Magnetic precessions in $\varphi_0$ junction along IV-characteristics.

We show that a current sweep along current-voltage (IV) characteristic curves of a $\varphi_0$-junction may lead to both regular and chaotic magnetization dynamics with a series of specific phase trajectories. The origin of these trajectories is related to the direct coupling between the magnetic moment and the Josephson current realized in these junctions. 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. Our numerical results also demonstrate appearance of DC component of superconducting current in accordance with earlier theoretical prediction [Phys. Rev. Lett. {\bf 102} , 017001 (2009)]. An excellent agrement of numerical and analytical results has been found describing the ferromagnetic resonance as Josephson frequency is tuned to the ferromagnetic one. We discuss experiments which can test our results.