An analytic proof of the bifurcation condition for localized bulging in an inflated hyperelastic tube

We provide a simple proof of the bifurcation condition for localized bulging in a hyperelastic tube of arbitrary thickness that is subjected to combined axial loading and internal pressure. Using analytic tools, we prove that the bifurcation condition is equivalent to the vanishing of the Jacobian of the internal pressure $P$ and the resultant axial force $N$, with each of them viewed as a function of the azimuthal stretch on the inner surface and the axial stretch. Previously this was only established by numerical calculations. The method should be applicable to any bifurcations that depend on a slowly varying variable, concluding that they share the same bifurcation conditions with bifurcations into uniform/homogeneous states as long as the equations determining the bifurcation condition are not trivially satisfied by uniform/homogeneous solutions.