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Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system
Peter Ashwin, Sebastian Wieczorek, Renato Vitolo, Peter Cox
TL;DR
Climate tipping has commonly been linked to bifurcations or noise, but the paper examines whether rapid changes in inputs or parameters provide a distinct mechanism. Using open-system models and examples including a global energy-balance model, it distinguishes B-, N-, and R-tipping and shows that each can produce tipping responses, including R-tipping without noise or bifurcations.
Problem
Existing climate-tipping explanations centered on bifurcations or noise do not fully cover rate-dependent tipping, which is identified as an important distinct mechanism.
Method
The paper develops an open-system framework, introduces a linear model with a tipping radius, and studies illustrative normal-form, slow-fast, and climate energy-balance examples.
Results
Each of B-tipping, N-tipping, and R-tipping can independently produce a tipping response, while R-tipping can occur without noise or bifurcations.
Takeaways & Limitations
Rate-dependent tipping is a mechanism that climate-system subsystems may exhibit independently of bifurcation- and noise-induced tipping.
Takeaways & Limitations
The general parameter-drift rate is difficult to define coordinate-independently, and the paper notes that predictive techniques for R-tipping remain under investigation.
Abstract
from arXiv · showhide
Tipping points associated with bifurcations (B-tipping) or induced by noise (N-tipping) are recognized mechanisms that may potentially lead to sudden climate change. We focus here a novel class of tipping points, where a sufficiently rapid change to an input or parameter of a system may cause the system to "tip" or move away from a branch of attractors. Such rate-dependent tipping, or R-tipping, need not be associated with either bifurcations or noise. We present an example of all three types of tipping in a simple global energy balance model of the climate system, illustrating the possibility of dangerous rates of change even in the absence of noise and of bifurcations in the underlying quasi-static system.
1 Tipping points - not just bifurcations
Climate tipping need not arise solely from bifurcations: noise can induce departures, and sufficiently rapid parameter variation can cause rate-dependent tipping without noise or bifurcations. The paper organizes these mechanisms as B-, N-, and R-tipping in open systems.
- Climate transitions may abruptly move the system between regimes, motivating study of tipping beyond conventional gradual change.
- Noise-induced tipping can drive a system away from an attractor without any bifurcation, although bifurcation-based predictions can remain practically useful.
- B-tipping follows a bifurcation of a quasi-static attractor, whereas N-tipping follows noisy fluctuations that depart from its neighbourhood.
- R-tipping occurs when a system fails to track a continuously changing quasi-static attractor during sufficiently rapid input or parameter variation.
- The framework treats real systems as open, analyzes low-dimensional subsystems under time-varying inputs, and allows tipping mechanisms to occur individually or in combination.
2 R-tipping: a linear model
The linear model defines R-tipping as failure to track a moving quasi-static equilibrium when the state leaves a prescribed tipping radius. It separates instantaneous lag from history-dependent error and develops criteria for avoiding or triggering R-tipping under steady and variable drift.
- Model assumptions: The model assumes a quasi-static equilibrium ˜x(λ), a tipping radius R, and initial conditions within that radius.The state is treated as tracking the equilibrium while its distance from ˜x(λ(t)) remains below R.
- Model assumptions: R-tipping occurs when the state reaches the tipping radius, marking failure of the adiabatic approximation.Because M is fixed and no noise is included, this model exhibits R-tipping without bifurcation or noise.
- Tracking decomposition: The solution follows the quasi-static equilibrium through a linear instantaneous lag and a history-dependent error term.The history-dependent term reflects the influence of prior equilibrium and parameter motion, while steady drift can eliminate it.
- Steady drift: Under steady drift, the tracking error reduces to M^-1r, yielding sufficient rate conditions for avoiding or inducing R-tipping.Avoidance requires the error norm to remain below R; a corresponding sufficient condition guarantees tipping when the error reaches the radius.
- General criteria: For variable drift, upper bounds can guarantee avoidance, but the intermediate regime depends on the path through parameter space.The general matrix criterion may require structure-specific choices rather than relying only on a matrix norm.
- General criteria: The tipping-radius approach does not readily provide a sufficient R-tipping condition based only on the maximum variable drift rate.This limits direct extension of the steady-drift tipping criterion to general time-varying drift.
- Timescale interpretation: R-tipping is expected when the QSE motion timescale is comparable to the system’s slowest timescale, with the threshold also depending on equilibrium sensitivity.Large |d˜x/dλ| can produce R-tipping at parameter-change rates smaller than the system norm scale.
3 R-tipping: model examples
The paper develops three illustrative R-tipping examples: saddle-node and Hopf normal forms, plus a slow-fast system where tipping occurs despite a unique globally stable attractor. These examples show that sufficiently rapid parameter change can make trajectories lose the ability to track a quasi-static equilibrium.
- Overview: The examples cover saddle-node and Hopf normal forms, plus a slow-fast system with a single globally asymptotically stable attractor.The slow-fast case demonstrates R-tipping without multiple attractors.
- Saddle-node normal form: When r > µ in the scalar example, the tipping threshold disappears and trajectories from all initial conditions become unbounded.For 0 < r < µ, initial conditions below the threshold converge to the attracting invariant line, while those above it diverge.
- Saddle-node normal form: For the scalar saddle-node example, the critical rate depends on the initial condition through the rate-dependent tipping threshold.Depending on the initial position, the threshold either crosses the initial condition or disappears when attracting and saddle invariant lines meet.
- Hopf normal form: In the Hopf example with steady drift, the ability to track the QSE disappears through a saddle-node bifurcation for ω^2 < 1/4 or destabilises through a subcritical Hopf bifurcation for ω^2 > 1/4.The bifurcation concerns the co-moving tracking equilibrium, not the QSE itself.
- Hopf normal form: For unsteady drift, ρ < ρc allows QSE tracking, whereas ρ > ρc causes tipping near t = t0.The parameter ρ scales the maximum rate of change, and the QSE stability and basin size remain unchanged during the shift.
- Slow-fast system: In the slow-fast example, trajectories below the critical rate avoid the fold and approach the QSE, whereas trajectories above it reach the fold and diverge in the fast direction.For steady drift, the critical rate is obtained in the singular limit by analysing the projected reduced system; for unsteady drift, ρc depends on the initial condition.
4 B-, N- and R-tipping examples in a simple climate model
The Sutera–Fraedrich global energy-balance model incorporates radiative heating, ice–albedo feedback, deterministic dynamics, and stochastic forcing to illustrate B-, N-, and R-tipping under different parameter changes.
- Model formulation: The model represents average ocean surface temperature using a deterministic energy-conservation law with radiative heating and outgoing radiation.The ocean layer is 30 m deep and covers 70.8% of Earth’s surface.
- Model formulation: Ice–albedo feedback links temperature changes to planetary albedo through a quadratic relation controlled by a2 and b2.a2 controls albedo magnitude, while b2 controls the slope of the albedo–temperature relation.
- Model formulation: For µ > µc, the deterministic system has stable T+ and unstable T− equilibria, which coalesce at a saddle-node bifurcation when µ = µc.The stochastic formulation adds Wiener-process forcing with amplitude parameter ν.
- Examples: The simulations independently realize R-tipping through changing parameters, N-tipping through added noise, and B-tipping through a steady downward drift of µ.For R-tipping, λ follows dλ/dt = ρλ(1 − λ), with ρ scaling the transition rate between parameter values.
5 Summary and conclusions
The paper classifies tipping as bifurcation-induced, noise-induced, or rate-dependent, and argues that rate-dependent tipping can occur independently of noise and stability changes. It also highlights that realistic systems may combine mechanisms and exhibit new rate-dependent behaviors.
- 5 Summary and conclusions: The paper proposes three categories of tipping: bifurcation-induced, noise-induced, and rate-dependent tipping.They are denoted B-tipping, N-tipping, and R-tipping, respectively.
- 5 Summary and conclusions: Figure 7 illustrates pure B-, N-, and R-tipping independently in the Sutera–Fraedrich model.The trajectories use years on the horizontal axis and Kelvin on the vertical axis.
- 5 Summary and conclusions: For R-tipping, ρ = 0.18 yr−1 produces recovery to the quasi-static equilibrium, whereas ρ = 0.19 yr−1 produces unbounded behavior, giving ρc ≈ 0.185 yr−1.The N-tipping example adds noise with amplitude ν = 1.0 yr−1, while the B-tipping example decreases µ until the quasi-static equilibria coalesce.
- 5 Summary and conclusions: R-tipping can occur independently of the presence or absence of B-tipping and N-tipping.Neither N-tipping nor R-tipping requires a change of stability.
- 5 Summary and conclusions: Realistic systems may combine mechanisms, and changing the rate can suppress B-tipping or trigger R-tipping before a bifurcation is reached.The paper identifies these mixed-mechanism cases as a challenge for future analysis.
- 5 Summary and conclusions: The proposed classification may apply to other open systems influenced by noise and parameter changes, including mechanics, ecology, economics, and social sciences.The paper suggests rate-dependent bifurcation theory may also have applications to cell fate.