A study of the asymmetric Malkus waterwheel: the biased Lorenz equations.

In this work, the asymmetric case of the Malkus waterwheel is studied, where the water inflow to the system is biasing the system toward stable motion in one direction, like a Pelton wheel. The governing equations of this system, when expressed in Fourier space and decoupled to form a closed set, can be mapped into a four-dimensional space where they form a quasi-Lorenz system. This set of equations is analyzed in light of analogues of the Rayleigh Bernard convection and conclusions are drawn. The properties and behavior of the equations are studied and correlated to the physical model. Phase space behavior and linear stability analysis are used for this. Spectral analysis is used as a qualitative measure of chaos. Chaotic behavior is quantified through the calculation of the Lyapunov exponents and these are further correlated to the bifurcation diagrams for a conclusive analysis of the dynamical behavior of the system.