Period-Three Implies Chaos

Li-Yorke (1975): if a continuous map f: [a,b]→[a,b] has a period-3 orbit, it has orbits of ALL periods plus uncountably many aperiodic orbits.

Parameters

r (growth rate) 3.83

x₀ 0.5

Iterations 500

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Period: —

Lyapunov λ: —

f(x) = rx(1−x)

r ≈ 3.83: period-3 window.
Li-Yorke: period 3 → chaos.
Sharkovskii: 3≻5≻7≻…≻2·3≻…≻2ⁿ≻…≻1

Feigenbaum δ ≈ 4.669...
Period doublings accumulate at r∞≈3.5699