Estimating rate-induced tipping via asymptotic series and a Melnikov-like method

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.