the logistic map
/ low-JIS-tik map /
Take the simplest imaginable rule for next year's population: a fixed fraction of this year's, with a brake that kicks in when the population gets too crowded. Apply it once, get next year; apply it again, get the year after; keep iterating. This one-line discrete rule — not a differential equation but its step-by-step cousin — is the logistic map, and astonishingly, this toy generates the full drama of order collapsing into chaos.
The map is x_(n+1) = r x_n (1 - x_n), where x_n is a population fraction between 0 and 1 and r is a growth parameter, typically between 0 and 4. Each step you plug the current value in and get the next: the r x_n part is reproduction, the (1 - x_n) factor is the crowding brake that shrinks growth as x approaches 1. Now sweep r upward and watch the long-term behaviour. For small r (up to 3) the iterates settle to a single fixed value — a steady population. Past r = 3 that fixed point goes unstable and the values start alternating between two — a 2-cycle (boom-bust every other year). Past about 3.449 the 2-cycle splits into a 4-cycle, then 8, then 16, the splittings coming faster and faster. Beyond roughly r = 3.5699 the period becomes effectively infinite: the sequence never repeats and behaves chaotically.
The logistic map is the standard first encounter with chaos because it is utterly elementary — a quadratic you can iterate on a calculator — yet it contains the period-doubling cascade, sensitive dependence, and a fractal bifurcation diagram. Its deep surprise, shown by Feigenbaum, is universality: the rate at which the period-doublings accelerate (a constant near 4.669) is the same for the logistic map and for whole classes of utterly different systems with a single hump. So this nursery-simple equation revealed a quantitative law of how order turns into chaos that nature obeys far beyond the model itself.
Iterate x_(n+1) = 3.9 x_n (1 - x_n) from x_0 = 0.5: you get 0.975, 0.0951, 0.3354, 0.8687, 0.4453, ... — bounded in (0,1) but with no pattern, never settling and never repeating. Start instead at x_0 = 0.5001 and within a few dozen steps the two sequences look completely unrelated, the hallmark of chaos in a one-line formula.
At r = 3.9 the same quadratic that gave a steady population now produces bounded but unpredictable, never-repeating values.
The logistic map is a discrete iteration, not the logistic differential equation. The continuous logistic ODE x' = r x (1 - x/K) is tame and never chaotic — it is one-dimensional, so it can only rise or fall monotonically. The discreteness of the map is exactly what unlocks oscillation and chaos in a single variable.