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An in-depth stability and bifurcation analysis of self-oscillating quasi-periodic solutions is presented. It is based on the formal analysis of the frequency-domain characteristic system, with a high degree of complexity due to the repetition of singularities at the intermodulation frequencies of the quasi-periodic spectrum. The problem is tackled by relating the system singularities to the Lyapunov exponents so that equivalent singularities of the frequency-domain system are mapped into the same Lyapunov exponents. The study is illustrated by means of its application to a self-oscillating power amplifier, which is used here as a test bench. The main types of qualitative behavior versus relevant circuit parameters, such as the bias voltage and input power, are distinguished and analyzed in detail. The influence of the transistor biasing on the number of oscillatory solutions is studied, as well as the effect of these coexisting solutions on the circuit response versus the input power. Two types of hysteresis are identified and explained, as well as a co-dimensional 2 bifurcation, which leads to a qualitative change in the structure of the quasi-periodic solution curves. The analysis is validated with measurement results.