# New Chaotic Systems with Two Closed Curve Equilibrium Passing the Same Point: Chaotic Behavior, Bifurcations, and Synchronization

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## Abstract

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## 1. Introduction

## 2. A New Family with Two Closed Curve Equilibrium

## 3. Anti-Synchronization of New Systems

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Different shapes of equilibrium points of system (5), k = 1, 2, 3, 4, 5, from the interior to the outside, respectively, in the plane $w=0$.

**Figure 5.**3D view of the chaotic attractor and eye-shape of equilibrium points located in the plane w = 0 of system (7) for $\alpha =5,\beta =30$.

**Figure 6.**The projection of the trajectory of system (7) in (

**a**) u-v plane, (

**b**) u-w plane, (

**c**) v-w plane for $\alpha =5,\beta =30$.

**Figure 7.**The Poincaré section of system (7) for (

**a**) $z=0.2$, (

**b**) $y=0.2$, (

**c**) $x=-0.2$ for $\alpha =5,\beta =30$.

**Figure 8.**Bifurcation plot of system (7) for (

**a**) $\alpha =5,\beta \in [28,48]$ and (

**b**) $\beta =30,\alpha \in [3,5.5]$.

**Figure 11.**Periodic behavior of system (7) in the u-w plane: (

**a**) period-1 $(\beta =45)$, (

**b**) period-2 $(\beta =38)$, (

**c**) period-4 $(\beta =36)$.

**Figure 12.**Anti-synchronization of the driver-response system: (

**a**) $u,{u}_{1}$, (

**b**) $v,{v}_{1}$, (

**c**) $w,{w}_{1}$, the driver system (

**dashed lines**), the response system (

**solid lines**).

**Figure 13.**Time history of the anti-synchronization of the state errors system: (

**a**) ${e}_{1}-t$, (

**b**) ${e}_{2}-t$, (

**c**) ${e}_{3}-t$.

System | Equilibrium | Closed Curve Equilibrium | Paper |
---|---|---|---|

Gotthans, Sprott, and Petrzela | ${u}^{2}+{v}^{2}-1=0$ | Circle | [11] |

Zhu and Du | ${\left|u\right|}^{k}+{\left|v\right|}^{k}-1=0$ | Circle, Square, etc | [13] |

Wang, Pham, and Volos | ${u}^{2}-\left|uv\right|+{v}^{2}-1=0$ | Cloud-shaped | [19] |

New system (see below) | ${u}^{2}-\left|u\right|+\left|v\right|+{v}^{2}=0$ | Eye-shaped | This work |

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**MDPI and ACS Style**

Zhu, X.; Du, W.-S.
New Chaotic Systems with Two Closed Curve Equilibrium Passing the Same Point: Chaotic Behavior, Bifurcations, and Synchronization. *Symmetry* **2019**, *11*, 951.
https://doi.org/10.3390/sym11080951

**AMA Style**

Zhu X, Du W-S.
New Chaotic Systems with Two Closed Curve Equilibrium Passing the Same Point: Chaotic Behavior, Bifurcations, and Synchronization. *Symmetry*. 2019; 11(8):951.
https://doi.org/10.3390/sym11080951

**Chicago/Turabian Style**

Zhu, Xinhe, and Wei-Shih Du.
2019. "New Chaotic Systems with Two Closed Curve Equilibrium Passing the Same Point: Chaotic Behavior, Bifurcations, and Synchronization" *Symmetry* 11, no. 8: 951.
https://doi.org/10.3390/sym11080951