I plotted the variable x versus the parameter b,
b is varying between 0 and 1.

inclreasing l
induces a series of period-doubling bifurcation where a stable point drive
to cycles of increasing period.

__zoom__

here we can notice the first period-doubling bifurcation
before a chaotic behaviour.

when b is lower than 0.4 we have a convergence to one
point then for 0.4< b <0.59 it converge to 2 points then 4 then 8
then ... until a chaotic behavior, and we can notice in the same times
that for b between 0.65 and 0.9 the system is chaotic.