Delayed Hopf bifurcation of a ferrofluid interface subject to a time-dependent magnetic field

Previously, we demonstrated that the combination of a radial and azimuthal static magnetic field deforms a confined ferrofluid droplet in Hele-Shaw cell into a stably spinning "gear". Weakly nonlinear analysis was used to predict the evolution. With the azimuthal field fixed, the traveling wave solution bifurcates from the trivial solution, as the radial field strength is increased. We show that this is a Hopf bifurcation at the critical growth rate: a positive linear growth rate allows for a stably spinning "gear", while a negative one leads to a stationary state. A center manifold reduction is applied to show the geometrical equivalence between a two-harmonic-mode coupled ODE system and the Hopf bifurcation. Inspired by the well-known delay behavior of time-dependent Hopf bifurcations, we design a slowly-time-varying magnetic field such that the timing of the emergence of the spinning "gear" can be controlled. The time-varying parameters and initial perturbation are determined through an amplitude equation derived from a multiple-time-scale expansion. This amplitude equation also reveals hysteresis in the time-dependent field manipulation.