Inverse-square law between time and amplitude for crossing tipping thresholds

Paul Ritchie, Özkan Karabacak, Jan Sieber

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1 Citation (Scopus)

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.

OriginalsprogEngelsk
Artikelnummer20180504
TidsskriftProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Vol/bind475
Udgave nummer2222
Antal sider19
ISSN1364-5021
DOI
StatusUdgivet - feb. 2019

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Dynamical systems
thresholds
dynamical systems
Dynamical system
Forcing
Fold
Feedback
positive feedback
Positive Feedback
monsoons
Feedback Loop
Approximation
guy wires
approximation
reaction time
escape
summer
Reverse
Exceed
High-dimensional

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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 tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

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AU - Ritchie, Paul

AU - Karabacak, Özkan

AU - Sieber, Jan

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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.

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KW - Overshoot

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