Resumé
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.
| Originalsprog | Engelsk |
|---|---|
| Artikelnummer | 20180504 |
| Tidsskrift | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |
| Vol/bind | 475 |
| Udgave nummer | 2222 |
| Antal sider | 19 |
| ISSN | 1364-5021 |
| DOI | |
| Status | Udgivet - feb. 2019 |
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Inverse-square law between time and amplitude for crossing tipping thresholds. / Ritchie, Paul; Karabacak, Özkan; Sieber, Jan.
I: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Bind 475, Nr. 2222, 20180504, 02.2019.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › peer review
TY - JOUR
T1 - Inverse-square law between time and amplitude for crossing tipping thresholds
AU - Ritchie, Paul
AU - Karabacak, Özkan
AU - Sieber, Jan
PY - 2019/2
Y1 - 2019/2
N2 - 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.
AB - 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.
KW - Bifurcation
KW - Overshoot
KW - Tipping point
UR - http://www.scopus.com/inward/record.url?scp=85062687617&partnerID=8YFLogxK
U2 - 10.1098/rspa.2018.0504
DO - 10.1098/rspa.2018.0504
M3 - Journal article
VL - 475
JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
SN - 1364-5021
IS - 2222
M1 - 20180504
ER -