What is deterministic chaos?
A deterministic system is chaotic when arbitrarily small differences in the initial conditions grow exponentially, so long-term prediction becomes impossible even though the equations are exact and contain no randomness. This is the butterfly effect. The lab shows two textbook examples: the Lorenz attractor and the double pendulum.
The Lorenz attractor
Edward Lorenz's 1963 three-equation model of atmospheric convection produces a bounded, non-repeating trajectory shaped like a pair of butterfly wings — a strange attractor with fractal structure. Two trajectories that start a hair apart separate exponentially, a rate captured by a positive Lyapunov exponent.
The double pendulum
A pendulum hung from the end of another is one of the simplest mechanical systems that is chaotic. Its equations are nonlinear and coupled, so two releases from almost the same angle track together for a moment and then swing into completely different motions.
Note: the trajectories are produced by numerically integrating the exact equations of motion; the apparent unpredictability is real chaos, not numerical noise. The Lorenz parameters and pendulum lengths are shown in dimensionless model units.
Frequently asked questions
What is the butterfly effect?
It is the popular name for sensitive dependence on initial conditions: a difference too small to measure can, after enough time, lead to a completely different outcome. The term comes from Edward Lorenz's weather modeling.
Is chaos the same as randomness?
No. Chaotic systems are fully deterministic: the same initial state always produces the same trajectory. They only look random because the error between nearby states grows exponentially, so prediction fails in practice.
What is a strange attractor?
A set toward which a chaotic system settles that has a fractal structure and never repeats, like the Lorenz attractor. The trajectory stays bounded forever but never closes on itself.
Why is the double pendulum chaotic?
Its equations of motion are nonlinear and coupled, so energy is exchanged unpredictably between the two arms. A minute change in the release angle leads to a totally different swing within a few seconds.