# A High Spectral Entropy (SE) Memristive Hidden Chaotic System with Multi-Type Quasi-Periodic and its Circuit

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## Abstract

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## 1. Introduction

## 2. A New 5-D Memristive Chaotic System

#### 2.1. Description of the New Memristive Chaotic System

#### 2.2. Bifurcation Diagram with $a$ as Varying Parameter

#### 2.3. Analysis of Multi-Stability

#### 2.4. Analysis of Transient Chaos

## 3. Entropy Analysis for Memristive Chaotic Systems

#### 3.1. SE Analysis Depending on Parameters

#### 3.2. Entropy Analysis of Chaotic Behavior

## 4. Circuitry Realization of Memristor-Based Chaotic System

#### 4.1. Equivalent Circuit implementation for Memristor

#### 4.2. Circuit of Memristive Chaotic System

#### 4.3. Circuit simulation of Memristive Chaotic System

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Memristor model: (

**a**) the relationship of magnetic flux and charge; (

**b**) the I-V characteristic curve.

**Figure 2.**3-D chaotic attractor of system (2): (

**a**) $x-z-w$ space, (

**b**) $x-y-u$ space, (

**c**) $x-w-u$ space, and (

**d**) $z-w-u$ space.

**Figure 3.**2-D chaotic attractor of system: (

**a**) $y-w$ plane; (

**b**) $z-w$ plane; (

**c**) $w-u$ plane; (

**d**) $x-\mathrm{u}$ plane; (

**e**) $z-u$ plane; and (

**f**) $\mathrm{w}-i$ plane of memristor model (1).

**Figure 4.**Frequency spectrum and time series of $x$ variable for system (2): (

**a**) frequency spectrum; (

**b**) time series.

**Figure 5.**Poincaré map of system (2): (

**a**) $x-y$ plane; (

**b**) $y-u$ plane; (

**c**) $y-w$ plane; (

**d**) $x-w$ plane.

**Figure 6.**Bifurcation diagram of $x$ versus $a$ for system (2) when $b=0.05$, $d=0.1$, $c=e=g=1$, $({x}_{0},{y}_{0},{z}_{0},{w}_{0},{u}_{0})=(-1,-1,0,0,1)$, and $a\in (0,4)$.

**Figure 7.**Les of system (2) versus $a$, when $b=0.05$, $d=0.1$, $c=e=g=1$, $({x}_{0},{y}_{0},{z}_{0},{w}_{0},{u}_{0})=(-1,-1,0,0,1)$, and $a\in (0,4)$.

**Figure 8.**Projections of 2-D phase diagram with parameter $a=0.3$: (

**a**) attractor on the $x-z$ plane; (

**b**) attractor on the $y-z$ plane; (

**c**) attractor on the $y-w$ plane.

**Figure 9.**Projections of a 2-D phase diagram with parameter $a=0.4$: (

**a**) attractor on the $x-z$ plane; (

**b**) attractor on the $y-z$ plane; (

**c**) attractor on the $y-w$ plane.

**Figure 10.**Projections of 2-D phase diagram with parameter $a=0.70$: (

**a**) attractor on the $x-z$ plane; (

**b**) attractor on the $y-z$ plane; (

**c**) attractor on the $y-w$ plane.

**Figure 11.**Projections of 2-D phase diagram with parameter $a=0.75$: (

**a**) attractor on the $x-z$ plane; (

**b**) attractor on the $y-z$ plane; (

**c**) attractor on the $y-w$ plane.

**Figure 12.**Projections of 2-D phase diagram with parameter $a=1.17$: (

**a**) attractor on the $x-z$ plane; (

**b**) attractor on the $y-z$ plane; (

**c**) attractor on the $y-w$ plane.

**Figure 13.**Projections of 2-D phase diagram with parameter $a=1.28$: (

**a**) attractor on the $x-z$ plane; (

**b**) attractor on the $y-z$ plane; (

**c**) attractor on the $y-w$ plane.

**Figure 14.**Projections of 2-D phase diagram with parameter $a=1.891$: (

**a**) attractor on the $x-z$ plane; (

**b**) attractor on the $y-z$ plane; (

**c**) attractor on the $y-w$ plane.

**Figure 15.**Projections of 2-D phase diagram with parameter $a=1.89678$: (

**a**) attractor on the $x-z$ plane; (

**b**) attractor on the $y-z$ plane; (

**c**) attractor on the $y-w$ plane.

**Figure 16.**Projections of 2-D phase diagram with parameter $a=2.56$: (

**a**) attractor on the $x-z$ plane; (

**b**) attractor on the $y-z$ plane; (

**c**) attractor on the $y-w$ plane.

**Figure 17.**Projections of 2-D phase diagram with parameter $a=2.65$: (

**a**) attractor on the $x-z$ plane; (

**b**) attractor on the $y-z$ plane; (

**c**) attractor on the $y-w$ plane.

**Figure 18.**Bifurcation diagram of $y$ and Les versus $u$ for system (2) with initial value ${O}_{0}=({x}_{0},{y}_{0},{z}_{0},{w}_{0},{u}_{0})=(u,0,0,0,0)$, and $u\in [0,4]$: (

**a**) bifurcation diagram; (

**b**) Les graph.

**Figure 19.**Bifurcation diagram of $y$ and Les versus $u$ for system (2) with initial value ${O}_{1}=({x}_{0},{y}_{0},{z}_{0},{w}_{0},{u}_{0})=(u,u,0,0,0)$, and $u\in [0,4]$: (

**a**) bifurcation diagram; (

**b**) Les graph.

**Figure 20.**Phase diagram on the $y-z$ plane for system (2) with different initial values: (

**a**) $\left(0.51,0.51,0,0,0\right)$ (blue), $\left(3.34,3.34,0,0,0\right)$ (red), $\left(0.01,0,0,0,0\right)$ (green); (

**b**) $\left(3,3,0,0,0\right)$ (blue), $\left(0.9711,0.9711,0,0,0\right)$ (red), $\left(1.922,0,0,0,0\right)$ (green).

**Figure 21.**Multi-stable nonlinear dynamic behavior distribution of chaotic for memristive system with different initial values.

**Figure 22.**Time series and phase diagram of attractors for transient chaotic: (

**a**) time series of $z$ when $t\in \left(0,8000\right)$; (

**b**) phase diagrams in the $x-y-u$ space when $t\in \left(0,2000\right)$; (

**c**) phase diagrams on the $x-u$ plane when $t\in \left(0,2000\right)$; (

**d**) phase diagrams in the $x-y-u$ space when $t\in \left(2000,8000\right)$; (

**e**) phase diagrams on the $x-u$ plane when $t\in \left(2000,8000\right)$.

**Figure 24.**SE distribution of the system under different conditions: (

**a**) the interaction of parameters $a$ and $b$; (

**b**) the interaction of initial values ${x}_{0}$ and ${y}_{0}$.

**Figure 28.**Phase diagrams observed by oscilloscope: (

**a**) ${u}_{y}-{u}_{w}$ plane; (

**b**) ${u}_{z}-{u}_{w}$ plane; (

**c**) ${u}_{w}-{u}_{u}$ plane; (

**d**) ${u}_{x}-{u}_{u}$ plane; (

**e**) ${u}_{z}-{u}_{u}$ plane; (

**f**) ${u}_{x}-{u}_{z}$ plane.

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**MDPI and ACS Style**

Liu, L.; Du, C.; Liang, L.; Zhang, X.
A High Spectral Entropy (SE) Memristive Hidden Chaotic System with Multi-Type Quasi-Periodic and its Circuit. *Entropy* **2019**, *21*, 1026.
https://doi.org/10.3390/e21101026

**AMA Style**

Liu L, Du C, Liang L, Zhang X.
A High Spectral Entropy (SE) Memristive Hidden Chaotic System with Multi-Type Quasi-Periodic and its Circuit. *Entropy*. 2019; 21(10):1026.
https://doi.org/10.3390/e21101026

**Chicago/Turabian Style**

Liu, Licai, Chuanhong Du, Lixiu Liang, and Xiefu Zhang.
2019. "A High Spectral Entropy (SE) Memristive Hidden Chaotic System with Multi-Type Quasi-Periodic and its Circuit" *Entropy* 21, no. 10: 1026.
https://doi.org/10.3390/e21101026