  In this paper, we studied Rayleigh autonomous oscillator by searching its
limit cycle and by studying the stability of the cycle. Through this study, we
studied the fixed points of the equation of Rayleigh oscillator and we realized
that in reality this oscillator has only one fixed point. What appears new
according to our point of view is that we found the cycle limit of the
oscillator using a form of Poincar\'e-Bendixson theorem. Then, we studied the
stability of this limit cycle by the method of multiple scales. Finally, the
most important is that in this paper we have shown analytically and confirmed
by numerical simulation using Mathematica that the Rayleigh oscillator exhibits
a bifurcation of PAH
