How to have passively Q-switched lasers self-pulsate

The rate equations of passively Q-switched lasers are considered. These equations represent the dynamics of the photon number in the laser cavity, the population inversion of the laser, and the ground-state population of the saturable absorber, when excited by an input (pumping rate). Having the rate equations, (i) it is shown that the laser is the bounded-input bounded-state (BIBS) and the bounded-input bounded-output (BIBO) stable; (ii) the amplitudes of step inputs are determined for which all equilibrium points of the laser are unstable. The boundedness of the laser output and the instability of its equilibrium points imply that the laser can have a periodic, a quasi-periodic, or a chaotic output. When the output is periodic, the laser is self-pulsating, which is the desirable behavior of the laser