What to notice
Explore · Predict · Explain
Order & Chaos
When does a simple repeating rule become unpredictable?
Follow the values one step at a time
The idea
One rule, many behaviors
Apply xₙ₊₁ = rxₙ(1 − xₙ) repeatedly. Here x is a normalized value, not a measured population. A fixed point repeats one value; a cycle repeats several.
Nearby starts can separate in chaotic regimes even though both follow the same deterministic rule. The full map plots late iterations for many values of r. Finite sampling is not a proof that an orbit is chaotic.