Abstract: For passive Q-switched solid state lasers including excite state absorption (ESA),the coupled rate equations is presented. By means of the adiabatic elimination approximation (AEA) and the Hopf bifurcation conditions,the series of multi-pulses produced by passive Q-switch correspond to the limit cycle on the phase plane of optic intensity and population inversion density. Threshold formula for giant pulses and analytical solution for pulses period or repetitive frequency are obtained. The phenomena about Cr4+:YAG saturable absorber Q-switched laser are explained. The calculated results accord with the experiment.