  We demonstrate the manifestations of the nonlinear features in magnetic
dynamics and IV-characteristics of the $\varphi_0$ Josephson junction in the
ferromagnetic resonance region. We show that at small values of system
parameters, namely, damping, spin-orbit interaction, and Josephson to magnetic
energy ratio, the magnetic dynamics is reduced to the dynamics of the scalar
Duffing oscillator, driven by the Josephson oscillations. The role of
increasing superconducting current in the resonance region is clarified.
Shifting of the ferromagnetic resonant frequency and the reversal of its
damping dependence due to nonlinearity are demonstrated by the full
Landau-Lifshitz-Gilbert-Josephson system of equations, and in its different
approximations. Finally, we demonstrate the negative differential resistance in
the IV--characteristics, and its correlation with the foldover effect.
