The present study investigates the influence of actuation frequency and diaphragm displacement of the rectangular shaped synthetic jet and reports the occurrence of vortex ring bifurcation for a rectangular orifice, along with other reported modes such as axial switching and vortex suction. The novel finding of vortex ring bifurcation of rectangular synthetic jets has been observed without any other mode of excitation except the periodic axial actuation. The experiments on synthetic jets have been conducted at different actuation frequencies and both qualitative and quantitative characterization of the flow structures has been carried out using Laser Induced Fluorescence and Laser Doppler Velocimetry, respectively. LIF flow visualization provides insights into the size of the vortex and the vortex evolution with respect to time, enabling us to propose the flow physics behind the axial switching and the vortex bifurcation processes for rectangular synthetic jets. The proposed flow physics is then quantitatively evidenced by the time-averaged velocity measurements. Vortex splitting or bifurcation is found to occur in the minor axis plane of orifice and their divergence angle depends on the actuation frequency and average velocity of fluid expelled through the orifice in the forward stroke of diaphragm. In the case of occurrence of axial switching, a maximum of three axial switching events are observed before vortex breakup. Finally, by systematically carrying out experiments across a wide range of operational parameters, a narrow region corresponding to the vortex bifurcation has been identified on an Re-St map, along with other modes such as axial switching regime and the vortex suction regime.