Study of oscillators with a non-negative real-power restoring force and quadratic damping

Oscillators with a non-negative real-power restoring force and quadratic damping are considered in this paper. The equation of motion is transformed into a linear first-order differential equation for the kinetic energy. The expressions for the energy-displacement function are derived as well as the closed form exact solutions for the relationship between subsequent amplitudes. They are expressed in terms of incomplete Gamma functions. On the basis of these results, expressions for the phase trajectories and the loci of maximal velocities are obtained. It is also demonstrated that the time difference between two consecutive relative maxima and minima of the displacement can both increase and decrease with time.