  A time dependent thermal model for a superconducting constriction based
weak-link (WL) is discussed for investigating the deterministic dynamics of its
temperature and phase. A new dynamic regime is found where a non-zero voltage
exists across the WL and its temperature stabilizes between the bath
temperature and superconductor's critical temperature. This regime exists over
a limited bias current range and gives rise to a new hysteretic regime in
current-voltage characteristics. We also discuss the effect of fluctuations on
the current-voltage characteristics and experimental implications of this
dynamic regime.
