  Experimental studies of magnetoresistance in thin superconducting strips
subject to a perpendicular magnetic field B exhibit a multitude of transitions,
from superconductor to insulator and vice versa alternately. Motivated by this
observation, we study a theoretical model for the transport properties of a
ladder--like superconducting device close to a superconductor--insulator
transition. In this regime, strong quantum fluctuations dominate the dynamics
of the vortex chain forming along the device. Utilizing a mapping of the vortex
system at low energies to one-dimensional (1D) Fermions at a chemical potential
dictated by B, we find that a quantum phase transition of the Ising type occurs
at critical values of the vortex filling, from a superconducting phase near
integer filling to an insulator near 1/2-filling. The current--voltage (I-V)
characteristics of the weakly disordered device in the presence of a d.c.
current bias I is evaluated, and investigated as a function of B, I, the
temperature T and the disorder strength. In the Ohmic regime (I/e << T), the
resulting magnetoresistance R(B) exhibits oscillations similar to the
experimental observation. More generally, we find that the I-V characteristics
of the system manifests a dramatically distinct behavior in the superconducting
and insulating regimes.
