectrum of 2D-CLS with initial values ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$ under the condition of parameter $(\mu ,a,b,\u03f5)=(3,0.5,\pi ,0.03)$. (

**a**) Phase diagram. (

**b**) Temporal diagram and frequency spectrum.

**Figure 6.**Phase diagram, temporal diagram, and frequency spectrum of 2D-MCLS with initial values ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$ under the condition of parameter $(\mu ,a,b,\u03f5)=(3,0.5,\pi ,1.13)$. (

**a**) Phase diagram. (

**b**) Temporal diagram and frequency spectrum.

**Figure 7.**Phase diagram, temporal diagram, and frequency spectrum of 2D-MCLS with initial values ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$ under the condition of parameter $(\mu ,a,b,\u03f5)=(0.02,0.3,\pi ,0.03)$. (

**a**) Phase diagram. (

**b**) Temporal diagram and frequency spectrum.

**Figure 8.**Phase diagram, temporal diagram, and frequency spectrum of 2D-MCLS with initial values ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$ under the condition of parameter $(\mu ,a,b,\u03f5)=(3,0.5,\pi ,0.03)$. (

**a**) Phase diagram. (

**b**) Temporal diagram and frequency spectrum.

**Figure 9.**Lyapunov exponents and bifurcation diagram of 2D-CLS changing with parameter $\mu $, when the initial values are ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$ and the fixed parameters are $a=0.5,b=\pi ,\u03f5=0.03$. (

**a**) Change of LEs. (

**b**) Change of bifurcation diagram.

**Figure 10.**Lyapunov exponents and bifurcation diagram of 2D-MCLS changing with parameter $\mu $, when the initial values are ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$ and the fixed parameters are $a=0.5,b=\pi ,\u03f5=0.03$. (

**a**) Change of LEs. (

**b**) Change of bifurcation diagram.

**Figure 11.**Lyapunov exponents and bifurcation diagram of 2D-CLS changing with parameter a, when the initial values are ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$ and the fixed parameters are $\mu =3,b=\pi ,\u03f5=0.03$. (

**a**) Change of LEs. (

**b**) Change of bifurcation diagram.

**Figure 12.**Lyapunov exponents and bifurcation diagram of 2D-MCLS changing with parameter a, when the initial values are ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$ and the fixed parameters are $\mu =3,b=\pi ,\u03f5=0.03$. (

**a**) Change of LEs. (

**b**) Change of bifurcation diagram.

**Figure 13.**Lyapunov exponents and bifurcation diagram of 2D-CLS changing with parameter b, when the initial values are ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$ and the fixed parameters are $\mu =3,a=0.5,\u03f5=0.03$. (

**a**) Change of LEs. (

**b**) Change of bifurcation diagram.

**Figure 14.**Lyapunov exponents and bifurcation diagram of 2D-MCLS changing with parameter b, when the initial values are ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$ and the fixed parameters are $\mu =3,a=0.5,\u03f5=0.03$. (

**a**) Change of LEs. (

**b**) Change of bifurcation diagram.

**Figure 15.**Lyapunov exponents and bifurcation diagram of 2D-CLS changing with parameter $\u03f5$, when the initial values are ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$ and the fixed parameters are $\mu =3,a=0.5,b=\pi $. (

**a**) Change of LEs. (

**b**) Change of bifurcation diagram.

**Figure 16.**Lyapunov exponents and bifurcation diagram of 2D-MCLS changing with parameter $\u03f5$, when the initial values are ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$ and the fixed parameters are $\mu =3,a=0.5,b=\pi $. (

**a**) Change of LEs. (

**b**) Change of bifurcation diagram.

**Figure 17.**Lyapunov exponents of 2D-CLS and 2D-MCLS with changing initial values. (

**a**) Change of the first LE of CLS. (

**b**) Change of the first LE of MCLS. (

**c**) Change of the second LE of CLS. (

**d**) Change of the second LE of MCLS.

**Figure 18.**PE of 2D-CLS and 2D-MCLS with ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$. (

**a**) Change of PE with parameter $\mu $ when $a=0.5$, $b=\pi $, $\u03f5=0.03$. (

**b**) Change of PE with parameter a when $\mu =3$, $b=\pi $, $\u03f5=0.03$. (

**c**) Change of PE with parameter b when $\mu =3$, $a=0.5$, $\u03f5=0.03$. (

**d**) Change of PE with parameter $\u03f5$ when $\mu =3$, $a=0.5$, $b=\pi $.

**Figure 19.**SE of 2D-CLS with ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$. (

**a**) Change of SE with parameter $\mu $ and $\u03f5$ when $a=0.5$, $b=\pi $. (

**b**) Change of SE with parameter a and $\u03f5$ when $\mu =3$, $b=\pi $. (

**c**) Change of SE with parameter b and $\u03f5$ when $\mu =3$, $a=0.5$. (

**d**) Change of SE with parameter a and b when $\mu =3$, $\u03f5=0.03$.

**Figure 20.**SE of 2D-MCLS with ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$. (

**a**) Change of SE with parameter $\mu $ and $\u03f5$ when $a=0.5$, $b=\pi $. (

**b**) Change of SE with parameter a and $\u03f5$ when $\mu =3$, $b=\pi $. (

**c**) Change of SE with parameter b and $\u03f5$ when $\mu =3$, $a=0.5$. (

**d**) Change of SE with parameter a and b when $\mu =3$, $\u03f5=0.03$.

**Figure 21.**The $p-s$ diagram of 2D-CLS and 2D-MCLS with ${x}_{1}\left(0\right)=0.3$, ${x}_{2}\left(0\right)=0.4$. (

**a**) $p-s$ diagram of CLS. (

**b**) $p-s$ diagram of MCLS.

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**MDPI and ACS Style**

Wei, C.; Li, G.; Xu, X.
Design of a New Dimension-Changeable Hyperchaotic Model Based on Discrete Memristor. *Symmetry* **2022**, *14*, 1019.
https://doi.org/10.3390/sym14051019

**AMA Style**

Wei C, Li G, Xu X.
Design of a New Dimension-Changeable Hyperchaotic Model Based on Discrete Memristor. *Symmetry*. 2022; 14(5):1019.
https://doi.org/10.3390/sym14051019

**Chicago/Turabian Style**

Wei, Chengjing, Guodong Li, and Xiangliang Xu.
2022. "Design of a New Dimension-Changeable Hyperchaotic Model Based on Discrete Memristor" *Symmetry* 14, no. 5: 1019.
https://doi.org/10.3390/sym14051019