  The paper deals with the study of rate-induced tipping in asymptotically
autonomous scalar ordinary differential equations. We prove that, in such a
tipping scenario, a solution which limits at a hyperbolic stable equilibrium of
the past limit-problem loses uniform asymptotic stability and coincides with a
solution which limits at a hyperbolic unstable equilibrium of the future
limit-problem. We use asymptotic series to approximate such pairs of solutions
and characterize the occurrence of a rate-induced tipping by using only
solutions calculable on finite time intervals. Moreover, we show that a
Melnikov-inspired method employing the asymptotic series allows to
asymptotically approximate the tipping point.
