Uniform rotation and multi-period oscillation in the Malkus-Lorenz waterwheel

The Malkus-Lorenz waterwheel is a simple mechanical system whose equations of motion reduce to the Lorenz equations under simplifying assumptions. Our apparatus has 36 acrylic cylinders around the periphery of a wheel with a 9 in (0.229 m) radius. The wheel's shaft is held by two air bushings and a flat air bearing to support the axial force. An aluminum ring around the periphery of the wheel passes between the poles of variable gap magnets, producing a frictional torque approximately proportional to the angular velocity. The wheel's angular position is measured with a non-contact, two-axis Hall sensor, and phase space variables are obtained by differentiation via fast Fourier transformation. The wheel can exhibit all three types of motion seen in the Lorenz equations: uniform rotation, periodic reversals, and chaotic reversals. The first two types of motion, although conceptually simplest, nevertheless reveal intriguing details about the wheel and its dynamics. Minor variations in the uniform rotation can yield information about imperfections in the wheel's construction and departures from ideal behavior caused by the use of discrete cells to contain the water in the wheel. We also see within the periodic region bands of mostly period-3 motion that occur outside the chaotic region when using the brake strength as the bifurcation parameter. We also present progress toward computer simulations to model both types of motion in a more realistic manner than the usual simplifying assumptions.