# Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode

## Abstract

**:**

## 1. Introduction

## 2. Formulation of a Modified Chaotic Jerk Circuit with Chua’s Diode to a System of ODEs

## 3. Analysis for Chaotic Behavior of the System

#### 3.1. Case 1: $-1<x<1$

#### 3.2. Case 2: $-\frac{{v}_{f}}{{v}_{e}}\u2a7dx\u2a7d-1$

#### 3.3. Case 3: $1\u2a7dx\u2a7d\frac{{v}_{f}}{{v}_{e}}$

#### 3.4. Type of Equilibrium Points

**Theorem**

**1**

**Theorem**

**2.**

#### 3.5. Localization of a Hidden Attractor of The System

**Theorem**

**3**

## 4. Numerical Experiment

**Remark**

**1.**

#### 4.1. Mathematical Analysis of the System

#### 4.2. Time Waveforms and Trajectories of The System

#### 4.3. Effects of Changing Initial Points

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Chua’s circuit [14].

**Figure 2.**A chaotic jerk circuit [19].

**Figure 14.**Effects of changing initial points in X-axis from $I1=(0,-0.7,0)$ to $I2=(0.0001,-0.7,0)$.

**Figure 16.**Effects of changing initial points in Z-axis from $I1=(0,-0.7,0)$ to $I2=(0,-0.7,0.0001)$.

No. | Terms of Comparison | Ref. [19] | Ref. [14] | This Paper |
---|---|---|---|---|

1 | Number of equilibrium points | 1 | 3 | 3 |

2 | Number of eigenvalues | 3 | 9 | 9 |

3 | Types of trajectories | 1 saddle focus node | 1 stable focus node and 2 saddle foci | 3 saddle foci |

4 | Number of components | 14 | 5 | 15 |

5 | Positions of equilibrium points | a point | 3 symmetric points | 3 symmetric points |

6 | Jerk-circuit type | yes | no | yes |

7 | Existence of Chua’s diode | no | yes | yes |

8 | Existence of chaotic attractors | yes | yes | yes |

9 | Sensitivity to initial conditions | $\surd \surd $ | $\surd $ | $\surd \surd \surd $ |

10 | Nonlinear system | yes | yes | yes |

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Chansangiam, P.
Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode. *Symmetry* **2020**, *12*, 1803.
https://doi.org/10.3390/sym12111803

**AMA Style**

Chansangiam P.
Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode. *Symmetry*. 2020; 12(11):1803.
https://doi.org/10.3390/sym12111803

**Chicago/Turabian Style**

Chansangiam, Pattrawut.
2020. "Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode" *Symmetry* 12, no. 11: 1803.
https://doi.org/10.3390/sym12111803