One equation, one knob. The logistic map x → r·x·(1−x) doubles its rhythm again and again, then dissolves into chaos — and the same pattern of period-doubling appears across all of nature. Drag the r slider.
Below r = 3 the population settles on one steady value. Just past 3 it starts bouncing between two values, then four, then eight — each doubling coming twice as fast. At r ≈ 3.57 it tips into chaos: no pattern repeats, yet tiny windows of order reappear inside the chaos. The bifurcation diagram (bottom half) plots every long-term value against r, and that perfect branching tree is why ecologists, economists and physicists all study this one map.