  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.
