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

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 ($x_{n+1} = μx_{n}^{1/2}$), 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.