Numerical Study of the Oscillatory Convergence to the Attractor at the Edge of Chaos

TL;DR

This study investigates the convergence behavior of different chaos onset types in logistic maps, analyzing oscillations, scaling, and entropy to understand the dynamics near the edge of chaos.

Contribution

It compares three chaos transition types in logistic maps and explores how Tsallis entropy and escort distributions influence convergence and oscillations.

Findings

Different regimes show distinct log-log oscillations and power-law scalings.

Tsallis entropy's escort distribution can tune oscillations and crossover times.

Convergence behaviors vary depending on initial conditions and bifurcation types.

Abstract

This paper compares three different types of ``onset of chaos'' in the logistic and generalized logistic map: the Feigenbaum attractor at the end of the period doubling bifurcations; the tangent bifurcation at the border of the period three window; the transition to chaos in the generalized logistic with inflection 1/2 ($x_{n+1} = \mu x_{n}^{1/2}$), in which the main bifurcation cascade, as well as the bifurcations generated by the periodic windows in the chaotic region, collapse in a single point. The occupation number and the Tsallis entropy are studied. The different regimes of convergence to the attractor, starting from two kinds of far-from-equilibrium initial conditions, are distinguished by the presence or absence of log-log oscillations, by different power-law scalings and by a gap in the saturation levels. We show that the escort distribution implicit in the Tsallis entropy may tune the log-log oscillations or the crossover times.