Robert May, 1976 — one line of arithmetic covering the whole road from stability to chaos
Before the 1970s there was a default correspondence in the sciences: simple equations give simple behaviour, and complicated behaviour must come from complicated causes or from outside random shocks. Population ecology worked exactly this way — violent fluctuations in the data were assigned to weather, disease, predators.
Robert May's question was: can a rule with one parameter, simple enough for a schoolchild to compute, generate behaviour that looks entirely random all by itself? The logistic map is his answer, and a counterexample prepared specifically for the intuition that complexity must come from complexity.
The two factors each do one job: r × x is growth (more now, more next step), and (1 − x) is the crowding discount (the closer to full, the harsher). Multiply them and you get a downward parabola — a single maximum, what mathematicians call unimodal. That single hump is not decoration; it is the precondition for everything that follows.
The cheapest way to see it is one picture: this year's x on the horizontal axis, next year's on the vertical. The rule itself is the parabola. Add a 45° line, which stands for "next year equals this year" — where the two curves cross is a fixed point, the candidate stable value.
The iteration can be walked directly on this picture: go up from the axis to the parabola (that is next year's value), across to the 45° line (which turns next year into this year), then up again. The resulting zig-zag is a cobweb diagram.
Sweep r from small to large and four regimes appear in order:
r < 1: everything goes to 0 whatever the start — extinction.
1 < r < 3: it settles on a nonzero stable value, (r−1)/r, the same every year.
3 < r < 3.5699: the stable value loses stability; behaviour becomes 2 alternating values, then 4, 8, 16… These period-doubling bifurcations arrive at intervals shrinking by a factor of about 4.669, so infinitely many of them complete before r∞ ≈ 3.5699.
r > 3.5699: over most of the range no period is visible at all, shot through with infinitely many narrow periodic windows (the most conspicuous being the period-3 window opening at r = 1+√8 ≈ 3.8284).
The most famous property of the chaotic region is sensitive dependence on initial conditions: two starting points differing by a hair separate into unrelated trajectories within a handful of steps.
One counterintuitive detail is worth keeping: at r = 4 the rule can be solved exactly — substitute x = sin²(πθ) and the iteration becomes "double θ each step and keep the fractional part." So the chaos here is both fully determined and expressible in closed form. Chaos and "unsolvable" are two different things.
The logistic map is almost never used to forecast a specific system. Its value is that it retires a whole class of inference: irregular fluctuation is not evidence of an irregular external cause. A fixed rule with no random component at all is enough to produce sequences that are hard even to distinguish from noise.
Where it lands is the family of structures with lagged self-regulation: population and fishery quotas set from last period's count, agricultural supply set from last year's price, production and delivery load set from last cycle's shortfall. What these share is a shape — the stronger the response, the easier it is to cross the boundary of stability, and past that boundary, responding harder makes it worse.
More important is the universality result it carries: any unimodal map with a smooth peak takes the quantitatively identical road to chaos, with the Feigenbaum constant δ = 4.6692 independent of the equation. That has been measured in convecting liquid helium, in nonlinear circuits and in chemical reactions. Which means you can extrapolate the critical point from the first two or three bifurcations without ever knowing the equation.