Eur. Phys. J. B 50, 355-359 (2006)
https://doi.org/10.1140/epjb/e2006-00064-6

Numerical study of the oscillatory convergence to the attractor at the edge of chaos

¹ Physics Department and INFN, University of Cagliari, Cagliari, Italy
² SLACS Laboratory, Physics Department, University of Cagliari, Cagliari, Italy

Corresponding authors: ^a roberto.tonelli@dsf.unica.it - ^b massimo.coraddu@ca.infn.it

Received: 21 October 2005
Revised: 2 December 2005
Published online: 20 February 2006

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 = 1-μx_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.