the logistic map

Imagine a population of insects that breeds once a year: more parents mean more offspring, but crowd too many together and they exhaust their food, so the next generation crashes. The simplest equation capturing both growth and self-limiting crowding, iterated year after year, is the logistic map, a single line of arithmetic that nonetheless contains the whole story of the route to chaos.

The logistic map is the one-dimensional discrete-time map x_{n+1} = r x_n (1 - x_n), where x_n is a normalized population between 0 and 1 and r (between 0 and 4) is the growth rate. The factor r x_n is unchecked growth; the factor (1 - x_n) is the brake that kills growth as the population fills its niche. Despite being about as simple as a nonlinear equation can be, its long-term behavior as r increases runs through the entire menagerie: a single stable fixed point for r < 3, a period-doubling cascade (2, 4, 8, ...) for 3 < r < 3.5699..., and chaos beyond, riddled with periodic windows (a famous period-3 window opens near r = 3.828).

Robert May's 1976 study of this map was a landmark: it showed that complex, unpredictable, chaotic behavior does not require complicated equations or many variables, since one quadratic recursion suffices. The map is the standard laboratory for period doubling, the Feigenbaum constants, sensitive dependence, and the very definition of deterministic chaos. Honest caveat: it is a caricature of real population dynamics, not a faithful ecological model; its value is conceptual, as the simplest concrete system in which the universal features of chaos can be seen and computed by hand.