  In the well known logistic map, the parameter of interest is weighted by a
coefficient that decreases linearly when this parameter increases. Since such a
linear decrease forms a specific case, we consider the more general case where
this coefficient decreases nonlinearly as in a hyperbolic tangent relaxation of
a system toward equilibrium. We show that, in this latter case, the asymptotic
values obtained via iteration of the logistic map are considerably modified. We
demonstrate that both the steepness of the nonlinear decrease as well as its
upper and lower boundaries significantly alter the bifurcation diagram. New
period doubling features and transitions to chaos appear, possibly leading to
regimes with small periods. Computations with a variety of parameter values
show that the logistic map can be significantly reordered in the case of a
nonlinear growth rate.
