The bifurcation structure of a thin superconducting loop with small variations in its thickness

We study bifurcations between the normal and superconducting states, and between superconducting states with different winding numbers, in a thin loop of superconducting wire, of uniform thickness, to which a magnetic field is applied. We then consider the response of a loop with small thickness variations. We find that close to the transition between normal and superconducting states lies a region where the leading order problem has repeated eigenvalues. This leads to a rich structure of possible behaviours. A weakly nonlinear stability analysis is conducted to determine which of these behaviours occur in practice