In recent years, the Solar Stirling Thermal Oscillator (SSTO) has emerged as one of the most effective systems for converting solar energy into mechanical and electrical energy, attracting significant interest from researchers. The main challenge in harnessing these oscillators is to ensure the establishment of stable oscillations in their nonlinear dynamics for consistent and reliable performance. The main contribution of this work is the introduction and application of bifurcation theory for the first time to analyze the nonlinear dynamic behavior of the SSTO and to precisely identify the critical points at which stable limit cycles emerge. In this study, by studying the effects of crucial parameters such as power piston mass, stiffness spring of the displacer piston, and the hot source temperature, the critical ranges related to the onset of stable oscillations in the SSTO have been determined. The outcomes demonstrate that bifurcation analysis provides a novel and powerful tool for assessing and optimizing the design of SSTOs, enabling reliable prediction and assurance of stable dynamic performance before practical implementation. This method offers an effective and new pathway for the reliable and cost-efficient design of SSTOs.