Theoretical and numerical studies of nonlinear acoustic effects arising from the propagation of initially harmonic waves in polycrystalline solids with saturation of amplitude-dependent internal friction have been carried out. Two main types of hysteresis are considered: elastic and inelastic. A comparative analysis of the regularities of nonlinear effects is carried out and the characteristics of nonlinear quasi-harmonic waves in such media are determined: amplitude-dependent losses and changes in propagation velocity, as well as amplitudes of higher harmonics. A method for determining the type of polycrystal hysteresis based on the analysis of theoretical and experimentally established amplitude dependences of nonlinear effects is proposed. Keywords: elastic waves, amplitude-dependent internal friction.