On stability in a case of oscillations of a pendulum with a mobile point mass

Motion in a uniform gravitational field of a modified pendulum in the form of a thin, uniform rod, one end of which is attached by a hinge, is investigated. A point mass (for example, a washer mounted on the rod) can move without friction along the rod. From time to time, the point mass collides with the other end of the rod (if, for example, at this end of the rod a rigid plate of negligibly small mass is attached perpendicular to it). The collisions are assumed to be perfectly elastic. There exists such a motion of the pendulum in which the rod is at rest (it hangs) along the vertical passing through its suspension point, but the point mass moves along the rod, periodically bouncing up from its lower end to some height not exceeding the rod length. The nonlinear problem of the orbital stability of this periodic motion of the pendulum is investigated. In the space of two dimensionless parameters of the problem, stability and instability regions are found.