Michael Bradford Williams

Home / Programming / Bifurcations in the Logistic Map

The logistic map is a seemingly simple non-linear recurrence relation: xn+1=rxn(1xn) 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 xn as n, and in particular how it depends on the parameter r. It turns out that the behavior is quite complicated, indeed chaotic. When r[0,3], the sequence has a unique limit for any initial data x0[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.

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