A classical scenario for tipping is that a dynamical system experiences a
slow parameter drift across a fold tipping point, caused by a run-away positive
feedback loop. We study what happens if one turns around after one has crossed
the threshold. We derive a simple criterion that relates how far the parameter
exceeds the tipping threshold maximally and how long the parameter stays above
the threshold to avoid tipping in an inverse-square law to observable
properties of the dynamical system near the fold.
For the case when the dynamical system is subject to stochastic forcing we
give an approximation to the probability of tipping if a parameter changing in
time reverses} near the tipping point.
The derived approximations are valid if the parameter change in time is
sufficiently slow. We demonstrate for a higher dimensional system, a model for
the Indian Summer Monsoon, how numerically observed escape from the equilibrium
converge to our asymptotic expressions. The inverse-square law between peak of
the parameter forcing and the time the parameter spends above a given threshold
is also visible in the level curves of equal probability when the system is
subject to random disturbances.Comment: 20 pages, 6 figures, 1 table, Supplementary Material found at
https://figshare.com/articles/Monsoon_supplementary_material_final_pdf/760582