  The onset of turbulent flow around an oscillating sphere in superfluid $^4$He
is known to occur at a critical velocity $v_c \sim \sqrt{\kappa\omega}$ where
$\kappa$ is the circulation quantum and $\omega$ is the oscillation frequency.
But it is also well known that initially in a first up-sweep of the oscillation
amplitude, $v_c$ can be considerably exceeded before the transition occurs,
thus leading to a strong hysteresis in the velocity sweeps. The velocity
amplitude $v_c^*
  > v_c$ where the transition finally occurs is related to the density $L_0$ of
the remanent vortices in the superfluid. Moreover, at temperatures below ca.
0.5 K and in a small interval of velocity amplitudes between $v_c$ and a
velocity that is about 2% larger, the flow pattern is found to be unstable,
switching intermittently between potential flow and turbulence. From time
series recorded at constant temperature and driving force the distribution of
the excess velocities $\Delta v = v_c^* - v_c$ is obtained and from that the
failure rate. Below 0.1 K we also can determine the distribution of the
lifetimes of the phases of potential flow. Finally, the frequency dependence of
these results is discussed.
