Detecting bifurcations in numerical simulation of fluid flow

We present a general method for detecting bifurcations in numerical simulations of dissipative fluid flow whose solutions possess attracting sets for certain values of the parameters of the system. We describe the method as 'fast-slow' steering. The parameters governing system behaviour are slowly steered within a chosen range of interest, as the system evolves in time. The parameter changes are continuous and are 'slow' relative to the 'fast' rate of the time stepping of the system. This allows the simulated system to evolve continuously through a series of states sufficiently close to a trajectory of attractors of the system held fixed at each successive value on the trajectory. We present the method in its general form which can be applied to any suitable dynamical system. We then present a detailed unfolding of a bifurcation sequence in the numerical solutions of magnetohydrodynamic equations modelling dynamo acting in a toroidal volume. Torus-shaped discs have been observed in astrophysical contexts around collapsed objects (e.g. black holes). Our results conform to those obtained using the traditional computational approach of 'quasi-static' steering, and are also compatible with the mathematical analysis.