The melt and solution spinning of polymeric fibers suffers from hydrodynamic instability known as “draw resonance” which occurs when the draw ratio exceeds a certain critical value. Above transition draw ratio, the disturbances grow rendering the flow unstable with periodic oscillations in the fiber diameter. In present study, weakly nonlinear stability analysis is carried for polymer fiber spinning to estimate the nature of bifurcation and to construct the finite amplitude branch in the neighborhood of the transition point. We employ the eXtended Pom-Pom (XPP) model which describes the nonlinear rheology of polymer melt. Using 1D model, linear stability analysis provides the stability map in DRc-De plane. In the unstable regime, the nonlinearities in the governing equations saturate the disturbance amplitude to an equilibrium value. Weakly nonlinear analysis is, therefore, carried out to obtain the equilibrium amplitude in the unstable region. For flows at small De, the Landau constant, which is the nonlinear correction to linear growth rate, is found to be negative, indicating supercritical Hopf bifurcation at the transition point. The finite amplitude branch is constructed by calculating the equilibrium amplitude in the unstable regime. This stable branch, in fact, shows a limit cycle behavior. As the fluid elasticity is increased, initially the equilibrium amplitude is found to decrease below its Newtonian value and reaches the lowest value for Deborah number when the strain hardening is maximum. Further increasing elasticity, the material undergoes strain softening behavior which leads to increase in equilibrium amplitude of the oscillations. At very high Deborah number, the flow becomes subcritically unstable, indicated by positive Landau constant. The cross-over from supercritical to subcritical bifurcation at high De means that the flow can become unstable at draw ratio below its critical value if the disturbance amplitude is above a threshold amplitude.