The logistic map is a seemingly simple non-linear recurrence relation: \[ x_{n+1} = r x_n (1 - x_n) \] where \(r\) 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 \(x_n\) as \(n \rightarrow \infty\), and in particular how it depends on the parameter \(r\). It turns out that the behavior is quite complicated, indeed chaotic. When \(r \in [0, 3]\), the sequence has a unique limit for any initial data \(x_0 \in [0, 1]\), but for \(r > 3\), 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 \(r = 3.56\)). There is also chaotic behavior for negative values of \(r\).
These phenomena can be explored in the plot below. Click to zoom in, and shift-click to zoom out.