  We present a numerical study of the interaction quench dynamics in a
superfluid ultracold Fermi gas confined in a three-dimensional cigar-shaped
harmonic trap. In the present paper we investigate the amplitude mode of the
superfluid order parameter after interaction quenches which start deep in the
BCS phase and end in the BCS-BEC crossover regime. To this end, we exploit the
Bogoliubov-de Gennes formalism which takes the confinement potential explicitly
into account and provides a microscopic fully coherent description of the
system. We find an anharmonic nonlinear oscillation of the modulus of the
superfluid order parameter, i.e., of the Higgs mode. This oscillation persists
for large times with only a small amplitude modulation being visible. We
connect the frequency and the mean value of this oscillation with the breaking
of Cooper pairs in the superfluid phase. Additionally, we demonstrate that the
occurrence of this persistent oscillation is connected to the onset of chaotic
dynamics in our model. Finally, we calculate an interaction quench phase
diagram of the Higgs mode for quenches on the BCS side of the BCS-BEC crossover
and discuss its properties as a function of the aspect ratio of the
cigar-shaped trap.
