  The Josephson $\varphi_0$ junctions with the current-phase relation $I = I_c
\sin (\varphi-\varphi_0)$, where the phase shift $\varphi_0$ is proportional to
the magnetic moment perpendicular to the gradient of the asymmetric spin-orbit
potential, demonstrate a number of unique features important for
superconducting spintronics and modern informational technologies. Here we show
that a current sweep along IV-characteristic of the $\varphi_0$ junction may
lead to regular magnetization dynamics with a series of specific phase
trajectories. The origin of these trajectories is related to a direct coupling
between the magnetic moment and the Josephson oscillations in these junctions,
and ferromagnetic resonance when Josephson frequency coincides with the
ferromagnetic one. We demonstrate that an external electromagnetic field can
control the dynamics of magnetic moment within a current interval corresponding
to a Shapiro step and produce topological transformation of specific precession
trajectories. We demonstrate the appearance of the DC component of
superconducting current and clarify its role in the transformation of
IV-characteristics in the resonance region. Good agreement between numerical
and analytical results has been found in the ferromagnetic resonance region.
The presented results might be used for developing novel resonance methods of
determination of the spin-orbit coupling parameter in the non-centrosymmetric
materials. We discuss experiments which can test our results.
