Fractional damping effects on the transient dynamics of the Duffing oscillator

We consider the nonlinear Duffing oscillator in presence of fractional damping which is characteristic different physical situations. The system studied with a smaller and larger parameter value, that we call underdamped overdamped regimes. In both have relation between parameter, amplitude oscillations times to reach asymptotic behavior, called times. regime, study shows that, also here, there are for order derivatives their amplitudes can suddenly change small variations parameter. addition, this latter resonant-like behavior take place suitable values parameters system. These results corroborated by calculating corresponding Q?factor. expect these be useful better understanding dynamics its possible applications as modeling kind materials normally need complicated terms.