# A Nonlinear Five-Term System: Symmetry, Chaos, and Prediction

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## Abstract

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## 1. Introduction

## 2. System without Linearity

## 3. System’s Entropy

## 4. Chaos Prediction

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Bifurcation diagram. We change parameter a while keeping $b=0.05$, and initial conditions $(0.5,1,0.5)$.

**Figure 3.**Maximum Lyapunov exponents. We change a while keeping $b=0.05$, and initial conditions $(0.5,1,0.5)$.

**Figure 4.**Attractors observed in three planes illustrating chaos in system (1) for $a=1$: (

**a**) $x-y$, (

**b**) $x-z$, and (

**c**) $y-z$.

**Figure 6.**Bifurcation diagrams for initial conditions $(0.5,1,0.5)$ (black) and $(-0.5,1,-0.5)$ (red) while keeping $b=0.05$.

**Figure 7.**Coexisting attractors in system (1) for $\left(x\right(0),y(0),z(0)=(0.5,1,0.5)$ (black) and $\left(x\right(0),y(0),z(0)=(-0.5,1,-0.5)$ (red): (

**a**) $a=1.23$, (

**b**) $b=1.6$, (

**c**) $b=1.845$, and (

**d**) $a=1.93$.

**Figure 9.**Neuron network includes four layers. Data set is provided by system (1) and is used for training.

**Figure 10.**Signals $x,y,z$ of system (1) (black color) and desired signals $X,Y,Z$ at the output of the neural network (red color): (

**a**) x and X, (

**b**) y and Y, (

**c**) z and Z.

System | Linear Term | Nonlinear Term | Total Term |
---|---|---|---|

[17] | 6 | 4 | 10 |

[1] | 6 | 2 | 8 |

[20] | 1 | 7 | 8 |

[27] | 0 | 8 | 8 |

[16] | 3 | 4 | 7 |

[28] | 2 | 5 | 7 |

[11] | 2 | 4 | 6 |

[9] | 2 | 3 | 5 |

This work | 0 | 5 | 5 |

Cases | a | ApEn |
---|---|---|

1 | 1 | 0.2162 |

2 | 1.5 | $7.418\times {10}^{-7}$ |

3 | 2 | 0.3526 |

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**MDPI and ACS Style**

Thoai, V.P.; Kahkeshi, M.S.; Huynh, V.V.; Ouannas, A.; Pham, V.-T.
A Nonlinear Five-Term System: Symmetry, Chaos, and Prediction. *Symmetry* **2020**, *12*, 865.
https://doi.org/10.3390/sym12050865

**AMA Style**

Thoai VP, Kahkeshi MS, Huynh VV, Ouannas A, Pham V-T.
A Nonlinear Five-Term System: Symmetry, Chaos, and Prediction. *Symmetry*. 2020; 12(5):865.
https://doi.org/10.3390/sym12050865

**Chicago/Turabian Style**

Thoai, Vo Phu, Maryam Shahriari Kahkeshi, Van Van Huynh, Adel Ouannas, and Viet-Thanh Pham.
2020. "A Nonlinear Five-Term System: Symmetry, Chaos, and Prediction" *Symmetry* 12, no. 5: 865.
https://doi.org/10.3390/sym12050865