# Chaotic Dynamics of the Fractional-Love Model with an External Environment

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Love Model

## 3. Chaotic Dynamics of the Fractional Order Love Model with the External Environment

#### 3.1. Analysis of the Systemic Dynamics of the Fixed Parameters (a = −1.5, b = −2, and c = d = 1)

#### 3.1.1. Time Series

#### 3.1.2. Phase Portrait

#### 3.1.3. Power Spectrum

#### 3.1.4. Poincaré Map

- (1)
- The periodic motion leaves a limited number of discrete points on this cross section;
- (2)
- The quasi-periodic motion leaves a closed curve on the cross section;
- (3)
- The chaotic motion is along a line or a curved-arc distribution point that is set on the cross section.

#### 3.1.5. Maximal Lyapunov Exponent

_{0}→

_{0}can make sure of the validity of the linear approximation at any time.

#### 3.2. Analysis of the Systemic Dynamics of the Fixed Fractional Orders (α = β = 0.85)

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Time series of the system with different fractional order q values, and subfigures (

**a**–

**k**) are the fractional order q changed from 1 to 0.5 by 0.05 steps.

**Figure 2.**Phase portrait of the system with different fractional order q values, and subfigures (

**a**–

**k**) are the fractional order q changed from 1 to 0.5 by 0.05 steps.

**Figure 3.**Power spectrum of the system with different fractional order q values, and subfigures (

**a**–

**k**) are the fractional order q changed from 1 to 0.5 by 0.05 steps.

**Figure 4.**Poincaré map of the system with different fractional order q values, and subfigures (

**a**–

**k**) are the fractional order q changed from 1 to 0.5 by 0.05 steps.

**Figure 5.**Time series (

**a**), phase portrait (

**b**), power spectrum (

**c**), and Poincaré map (

**d**) of the system when a = −5.

**Figure 6.**Time series (

**a**), phase portrait (

**b**), power spectrum (

**c**), and Poincaré map (

**d**) of the system when a = −2.42.

**Figure 7.**Time series (

**a**), phase portrait (

**b**), power spectrum (

**c**), and Poincaré map (

**d**) of the system when a = −2.

**Figure 8.**Time series (

**a**), phase portrait (

**b**), power spectrum (

**c**), and Poincaré map (

**d**) of the system when a = −1.76.

**Figure 9.**Time series (

**a**), phase portrait (

**b**), power spectrum (

**c**), and Poincaré map (

**d**) of the system when a = −1.53.

**Figure 10.**Time series (

**a**), phase portrait (

**b**), power spectrum (

**c**), and Poincaré map (

**d**) of the system when a = −1.45.

**Table 1.**The value of parameter “a” that can produce chaotic motion with different fractional order.

Fractional order α = β | α = β = 1 | α = β = 0.9 | α = β = 0.8 | α = β = 0.7 | α = β = 0.6 | α = β = 0.5 |

Parameter “a” | a = −2.88 | a = −2.22 | a = −1.8 | a = −1.65 | a = −1.45 | a = −1.22 |

**Table 2.**Maximal Lyapunov exponent (MLE) of the system with different fractional orders when the parameters are fixed.

Fractional Order (q) | MLE (λ) | Dynamic State |
---|---|---|

q = 1 | λ = −0.0458 | periodic |

q = 0.95 | λ = −0.0299 | periodic |

q = 0.9 | λ = 0.0711 | chaotic |

q = 0.85 | λ = 0.3467 | chaotic |

q = 0.8 | λ = −0.0331 | periodic |

q = 0.75 | λ = −0.0344 | periodic |

q = 0.7 | λ = 0.2456 | chaotic |

q = 0.65 | λ = 0.3585 | chaotic |

q = 0.6 | λ = 0.1835 | chaotic |

q = 0.55 | λ = 0.0754 | chaotic |

q = 0.5 | λ = 0.0357 | chaotic |

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**MDPI and ACS Style**

Huang, L.; Bae, Y.
Chaotic Dynamics of the Fractional-Love Model with an External Environment. *Entropy* **2018**, *20*, 53.
https://doi.org/10.3390/e20010053

**AMA Style**

Huang L, Bae Y.
Chaotic Dynamics of the Fractional-Love Model with an External Environment. *Entropy*. 2018; 20(1):53.
https://doi.org/10.3390/e20010053

**Chicago/Turabian Style**

Huang, Linyun, and Youngchul Bae.
2018. "Chaotic Dynamics of the Fractional-Love Model with an External Environment" *Entropy* 20, no. 1: 53.
https://doi.org/10.3390/e20010053