math / Patterns Over Time / free

Chaos & Dynamical Systems

Watch tiny changes grow into behavior that looks impossible to predict.

Follow the mechanism1 / 3

A state feeds back into its next state. The current value controls both the next increase and the next limitation, so the rule carries its own history forward.

iterated ruler = 3.20 · 2-cycle

Focus the idea

starting statenearby startlong-term values

control r3.20

initial x₀0.200

tail spread0.286

final separation0.000

Growth parameter r 3.20Initial state x₀ 0.200

At r = 3.20, the trajectory settles into a 2-cycle. The rule is deterministic, but changing the control parameter changes the long-term pattern.

Chaos is not the same as randomness: every next value still follows the rule. The difficulty is long-term prediction when tiny differences in starting state are repeatedly amplified.