Asymptotic study of damped one-dimensional oscillator with close to impact potential

Dynamics of linearly damped oscillator with strongly nonlinear elastic potential is ivestigated. The description of the oscillator dynamics is provided by means of matching asymptotic expansions for strongly nonlinear and quasilinear vibrations. The quasilinear vibrations are investigated by standard methods whereas the strongly nonlinear vibrations require special asymptotics based on exactly integrable partial case. The coincidence of the results with numerical simulation data is satisfactory for any combination of the regimes.