  A variation in the environment of a system, such as the temperature, the
concentration of a chemical solution or the appearance of a magnetic field, may
lead to a drift in one of the parameters. If the parameter crosses a
bifurcation point, the system can tip from one attractor to another
(bifurcation-induced tipping). Typically, this stability exchange occurs at a
parameter value beyond the bifurcation value. This is what we call here the
stability exchange shift. We study systematically how the shift is affected by
the initial parameter value and its change rate. To that end, we present
numerical and analytical results for different types of bifurcations and
different paradigmatic systems. Finally, we deduce the scaling laws governing
this phenomenon. We show that increasing the change rate and starting the drift
further from the bifurcation can delay the tipping process. Furthermore, if the
change rate is sufficiently small, the shift becomes independent of the initial
condition (no memory) and the shift tends to zero as the square root of the
change rate. Thus, the bifurcation diagram for the system with fixed parameters
is recovered.
