# The Application of Accurate Exponential Solution of a Differential Equation in Optimizing Stability Control of One Class of Chaotic System

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## Abstract

**:**

## 1. Introduction

## 2. Analysis of One Class of Chaos Phenomenon

#### 2.1. Introduction to System Model

#### 2.2. Simulation Analysis

## 3. The Application of Accurate Exponential Solution of a Differential Equation in Stability Control of One Class of Chaotic System with Known Parameters

#### 3.1. Methods Proposed

**Definition**

**1.**

**Theorem**

**1.**

**Proof**

**1.**

#### 3.2. Simulation Result

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

## 4. The Application of Accurate Exponential Solution of a Differential Equation in Stability Control of One Class of Chaotic System with Unknown Parameters

#### 4.1. Methods Proposed

**Theorem**

**2.**

**Proof**

**2.**

#### 4.2. Simulation Result

## 5. The Application of Ship’s Electric Chaotic System

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**Time response of system with known parameters (without disturbance, ${\lambda}_{1}=-1,{\lambda}_{2}=-2$).

**Figure 6.**Time response of system with known parameters (without disturbance, ${\lambda}_{1}=-2,{\lambda}_{2}=-3$).

**Figure 7.**Phase diagram of system with known parameters (without disturbance, ${\lambda}_{1}=-1,{\lambda}_{2}=-2$).

**Figure 8.**Time response of system with known parameters (with disturbance, ${\lambda}_{1}=-1,{\lambda}_{2}=-2$).

**Figure 9.**Phase diagram of system with known parameters (with disturbance, ${\lambda}_{1}=-1,{\lambda}_{2}=-2$).

**Figure 10.**Time response of system with unknown parameters (without disturbance, ${\lambda}_{1}=-1$).

**Figure 11.**Time response of system with unknown parameters (without disturbance, ${\lambda}_{1}=-2$).

**Figure 12.**Phase diagram of system with unknown parameters (without disturbance, ${\lambda}_{1}=-1$).

**Figure 13.**Identification of unknown parameters $\widehat{a}$ and $\widehat{b}$ (without disturbance, ${\lambda}_{1}=-1$).

**Figure 16.**Identification of unknown parameters $\widehat{a}$ and $\widehat{b}$ (with disturbance, ${\lambda}_{1}=-1$).

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**MDPI and ACS Style**

Jia, H.; Guo, C.
The Application of Accurate Exponential Solution of a Differential Equation in Optimizing Stability Control of One Class of Chaotic System. *Mathematics* **2020**, *8*, 1740.
https://doi.org/10.3390/math8101740

**AMA Style**

Jia H, Guo C.
The Application of Accurate Exponential Solution of a Differential Equation in Optimizing Stability Control of One Class of Chaotic System. *Mathematics*. 2020; 8(10):1740.
https://doi.org/10.3390/math8101740

**Chicago/Turabian Style**

Jia, Hao, and Chen Guo.
2020. "The Application of Accurate Exponential Solution of a Differential Equation in Optimizing Stability Control of One Class of Chaotic System" *Mathematics* 8, no. 10: 1740.
https://doi.org/10.3390/math8101740