The self-mode-locked (SML) operation at 946 nm can be achieved with a monolithic Nd:YAG crystal when the pump power is above the threshold of the multiple-longitudinal-mode generation. The SML output is further found to include two orthogonal polarization components with a beat frequency coming from the birefringence effect in the laser crystal. The beat frequency can be widely adjusted in the range of 5-220 MHz by controlling the cooling temperature. The present experiment also confirms the theoretical prediction that the two-mode operation generally exhibits the chaotic dynamics when the frequency difference is sufficiently close to the relaxation frequency.