Flux transitions in a superconducting ring

We consider a superconducting ring embedded in a magnetic field. The magnetic field is initially too high to support a superconducting state, but is ideally slowly decreased and then reversed. During this process, the number of fluxoids trapped by the ring exhibits a complex pattern of transitions, which depends qualitatively on the ratios of the geometric dimensions of the ring to the coherence length. We evaluate the current around the ring and find rough agreement with experiments. However, we also predict a new feature that has not shown up in experiments: in the vicinity of a critical field, giant jumps are expected.