Parameter shifts for nonautonomous systems in low dimension: bifurcation- and rate-induced tipping

Ashwin, Peter; Perryman, Clare; Wieczorek, Sebastian

doi:10.1088/1361-6544/aa675b

Cited by 98 publications

(186 citation statements)

References 39 publications

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“…In our numerical study this probability is P = 0.85, indicating that the Snowball Earth is the more probable outcome. This is in contrast to the way tipping transitions are usually discussed in deterministic systems, where as a result of a monotonous parameter drift, either all trajectories tip, or none of them do 40 .…”

Section: The Ensemble View: Visualizing the Snapshot Attractormentioning

confidence: 75%

The snowball Earth transition in a climate model with drifting parameters: Splitting of the snapshot attractor

Kaszás

Haszpra

Herein

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Using an intermediate complexity climate model (Planet Simulator), we investigate the so-called Snowball Earth transition. For certain values of the solar constant, the climate system allows two different stable states: one of them is the Snowball Earth, covered by ice and snow, and the other one is today's climate. In our setup, we consider the case when the climate system starts from its warm attractor (the stable climate we experience today), and the solar constant is decreased continuously in finite time, according to a parameter drift scenario, to a state, where only the Snowball Earth's attractor remains stable. This induces an inevitable transition, or climate tipping from the warm climate. The reverse transition is also discussed. Increasing the solar constant back to its original value on individual simulations, we find that the system stays stuck in the Snowball state. However, using ensemble methods i.e., using an ensemble of climate realizations differing only slightly in their initial conditions we show that the transition from the Snowball Earth to the warm climate is also possible with a certain probability. From the point of view of dynamical systems theory, we can say that the system's snapshot attractor splits between the warm climate's and the Snowball Earth's attractor. 1 arXiv:1906.00952v1 [physics.ao-ph] 31 May 2019 Ever since its discovery, the Snowball Earth, i.e. when the Earth's surface is nearly entirely frozen, received much attention within the climate science community. Much of the details of the transition to the planet's frozen state are still unexplored. Here, instead of focusing on the true Snowball events of Earth's history, we investigate the transition in an intermediate complexity climate model (PlaSim), with a continuously drifting solar constant (a hypothetical climate change scenario), in which a full return to the original value occurs. Using an ensemble based method we obtain both of the possible stable states as possible outcomes. We also show that the process is probabilistic and the probabilities of the corresponding outcomes are given by the ensemble's distribution. In addition, the third, unstable state (referred to as the edge state) is also recovered. I. INTRODUCTIONSnowball Earth refers to the planet's coldest possible global climate. In this state, the whole Earth, from the poles to the Equator, is covered in ice and snow. Since the thick ice covering the surface reflects much of the energy radiated by the Sun, the global average temperature is very low, around 220 K 1 .Modern findings suggest that during the Earth's history, there were periods, when such Snowball events occurred. For example, several traces of glacial activity point to the presence of glaciers along the so-called Paleoequator 2 .This suggests that also the current configuration of the Earth system may be bistable, the two stable states being the Snowball state and our current climate. To better understand the phenomenon, there are simple models available that only take the global energy balance...

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Section: The Ensemble View: Visualizing the Snapshot Attractormentioning

confidence: 75%

The snowball Earth transition in a climate model with drifting parameters: Splitting of the snapshot attractor

Kaszás

Haszpra

Herein

2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…The scenario (12) corresponds to an infinite rate of change in the parameter (the forcing amplitude u) at time t = 750 kyr BP. For this scenario Ashwin et al [3] developed the concept of breakdown of basin forward stability, which generalizes the scenario described above for forcing (12).…”

Section: Discussionmentioning

confidence: 99%

Effects of Periodic Forcing on a Paleoclimate Delay Model

Quinn¹,

Sieber²,

Heydt³

2019

SIAM J. Appl. Dyn. Syst.

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We present a study of a delay differential equation (DDE) model for the Mid-Pleistocene Transition (MPT). We investigate the behavior of the model when subjected to periodic forcing. The unforced model has a bistable region consisting of a stable equilibrium along with a large amplitude stable periodic orbit. We study how forcing affects solutions in this region. Forcing based on astronomical data causes a sudden transition in time and under increase of the forcing amplitude, moving the model response from a non-MPT regime to an MPT regime. Similar transition behavior is found for periodic forcing. A bifurcation analysis shows that the transition is not due to a bifurcation but instead to a shifting basin of attraction. While determining the basin boundary we demonstrate how one can accurately compute the intersection of a stable manifold of a saddle with a slow manifold in a DDE by embedding the algorithm for planar maps proposed by England et al. (SIADS 2004(3)) into the equation-free framework by Kevrekidis et al. (Rev. Phys. Chem. 2009 (60)).

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“…Since then, a number of papers have studied R-tipping and related effects either using the theory of fast-slow dynamical systems 12,13 or notions from nonautonomous stability theory 2,14,15 . In particular, it has been suggested that local pullback attractors (where typical initial conditions are chosen from some open region in the distant past) provide a suitable setting to describe such transitions 2 . Further studies have attempted to provide early warning indicators for this type of tipping points [16][17][18] .…”

Section: Introductionmentioning

confidence: 99%

Rate-induced tipping from periodic attractors: Partial tipping and connecting orbits

Alkhayuon

Ashwin

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We consider how breakdown of the quasistatic approximation for attractors can lead to rate-induced tipping, where a qualitative change in tracking/tipping behaviour of trajectories can be characterised in terms of a critical rate. Associated with rate-induced tipping (where tracking of a branch of quasistatic attractors breaks down), we find a new phenomenon for attractors that are not simply equilibria: partial tipping of the pullback attractor where certain phases of the periodic attractor tip and others track the quasistatic attractor. For a specific model system with a parameter shift between two asymptotically autonomous systems with periodic attractors, we characterise thresholds of rate-induced tipping to partial and total tipping. We show these thresholds can be found in terms of certain periodic-to-periodic and periodic-to-equilibrium connections that we determine using Lin's method for an augmented system.

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Parameter shifts for nonautonomous systems in low dimension: bifurcation- and rate-induced tipping

Cited by 98 publications

References 39 publications

The snowball Earth transition in a climate model with drifting parameters: Splitting of the snapshot attractor

The snowball Earth transition in a climate model with drifting parameters: Splitting of the snapshot attractor

Effects of Periodic Forcing on a Paleoclimate Delay Model

Rate-induced tipping from periodic attractors: Partial tipping and connecting orbits

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