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Power amplifiers (PAs) often exhibit instabilities giving rise to frequency divisions or spurious oscillations. The prediction of these instabilities requires a large-signal stability analysis of the circuit. In this paper, oscillations, hysteresis, and chaotic solutions, experimentally encountered in a high-efficiency class-E/Fodd PA with four transistors combined using a distributed active transformer, are studied through the use of stability and bifurcation analysis tools. The tools have enabled an in-depth comprehension of the different phenomena, which have been observed in simulation with good agreement with experimental results. The study of the mechanism generating the instability has led to a simplified equivalent circuit from which the optimum stabilization network has been determined. The network enables a global stabilization of the circuit for all the expected operating values of the amplifier bias voltage and input power. This has been achieved with negligible degradation of the amplifier performance in terms of drain efficiency and output power. The stable behavior obtained in simulation has been experimentally confirmed.