*Article* **Fractional Dynamics in Soccer Leagues**

#### **António M. Lopes 1,\* and Jose A. Tenreiro Machado 2**


Received: 27 January 2020; Accepted: 16 February 2020; Published: 1 March 2020

**Abstract:** This paper addresses the dynamics of four European soccer teams over the season 2018–2019. The modeling perspective adopts the concepts of fractional calculus and power law. The proposed model embeds implicitly details such as the behavior of players and coaches, strategical and tactical maneuvers during the matches, errors of referees and a multitude of other effects. The scale of observation focuses the teams' behavior at each round. Two approaches are considered, namely the evaluation of the team progress along the league by a variety of heuristic models fitting real-world data, and the analysis of statistical information by means of entropy. The best models are also adopted for predicting the future results and their performance compared with the real outcome. The computational and mathematical modeling lead to results that are analyzed and interpreted in the light of fractional dynamics. The emergence of patterns both with the heuristic modeling and the entropy analysis highlight similarities in different national leagues and point towards some underlying complex dynamics.

**Keywords:** Fractional dynamics; Power law behavior; Complex systems; Soccer
