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. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 200605.10.-a Computational methods in statistical physics and nonlinear dynamics, 05.45.-a Nonlinear dynamics and nonlinear dynamical systems, 05.45.Pq Numerical simulations of chaotic systems ,
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