The Lyapunov FractalA population that grows by the rule "next = rate × now × (1 − now)" can settle to a steady value or thrash about chaotically, depending on the rate. Here the rate flips back and forth between two values a and b following a fixed pattern (say a,b,b,a,b). Each pixel is one (a,b) pair; warm gold means the population locks into a calm cycle, deep blue-black means it never settles. The journey: we measure how fast two almost-identical populations drift apart — the Lyapunov exponent, the average of ln|rate · (1−2 now)| — and colour by its sign.