Free piston Stirling engines (FPSEs) are examples of closed cycle regenerative engines, which can be used to convert thermal energy into mechanical energy. These engines are multidegree-of-freedom dynamical systems, which are designed to operate in a periodic manner. Traditionally, for design purposes, linear models are used and the associated periodic orbits are meta-stable, making the system operations sensitive to disturbances. A preferred operating state would be a stable limit cycle, which can make the system dynamics robust to disturbances. To this end, in this article, it is investigated as to how to engineer Hopf bifurcations of an equilibrium solution in the β and double acting α FPSE configurations that could lead to attracting periodic solutions. Weakly nonlinear analyses are conducted and analytical relations governing the periodic motions are obtained and studied in the vicinity of Hopf bifurcation points. The analytical predictions are confirmed through numerical simulations that are based upon reported engine parameters. The overall analytical-numerical approach pursued here could serve as a tool for using nonlinearity in the design of FPSEs, thereby enhancing the robustness of device operations.