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We demonstrate unusual features of phase dynamics, IV-characteristics andmagnetization dynamics of the $\varphi_0$ Josephson junction at small values ofspin-orbit interaction, ratio of Josephson to magnetic energy and Gilbertdamping. In particular, an anomalous shift of the ferromagnetic resonancefrequency with an increase of Gilbert damping is found. The ferromagneticresonance curves show the Duffing oscillator behaviour, reflecting thenonlinear nature of Landau-Lifshitz-Gilbert (LLG) equation. Based on thenumerical analysis of each term in LLG equation we obtained an approximatedequation demonstrated both damping effect and Duffing oscillator features. Theresulting Duffing equation incorporates the Gilbert damping in a special wayacross the dissipative term and the restoring force. A resonance method for thedetermination of spin-orbit interaction in noncentrosymmetric materials whichplay the role of barrier in $\varphi_0$ junctions is proposed.

Anomalous Gilbert Damping and Duffing Features of the SFS {\boldmath $\varphi_0$} Josephson Junction