  We demonstrate unusual features of phase dynamics, IV-characteristics and
magnetization dynamics of the $\varphi_0$ Josephson junction at small values of
spin-orbit interaction, ratio of Josephson to magnetic energy and Gilbert
damping. In particular, an anomalous shift of the ferromagnetic resonance
frequency with an increase of Gilbert damping is found. The ferromagnetic
resonance curves show the Duffing oscillator behaviour, reflecting the
nonlinear nature of Landau-Lifshitz-Gilbert (LLG) equation. Based on the
numerical analysis of each term in LLG equation we obtained an approximated
equation demonstrated both damping effect and Duffing oscillator features. The
resulting Duffing equation incorporates the Gilbert damping in a special way
across the dissipative term and the restoring force. A resonance method for the
determination of spin-orbit interaction in noncentrosymmetric materials which
play the role of barrier in $\varphi_0$ junctions is proposed.
