Anomalous Gilbert damping and Duffing features of the superconductor-ferromagnet-superconductor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>φ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> Josephson junction

We study the phase dynamics, IV characteristics, and magnetization dynamics of the ${\ensuremath{\varphi}}_{0}$ Josephson junction at small values of spin-orbit interaction, ratio of the Josephson junction to magnetic energy, and Gilbert damping. We demonstrate that the coupled Landau-Lifshitz-Gilbert-Josephson dynamics is reduced to a scalar nonlinear Duffing oscillator. The resulting Duffing equation incorporates the Gilbert damping in a special way across the dissipative term and the restoring force. An anomalous shift of the ferromagnetic resonance frequency with decreasing Gilbert damping is found. We demonstrate that there is a critical damping value at which nonlinearity comes into play, and it changes the damping dependence of the ferromagnetic resonance.

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