SUMMARYSimulation codes for solving large systems of ordinary differential equations suffer from the disadvantage that bifurcation-theoretic results about the underlying dynamical system cannot be obtained from them easily, if at all. Bifurcation behaviour typically can be inferred only after significant computational effort, and even then the exact location and nature of the bifurcation cannot always be determined definitively. By incorporating relatively minor changes to an existing simulation code for the Taylor-Couette problem, specifically, by implementing the Newton-Picard method, we have developed a computational structure that enables us to overcome some of the inherent limitations of the simulation code and begin to perform bifurcation-theoretic tasks. While a complete bifurcation picture was not developed, three distinct solution branches of the Taylor-Couette problem were analysed. These branches exhibit a wide variety of behaviours, including Hopf bifurcation points, symmetry-breaking bifurcation points, turning points and bifurcation to motion on a torus. Unstable equilibrium and time-periodic solutions were also computed.