# Periodically Intermittent Control of Memristor-Based Hyper-Chaotic Bao-like System

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## Abstract

**:**

## 1. Introduction

- (i)
- A memristor-based hyper-chaotic Bao-like system is constructed, and its chaotic behavior is verified by designing an analog circuit;
- (ii)
- A novel control method called periodically intermittent control with variable control width is proposed, and the proposed hyper-chaotic system is controlled by this method.

**Notation**

**1.**

## 2. Construction of the New Hyper-Chaotic System

**Remark**

**1.**

## 3. Introduction of the Periodically Intermittent Control

**Remark**

**2.**

**Lemma**

**1**

**.**For any three real matrices ${\varphi}_{1},{\varphi}_{2,}{\varphi}_{3}\in {R}^{n\times m}$, $0<{\varphi}_{3}={\varphi}_{3}^{T}$, and a scalar $\mu \ge 0$, we have the following inequality:

**Lemma**

**2**

**.**Let ${\rm Y}\left(x\right)={{\rm Y}}^{T}\left(x\right),\Psi \left(x\right)={\Psi}^{T}\left(x\right)$, and

## 4. Main Results

**Theorem**

**1.**

- (i)
- $QB+{B}^{T}Q+QK+{K}^{T}Q+{\mu}_{1}QC{C}^{T}Q+{\mu}_{1}^{-1}L+{p}_{1}Q\le 0$;
- (ii)
- $QB+{B}^{T}Q+{\mu}_{2}QC{C}^{T}Q+{\mu}_{2}^{-1}L-{p}_{2}Q\le 0$;
- (iii)
- ${p}_{1}\ge {p}_{2}$.

**Proof.**

**Corollary**

**1.**

## 5. Numerical Simulation Examples

**Example**

**1.**

**Example**

**2.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Chaotic attractor phase diagram for the hyper-chaotic Bao-like system based on memristor with the initial condition $\mathbf{x}\left(0\right)={\left[10,10,10,10\right]}^{T}$. (

**a**) x-y-$\omega $ plane. (

**b**) x-y plane. (

**c**) x-z plane. (

**d**) y-z plane.

**Figure 3.**The Lyapunov exponential spectrum of the memristor-based hyper-chaotic Bao-like system with the initial condition $\mathbf{x}\left(0\right)={\left[10,10,10,10\right]}^{T}$.

**Figure 5.**Chaotic attractor phase diagram for the hyper-chaotic Bao-like system based on memristor with the initial condition $\mathbf{x}\left(0\right)={\left[1,1,1,1\right]}^{T}$. (

**a**) x-y plane. (

**b**) x-z plane. (

**c**) y-z plane.

**Figure 6.**Time diagram of state variable x with the initial condition $\mathbf{x}\left(0\right)={\left[5,-2,3,-3\right]}^{T}$.

**Figure 7.**The time response curves of the controlled system with periodically intermittent control with variable control width.

**Figure 8.**Chua’s oscillator produces a chaotic phenomenon with the initial condition $\mathbf{x}\left(0\right)={\left[2,0.3,-0.5\right]}^{T}$.

**Figure 9.**The time response curves of the controlled Chua’s oscillator with periodically intermittent control with variable control width.

Time | ${\mathit{LE}}_{1}$ | ${\mathit{LE}}_{2}$ | ${\mathit{LE}}_{3}$ | ${\mathit{LE}}_{4}$ |
---|---|---|---|---|

$t=0.3$ | 8.6789 | 2.9300 | −3.8563 | −23.7506 |

$t=0.6$ | 7.1126 | −1.2834 | −0.0935 | −21.7304 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

$t=59.7$ | 0.4827 | 0.1783 | −0.0079 | −16.3812 |

$t=60.0$ | 0.5369 | 0.1863 | −0.0077 | −16.4081 |

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**MDPI and ACS Style**

Li, K.; Li, R.; Cao, L.; Feng, Y.; Onasanya, B.O. Periodically Intermittent Control of Memristor-Based Hyper-Chaotic Bao-like System. *Mathematics* **2023**, *11*, 1264.
https://doi.org/10.3390/math11051264

**AMA Style**

Li K, Li R, Cao L, Feng Y, Onasanya BO. Periodically Intermittent Control of Memristor-Based Hyper-Chaotic Bao-like System. *Mathematics*. 2023; 11(5):1264.
https://doi.org/10.3390/math11051264

**Chicago/Turabian Style**

Li, Kun, Rongfeng Li, Longzhou Cao, Yuming Feng, and Babatunde Oluwaseun Onasanya. 2023. "Periodically Intermittent Control of Memristor-Based Hyper-Chaotic Bao-like System" *Mathematics* 11, no. 5: 1264.
https://doi.org/10.3390/math11051264