This article deals with the first pitchfork bifurcation of a stenotic flow within a global weakly nonlinear theory based on the method of multiple scales. For a stenosis constriction: S = 75%, the threshold Reynolds numbers Rec and characteristics of the destabilizing global mode are found to be in excellent agreement with the direct numerical simulations data of Sherwin & Blackburn (J. Fluid Mech., vol. 533, 2005,pp. 297-327). A first attempt is then made to extend the Landau amplitude equation up to the fifth order term with a hope to push the validity of the expansion further away from the threshold. For that purpose, the stenosis parameter is fixed and we assume that the Reynolds number is the only varying control parameter. The theoretical analysis recovers the subcritical nature of the bifurcation, consistent with the findings of Sherwin & Blackburn. In particular, the nonlinear interac tion between the second harmonic and the complex conjugate of the first harmonic is shown to play a decisive role in the bifurcation mechanism. The generation of higher harmonics and their contribution to the loss of axisymmetry are also examined. The prediction of the critical Reynolds number at the saddle‐node bifurcation is then discussed in light of DNS results from the literature. Finally, the theoretical analysis is applied to several stenosis parameters varying from S = 65% to S = 80%. The results indicate that, for all considered constrictions, the pitchfork bifurcation remains subcritical.