A theoretical investigation is carried out into the interpretation of the effect of fluid inertia on the complex viscosity function as measured on a controlled stress rheometer. The problem of non-unique solutions to the governing equations is considered for the parallel plate geometry. The locations of these solutions are investigated by considering the critical points of the complex mapping associated with the linear viscoelastic equations of motion. It is shown that these critical points play an important role in determining where convergence problems are likely to occur when applying numerical methods of solution to the governing equations. Analytical approximations based on a series expansion about a critical point are developed as an alternative approach to a numerical solution in the neighbourhood of a critical point. In order to verify the theoretical predictions a numerical simulation of the behaviour of a single element Maxwell fluid on a controlled stress rheometer is carried out for a parallel plate geometry.