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

Ignacio Sebastián Gomez, Guilherme Vieira Brito Junior, Samuel Silva Santos, Zolacir Trindade Oliveira Júnior

TL;DR

This paper investigates how the logistic map's dynamical regimes can be distinguished using probabilistic descriptions based on invariant densities. It computes Fisher information and Crámer–Rao complexity across the control parameter μ, uncovering regime-dependent patterns and a notable CR complexity peak near the Pomeau–Manneville onset. It also examines the temporal evolution of these quantities in the context of Frieden's informational interpretation of the Second Law and introduces a temperature notion via the Equipartition Theorem, with formal asymptotics for the temperature. The results offer a macroscopic dynamical signature and a framework to extend informational-thermodynamic reasoning to other discrete maps.

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.

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

TL;DR

This paper investigates how the logistic map's dynamical regimes can be distinguished using probabilistic descriptions based on invariant densities. It computes Fisher information and Crámer–Rao complexity across the control parameter μ, uncovering regime-dependent patterns and a notable CR complexity peak near the Pomeau–Manneville onset. It also examines the temporal evolution of these quantities in the context of Frieden's informational interpretation of the Second Law and introduces a temperature notion via the Equipartition Theorem, with formal asymptotics for the temperature. The results offer a macroscopic dynamical signature and a framework to extend informational-thermodynamic reasoning to other discrete maps.

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.

Paper Structure

This paper contains 16 sections, 16 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Probability density functions, discrete maps and invariant density
  4. Logistic map
  5. Fisher information and Crámer-Rao complexity
  6. Equipartition Theorem
  7. Statistical quantifiers and temperature for logistic map
  8. Fisher information and CR complexity of logistic map
  9. Evolution of Fisher information and CR complexity of logistic map
  10. Temperature associated to logistic map
  11. Numerical results
  12. Temperature as a function of time
  13. Some formal results
  14. Fisher information and CR complexity as a function of temperature
  15. Fisher information and CR complexity as a function of time
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Fisher information \ref{['Fisher-information']} of the invariant density \ref{['numerical-invariant-density']} of the logistic map as a function of the parameter $\mu$ for $\mu=0.05, 0.1,\ldots, 3.9, 3.95, 4$ with $N=10^6$ number of steps and $W=10^4$ bins.
  • Figure 3: CR complexity \ref{['CR']} as a function of the Fisher information \ref{['Fisher-information']} for the invariant density \ref{['numerical-invariant-density']} of the logistic map with $\mu=0.05, 0.1,\ldots, 3.9, 3.95, 4$ with $N=10^6$ number of steps and $W=10^4$ bins.
  • Figure 4: Averaged value of the map temperature \ref{['MP']} for $\mu=0.05, 0.1, \ldots, 3.9, 3.95, 4$ with $N=10^3$ number of steps and $M=10^5$ initial conditions.