The logistic map is a seemingly simple non-linear recurrence relation:
where is a real parameter. It is useful as a model of population dynamics, for example.
It is natural to ask about the long-term behavior of the values as ,
and in particular how it depends on the parameter . It turns out that the behavior is quite
complicated, indeed chaotic. When , the sequence has a unique limit for any initial data
, but for , strange things start to happen. There is no longer a unique limit,
instead the sequence values oscillate between multiple targets, and the number of targets changes as well:
from 1, to 2 (i.e., there is a bifurcation), to 4, to 8, and so on, but eventually there is no finite set
of "limits", and chaos takes over (slightly after ). There is also chaotic behavior for negative
values of .
These phenomena can be explored in the plot below. Click to zoom in, and shift-click to zoom out.