Two-dimensional dissipative maps at chaos threshold: Sensitivity to initial conditions and relaxation dynamics

TL;DR

This paper investigates the sensitivity to initial conditions and relaxation dynamics of two-dimensional maps at the chaos threshold, revealing dual entropic indexes and a scaling law, with implications for nonextensive statistical mechanics.

Contribution

It demonstrates the dual nature of the entropic index in two-dimensional maps and confirms a scaling law between sensitivity and relaxation indexes, extending previous findings to new maps.

Findings

Dual entropic indexes for the Henon map ($q_{sen}<1$, $q_{rel}>1$)

Scaling law between sensitivity and relaxation indexes

Sensitivity properties are consistent across $z$-logistic, Henon, and Lozi maps

Abstract

The sensitivity to initial conditions and relaxation dynamics of two-dimensional maps are analyzed at the edge of chaos, along the lines of nonextensive statistical mechanics. We verify the dual nature of the entropic index for the Henon map, one ($q_{sen}<1$) related to its sensitivity to initial conditions properties, and the other, graining-dependent ($q_{rel}(W)>1$), related to its relaxation dynamics towards its stationary state attractor. We also corroborate a scaling law between these two indexes, previously found for $z$-logistic maps. Finally we perform a preliminary analysis of a linearized version of the Henon map (the smoothed Lozi map). We find that the sensitivity properties of all these $z$-logistic, Henon and Lozi maps are the same, $q_{sen}=0.2445...$