Free-piston Stirling engines have attracted considerable attention as an efficient technology for converting thermal energy into mechanical and electrical power due to their simple structure, reduced mechanical friction, and ability to operate with various heat sources. However, the inherently nonlinear dynamic behavior of these engines makes the prediction of stable oscillations and operating frequency a challenging task in their analysis and design. In this study, the nonlinear dynamics of a free piston Stirling engine are analytically investigated using the homotopy method to obtain an approximate analytical solution of the governing nonlinear equations. In addition, a genetic algorithm is employed to determine and adjust the effective parameters of the system. Based on this framework, the oscillatory behavior of the engine is analyzed and the conditions required to achieve a stable limit cycle are examined. The main novelty of this work lies in proposing an integrated analytical approach based on the homotopy method combined with a genetic algorithm that simultaneously enables the analysis of nonlinear dynamics and the derivation of a parametric relation for predicting the nonlinear operating frequency of a free piston Stirling engine. The obtained results demonstrate that the proposed approach can effectively capture the nonlinear behavior of the system and provide a useful tool for the analysis, design, and optimization of free piston Stirling engines.