# Fractional Dynamics in Soccer Leagues

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## Abstract

**:**

## 1. Introduction

## 2. Modeling the Teams’ Dynamics

## 3. Entropy of the Spatio-Temporal Patterns of the Models’ Parameters

#### 3.1. The Entropy of the PL Model

#### 3.2. The Entropy of the Ho Model

## 4. Predicting the Teams’ Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Locus of the the models parameters for the 2018–2019 champions of ‘La Liga’ and ‘Premiership’: (

**a**) PL model for FC Barcelona; (

**b**) Ho model for FC Barcelona; (

**c**) PL model for Manchester City; (

**d**) Ho model for Manchester City. The point labels denote ${k}_{r}=3,\dots ,38$.

**Figure 4.**Histograms of the PL parameters $\{{a}_{i},{b}_{i}\}$ from the beginning, ${k}_{r}=3$, up to the end of the 2018–2019 season, ${k}_{r}=38$, and the leagues: (

**a**) ‘La Liga’; (

**b**) ‘Premiership’; (

**c**) ‘Serie A’; (

**d**) ‘Ligue 1’.

**Figure 5.**Evolution of the entropy $S\left({k}_{r}\right)$ versus ${k}_{r}=3,\dots ,38$, for the PL parameters and the ‘La Liga’, ‘Premiership’, ‘Serie A’ and ‘Ligue 1’, during the season 2018–2019.

**Figure 6.**The two-dimensional projections of the histograms of the Ho parameters, from the beginning, ${k}_{r}=3$, up to the end of the 2018–2019 season, and the leagues: (

**a**–

**c**) ‘La Liga’; (

**d**–

**f**) ‘Premiership’; (

**g**–

**i**) ‘Serie A’; (

**j**–

**l**) ‘Ligue 1’. The charts represent the combination of the pairs of parameters $\{{a}_{i},{b}_{i}\}$, $\{{a}_{i},{c}_{i}\}$ and $\{{b}_{i},{c}_{i}\}$.

**Figure 7.**Evolution of the entropy $S\left({k}_{r}\right)$ versus ${k}_{r}=3,\dots ,38$, for the Ho model parameters and the leagues ‘La Liga’, ‘Premiership’, ‘Serie A’ and ‘Ligue 1’, during the season 2018–2019.

SP | Qu | Hi | VP | PL | Ho | ||
---|---|---|---|---|---|---|---|

‘La Liga’ | $\mu $ | 1.3182 | 1.1832 | 1.5852 | 1.2051 | 1.4118 | 1.0782 |

$\sigma $ | 0.4986 | 0.5441 | 0.2592 | 0.5607 | 0.5746 | 0.4736 | |

‘Premiership’ | $\mu $ | 1.2354 | 1.1082 | 1.5061 | 1.1699 | 1.3368 | 1.0394 |

$\sigma $ | 0.5233 | 0.5529 | 0.2240 | 0.5984 | 0.5496 | 0.5112 | |

‘Serie A’ | $\mu $ | 1.1749 | 1.0574 | 1.5783 | 1.1596 | 1.3800 | 1.0045 |

$\sigma $ | 0.4426 | 0.4865 | 0.2966 | 0.5977 | 0.5963 | 0.4251 | |

‘Ligue 1’ | $\mu $ | 1.3035 | 1.2053 | 1.6562 | 1.2313 | 1.4357 | 1.1333 |

$\sigma $ | 0.5155 | 0.6144 | 0.3093 | 0.6080 | 0.5627 | 0.5813 |

SP | Qu | Hi | VP | PL | Ho | ||
---|---|---|---|---|---|---|---|

‘La Liga’ | $\mu $ | 2.0137 | 2.1177 | 1.9397 | 2.0604 | 2.1631 | 1.9172 |

$\sigma $ | 0.3807 | 0.4339 | 0.3076 | 0.3945 | 0.3684 | 0.4173 | |

‘Premiership’ | $\mu $ | 1.9765 | 1.9543 | 1.9936 | 2.0860 | 2.0926 | 1.9036 |

$\sigma $ | 0.5519 | 0.4019 | 0.2198 | 0.6336 | 0.5226 | 0.6672 | |

‘Serie A’ | $\mu $ | 1.8093 | 1.8717 | 1.8765 | 1.9462 | 2.0468 | 1.7322 |

$\sigma $ | 0.2930 | 0.2764 | 0.3042 | 0.4216 | 0.4139 | 0.5888 | |

‘Ligue 1’ | $\mu $ | 2.0407 | 2.1022 | 1.9912 | 2.1400 | 2.1935 | 1.9337 |

$\sigma $ | 0.3226 | 0.2835 | 0.2657 | 0.4683 | 0.4653 | 0.5536 |

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**MDPI and ACS Style**

Lopes, A.M.; Tenreiro Machado, J.A.
Fractional Dynamics in Soccer Leagues. *Symmetry* **2020**, *12*, 356.
https://doi.org/10.3390/sym12030356

**AMA Style**

Lopes AM, Tenreiro Machado JA.
Fractional Dynamics in Soccer Leagues. *Symmetry*. 2020; 12(3):356.
https://doi.org/10.3390/sym12030356

**Chicago/Turabian Style**

Lopes, António M., and Jose A. Tenreiro Machado.
2020. "Fractional Dynamics in Soccer Leagues" *Symmetry* 12, no. 3: 356.
https://doi.org/10.3390/sym12030356