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

Christian Kuehn; Iacopo P. Longo

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arXiv · 2011.04031

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

Christian Kuehn·Iacopo P. Longo

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Abstract

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.

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Christian Kuehn, Iacopo P. Longo. 2020-11-08. Estimating rate-induced tipping via asymptotic series and a Melnikov-like method. https://arxiv.org/abs/2011.04031

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