  We consider a dynamical system undergoing a saddle-node bifurcation with an
explicitly time dependent parameter~$p(t)$. The combined dynamics can be
considered as a dynamical systems where $p$ is a slowly evolving parameter.
Here, we investigate settings where the parameter features an overshoot. It
crosses the bifurcation threshold for some finite duration $t_e$ and up to an
amplitude $R$, before it returns to its initial value. We denote the overshoot
as safe when the dynamical system returns to its initial state. Otherwise, one
encounters runaway trajectories (tipping), and the overshoot is unsafe. For
shallow overshoots (small $R$) safe and unsafe overshoots are discriminated by
an inverse square-root border, $t_e \propto R^{-1/2}$, as reported in earlier
literature. However, for larger overshoots we here establish a crossover to
another power law with an exponent that depends on the asymptotics of $p(t)$.
For overshoots with a finite support we find that $t_e \propto R^{-1}$, and we
provide examples for overshoots with exponents in the range $[-1, -1/2]$. All
results are substantiated by numerical simulations, and it is discussed how the
analytic and numeric results pave the way towards improved risks assessments
separating safe from unsafe overshoots in climate, ecology and nonlinear
dynamics.
