Sharkovskii's Theorem (1964): Order the integers: 3 ▷ 5 ▷ 7 ▷ … ▷ 2·3 ▷ 2·5 ▷ … ▷ 4 ▷ 2 ▷ 1. If a continuous map has a period-n orbit, it has period-m orbits for all m that come after n.
Corollary (Li-Yorke 1975): Period 3 implies chaos (every period exists). For the logistic map f(x)=rx(1-x), a period-3 orbit first appears at r≈3.8284.
Left: Cobweb diagram showing orbit. Right: Bifurcation diagram.