Chaotic double pendulum simulation
A pendulum hanging from a pendulum is one of the simplest physical systems that is chaotic. Its equations are completely deterministic, yet two releases that differ by a hair soon do completely different things.
Interactive, free and runs in your browser. Part of the Dynamical Systems simulations.
What you will see
Eight double pendulums start almost exactly together, each a hundred-thousandth of a radian apart. For a few seconds they move as one, then they fan out across the screen. A live graph shows the logarithm of their separation climbing in a straight line: exponential growth.
Try this
- Watch the first few seconds closely, while the pendulums still move as one.
- Read the slope of the separation graph: that is the Lyapunov exponent.
- Wait for the 30-second replay and see if you can predict when they split.
Common misconception
A more accurate simulation would not make the double pendulum predictable for long. Because errors grow exponentially, each extra digit of precision only buys a fixed, short amount of extra time.
Exam-style question
Two releases differ by 10⁻⁵ rad and the separation grows as e^{λt} with λ ≈ 3 s⁻¹. Roughly when are they a full radian apart?
Solve 10⁻⁵ e^{3t} = 1: t = ln(10⁵)/3 ≈ 11.5/3 ≈ 3.8 s. Making the initial gap 1000 times smaller only adds ln(1000)/3 ≈ 2.3 s.
Key equations
δ(t) ≈ δ₀ e^{λt}λ = (1/t) ln(δ/δ₀)
Curriculum
Enrichment: nonlinear dynamics and chaos.