Notes on the videotape Nonlinear Dynamics and Chaos: Lab Demonstrations

A tabletop waterwheel, designed and built by Prof. Willem Malkus (Math. Dept., MIT), is used to demonstrate chaos in a mechanical analog of the Lorenz equations. The waterwheel’s rotational damping rate can be adjusted by tightening or loosening a brake. When the brake is not too tight, the wheel settles into a steady rotation. Either direction of rotation is possible. When the brake was tightened in an attempt to make the wheel go chaotic, unfortunately it was tightened too much. Instead of settling into the desired pattern of irregular reversals, the wheel fell into a simple pendulum-like motion, turning once to the left, then back to the right, and so on. After the brake was loosened slightly, the motion became chaotic. (And the crowd went wild...) The waterwheel is an exact mechanical analog of the Lorenz equations (see Exercise 9.1.3 in Strogatz (1994)). The corresponding Lorenz parameters are b = 1, σ ∝ ν, and r ∝ ν, where ν is the rotational damping rate. So as the brake is tightened, the parameters are constrained to move along a hyperbola rσ = C in the (r, σ) parameter space.