### Abstract

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.

Original language | English |
---|---|

Article number | 20180504 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 475 |

Issue number | 2222 |

Number of pages | 19 |

ISSN | 1364-5021 |

DOIs | |

Publication status | Published - Feb 2019 |

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### Keywords

- Bifurcation
- Overshoot
- Tipping point

### Cite this

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*,

*475*(2222), [20180504]. https://doi.org/10.1098/rspa.2018.0504

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*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 475, no. 2222, 20180504. https://doi.org/10.1098/rspa.2018.0504

**Inverse-square law between time and amplitude for crossing tipping thresholds.** / Ritchie, Paul; Karabacak, Özkan; Sieber, Jan.

Research output: Contribution to journal › Journal article › Research › 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

C2 - 30853839

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 -