Stability of inhomogeneous stationary states for the hot-spot model of a superconducting microbridge

The stability properties of inhomogeneous stationary states are discussed both for fixed current and for fixed voltage difference between the endpoints of the wire. We consider both Neumann as well as Dirichlet boundary conditions for the temperature distribution along the wire. It is found that states which are unstable for constant current are often stable at constant voltage. Analytic expressions for the minimum eigenvalue characterizing the time development of small perturbations around the stationary states are given. The resulting time rate of change of these perturbations is then discussed in detail. If the microbridge is “long” this time rate of change is often almost infinitely slow. If the microbridge is “sufficiently short” it is found for the case of Neumann boundary conditions and constant voltage that no stable stationary states exist along part of the voltage axis. It is conjectured that this corresponds to the existence of a Hopf bifurcation and limit cycle behaviour.