xₙ₊₁ = r × x × (1 − x). With just this equation, you can model how a population might change over time under limited resources. Change r slightly and watch the system shift from stable → oscillating → completely chaotic.
r acts like a "growth rate." When r is small, x settles at one value. As r increases, it alternates between two values (period 2), then 4, 8, and so on, eventually reaching a completely unpredictable chaotic state.
r (growth rate)2.8
r (compare in the chaotic region)3.90
💡 Even if the starting values differ by only 0.0001 (see the butterfly-effect tab), their trajectories become completely different over time in the chaotic region. This is the mathematical basis of the idea that "the flap of a butterfly's wings can change the weather."