  A ferrofluid droplet confined in a Hele-Shaw cell can be deformed into a
stably spinning ``gear,'' using crossed magnetic fields. Previously, fully
nonlinear simulation revealed that the spinning gear emerges as a stable
traveling wave along the droplet's interface bifurcates from the trivial
(equilibrium) shape. In this work, a center manifold reduction is applied to
show the geometrical equivalence between a two-harmonic-mode coupled system of
ordinary differential equations arising from a weakly nonlinear analysis of the
interface shape and a Hopf bifurcation. The rotating complex amplitude of the
fundamental mode saturates to a limit circle as the periodic traveling wave
solution is obtained. An amplitude equation is derived from a
multiple-time-scale expansion as a reduced model of the dynamics. Then,
inspired by the well-known delay behavior of time-dependent Hopf bifurcations,
we design a slowly time-varying magnetic field such that the timing and
emergence of the interfacial traveling wave can be controlled. The proposed
theory allows us to determine the time-dependent saturated state resulting from
the dynamic bifurcation and delayed onset of instability. The amplitude
equation also reveals hysteresis-like behavior upon time reversal of the
magnetic field. The state obtained upon time reversal differs from the state
obtained during the initial (forward-time) period, yet it can still be
predicted by the proposed reduced-order theory.
