  We study a two-dimensional low-dissipation dynamical system with a control
parameter that is swept linearly in time across a transcritical bifurcation. We
investigate the relaxation time of a perturbation applied to a variable of the
system and we show that critical slowing down may occur at a parameter value
well above the bifurcation point. We test experimentally the occurrence of
critical slowing down by applying a perturbation to the accessible control
parameter and we find that this perturbation leaves the system behavior
unaltered, thus providing no useful information on the occurrence of critical
slowing down. The theoretical analysis reveals the reasons why these tests fail
in predicting an incoming bifurcation.
