Oscillations with one degree of freedom and discontinuous energy

In 1995 for a linear oscillator, Myshkis imposed a constant impulse to the velocity, each moment the energy reaches a certain level. The main feature of the resulting system is that it defines a nonlinear discontinuous semigroup. In this note we study the orbital stability of a one-parameter family of periodic solutions and state the existence of a period-doubling bifurcation of such solutions.