The Logistic MapA toy model of a population that grows fast when small and starves when crowded: each year x becomes r x(1−x). Turn up the growth rate r (left to right) and the long-term population splits from one steady value into a 2-year cycle, a 4-year cycle, 8, 16… and then tips into full chaos, laced with narrow windows where order briefly returns. The journey: for every r the shader runs the iteration hundreds of times to shake off the transient, then builds a histogram of where the orbit lands — that density map is the picture. The doublings shrink by the universal Feigenbaum ratio 4.6692, marked in red.