Tsallis' q index and Mori's q phase transitions at edge of chaos

TL;DR

This paper explains why Tsallis statistics applies at the chaos threshold in logistic maps, linking Mori's q-phase transitions to the structure of the critical attractor and entropy rates.

Contribution

It reveals the connection between Mori's q-phase transitions, Feigenbaum's scaling function, and Tsallis entropy at the edge of chaos in logistic maps.

Findings

Identifies an infinite family of Mori's q-phase transitions at the chaos threshold.

Links the q-values of these transitions to Tsallis' entropic index.

Shows the q-Lyapunov coefficients match Tsallis entropy rates.

Abstract

We uncover the basis for the validity of the Tsallis statistics at the onset of chaos in logistic maps. The dynamics within the critical attractor is found to consist of an infinite family of Mori's $q$-phase transitions of rapidly decreasing strength, each associated to a discontinuity in Feigenbaum's trajectory scaling function $\sigma $. The value of $q$ at each transition corresponds to the same special value for the entropic index $q$, such that the resultant sets of $q$-Lyapunov coefficients are equal to the Tsallis rates of entropy evolution.