  Flux jumps observed in high-$J_c$ Nb$_3$Sn conductors are urgent problems to
construct high field superconducting magnets. The low-field instabilities
usually reduce the current-carrying capability and thus cause the premature
quench of Nb$_3$Sn coils at low magnetic field. In this paper, we explore
suppressing the flux jumps by ferromagnetic (FM) layer. Firstly, we
experimentally and theoretically investigate the flux jumps of Nb$_3$Sn/FM
hybrid wires exposed to a magnetic field loop with constant sweeping rate.
Comparing with bare Nb$_3$Sn and Nb$_3$Sn/Cu wires, we reveal two underlying
mechanisms that the suppression of flux jumps is mainly attributed to the
thermal effect of FM layer for the case of lower sweeping rate, whereas both
thermal and electromagnetic effects play a crucial role for the case of higher
sweeping rate. Furthermore, we explore the flux jumps of Nb$_3$Sn/FM hybrid
wires exposed to AC magnetic fields with amplitude $B_{a0}$ and frequency
$\rm\omega$. We build up the phase diagrams of flux jumps in the plane
$\rm\omega$-$B_{a0}$ for bare Nb$_{3}$Sn wire, Nb$_{3}$Sn/Cu wire and
Nb$_{3}$Sn/FM wire, respectively. We stress that the region of flux jumps of
Nb$_{3}$Sn/FM wire is much smaller than the other two wires, which indicates
that the Nb$_{3}$Sn/FM wire has significant advantage over merely increasing
the heat capacity. The findings shed light on suppression of the flux jumps by
utilizing FM materials, which is useful for developing new type of high-$J_c$
Nb$_{3}$Sn conductors.
