  We demonstrate that shunting of Superconductor-Insulator-Superconductor
Josephson junctions by Superconductor-Insulator-Normal metal (S-I-N) structures
having pronounced non-linear I-V characteristics can remarkably modify the
Josephson dynamics. In the regime of Josephson generation the phase behaves as
an overdamped coordinate, while in the superconducting state the damping and
current noise are strikingly small, that is vitally important for application
of such junctions for readout and control of Josephson qubits. Superconducting
Nb/AlO${_x}$/Nb junction shunted by Nb/AlO${_x}$/AuPd junction of S-I-N type
was fabricated and, in agreement with our model, exhibited non-hysteretic I-V
characteristics at temperatures down to at least 1.4 K.
