Deterministic Chaos in Malkus' Waterwheel : A simple dynamical system on the verge of low-dimensional chaotic behaviour

In 1972, W.V.R. Malkus invented and constructed the waterwheel that bears his name, along with a publication on toroidal convection that presents the same dynamics. The waterwheel's dynamics closely resembles those of the well-known Lorenz system, and therefore can be viewed as its mechanical analogy. The physical waterwheel is simple in its conception, yet not completely intuitive in its performance. A constant flow of water pours in at the top bucket of a simple circular symmetrical waterwheel. Low amounts of incoming water make the wheel roll permanently in the same direction. As we quasi-stationarily increase the incoming flow, the waterwheel enters a chaotic regime where it reverses in an apparently unpredictable way. A further increase in the incoming flow makes the waterwheel return to a periodic state, this time oscillating back and forth at fixed intervals. This thesis aims to study the waterwheel where most publications to this date have focused on its simplified analogies. Finally, we present numerical evidence on the degree of similarity between the waterwheel and its Lorenzian derivations.