Fisher information, Crámer-Rao complexity and temperature in logistic map

Abstract: In this work we study the dynamics of the logistic map based on a probabilistic characterization in terms of the invariant density. We analyze the relevant regimes of the dynamics (regular, oscillatory, onset chaotic and fully chaotic) in terms of the Fisher information and the Crámer-Rao (CR) complexity. We found that these informational quantifiers allow to distinguish the dynamical regions of the map, by maximizing the Fisher information in the regular behavior and with the CR complexity exhibiting variations and a maximum near to the Pomeau-Maneville scenario. Fisher information as a function of time is examined to the light of the Frieden's informational interpretation of the Thermodynamics Second Law. Finally, we apply the Equipartition Theorem to propose a definition of temperature for the logistic map, thus providing a macroscopic signature of the dynamics.