# Dynamic Analysis and FPGA Implementation of a New, Simple 5D Memristive Hyperchaotic Sprott-C System

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## Abstract

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## 1. Introduction

## 2. A Simple 4D Hyperchaotic System

#### 2.1. Equilibrium Point and Stability

#### 2.2. Lyapunov Exponents

#### 2.3. Nonlinear Dynamic Behavior Analysis

## 3. Sprott-C Hyperchaotic System Based on Memristor

#### 3.1. Divergence and Lyapunov Exponents

#### 3.2. Equilibrium Points and Stability

#### 3.3. Abundant Dynamic Behavior

#### 3.4. Coexistence of Attractors

## 4. FPGA Implementation

## 5. 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.**Ref. [58] System attractor dispersion diagram under initial conditions $(0.3,0.2,0.3,0.1)$, and phase diagrams of system (1) under initial conditions $(0.3,0.2,0.3,0.1)$. (

**a**) Ref. [58] System attractor diagram with $(a,b)=(0.1,0.09)$ in $x-u$ plane, (

**b**) System (1) attractor diagram with $(a,b)=(0.1,-0.1)$ in $x-y$ plane, (

**c**) System (1) attractor diagram with $(a,b)=(0.1,-0.1)$ in $x-z$ plane, (

**d**) System (1) attractor diagram with $(a,b)=(0.1,-0.1)$ in $x-y-z$ plane.

**Figure 3.**Bifurcation diagram of system (1) with initial conditions $(0.1,0.1,0.1,0.1)$. (

**a**) Bifurcation diagram of a with $b=-0.1$. (

**b**) Bifurcation diagram of b with $a=0.1$.

**Figure 4.**Lyapunov exponent spectra of System 1 with different initial conditions. (

**a**) Lyapunov exponent spectrum with $a\in [0,10]$ and $b=-0.01$ under $IC=(0.1,0.1,0.1,0.1)$. (

**b**) Lyapunov exponent spectrum with $b\in [-10,0]$ and $a=0.1$ under $IC=(0.1,0.1,0.1,0.1)$. (

**c**) Lyapunov exponent spectrum with $a\in [0,10]$ and $b=-0.01$ under $IC=(1,1,1,1)$. (

**d**) Lyapunov exponent spectrum with $b\in [-10,0]$ and $a=0.1$ under $IC=(1,1,1,1)$.

**Figure 6.**Phase diagrams with different c; Lyapunov exponent spectrum and the bifurcation diagram at $c\in (0,0.5]$. (

**a**) Period-1 attractor with $c=0.01$; (

**b**) chaotic attractor, $c=0.12$; (

**c**) hyperchaotic attractor, $c=0.5$; (

**d**) Lyapunov exponent spectrum with $c\in (0,0.5]$; (

**e**) bifurcation diagram, $c\in (0,0.5]$.

**Figure 7.**Coexistence of attractors with $(a,b,k,m,n)=(1,0.01,5,0.1,0.01)$ under initial conditions $(0.7,0.7,0.7,1,-1)$, $(-0.7,-0.7,0.7,1,-1)$, $(0.2,0.2,0.2,0.1,0.1)$ and $(-0.2,-0.2,0.2,0.1,0.1)$. (

**a**) $c=0.005$, period-1 of State 1 and State 2 coexists with period-1 of State 3 and State 4. (

**b**) $c=0.017$, period-1 of State 1 and State 2 coexists with the chaos of State 3 and State 4. (

**c**) $c=0.2$, chaos of State 1 and State 2 coexists with the chaos of State 3 and State 4.

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**MDPI and ACS Style**

Yu, F.; Zhang, W.; Xiao, X.; Yao, W.; Cai, S.; Zhang, J.; Wang, C.; Li, Y. Dynamic Analysis and FPGA Implementation of a New, Simple 5D Memristive Hyperchaotic Sprott-C System. *Mathematics* **2023**, *11*, 701.
https://doi.org/10.3390/math11030701

**AMA Style**

Yu F, Zhang W, Xiao X, Yao W, Cai S, Zhang J, Wang C, Li Y. Dynamic Analysis and FPGA Implementation of a New, Simple 5D Memristive Hyperchaotic Sprott-C System. *Mathematics*. 2023; 11(3):701.
https://doi.org/10.3390/math11030701

**Chicago/Turabian Style**

Yu, Fei, Wuxiong Zhang, Xiaoli Xiao, Wei Yao, Shuo Cai, Jin Zhang, Chunhua Wang, and Yi Li. 2023. "Dynamic Analysis and FPGA Implementation of a New, Simple 5D Memristive Hyperchaotic Sprott-C System" *Mathematics* 11, no. 3: 701.
https://doi.org/10.3390/math11030701