NOTE-1215: Ashwin, Wieczorek, Vitolo & Cox — Tipping points in open systems

Contribution

It names and separates a third mechanism of tipping beside bifurcation and noise: rate-dependent tipping (R-tipping), the failure of an open system to track an attractor that continues to exist. It gives a linear model with a “tipping radius” in which R-tipping occurs with no bifurcation and no noise, and sufficient conditions on the drift rate for it to occur or not (§2). It works three low-dimensional examples (saddle-node and subcritical Hopf normal forms, and a fast-slow system with a single globally stable equilibrium; §3), and shows each of B-, N- and R-tipping in isolation in the Fraedrich–Sutera global energy balance model (§4).

Key insight

An attractor can be lost without being destroyed. If the parameter moves, the quasi-static equilibrium x̃(λ(t)) moves with it, and the state lags behind by an amount set by how fast it moves relative to how fast the system relaxes. When the lag reaches the threshold for leaving, the system tips, although at every frozen value of λ the equilibrium is there, stable, and (in the Hopf example) of unchanged basin. “R-tipping that reduces to a bifurcation problem in a co-moving system should not be confused with B-tipping: observe that the bifurcation parameter r does not vary in time, and it is ‘the ability to track the QSE’, rather than the QSE itself, that bifurcates” (§3.2.1, p. 11).

Assumptions

• An open system dx/dt = f(x, λ(t)), eq. (1), p. 2, read as a low-dimensional subsystem forced by “observables” λ(y) of a high-dimensional system (Fig. 1, p. 4). “Real-world systems are never closed except to some degree of approximation” (§1.1, p. 4).
• The linear tipping-radius model (§2, p. 5): dx/dt = M(x − x̃(λ)) for |x − x̃(λ)| < R, eq. (4), with M “a fixed stable linear operator”. Beyond the radius the model is “unphysical”. The radius “may be related to the basin of attraction boundary for the nonlinear problem … but it need not be” (p. 5).
• For the integrations by parts (p. 6): M invertible and exponentially stable, an arbitrarily long past near the QSE, and dˡx̃/dλˡ and dˡλ/dtˡ bounded for l = 1, 2.
• Deterministic except in N-tipping. The R-tipping results assume no noise; the N-tipping example uses dT = f(T)dt + √ν dW, eq. (41).
• The setting is narrower than the title: every example is one- to three-dimensional, and the climate model is zero-dimensional. The paper makes no claim about any real climate subsystem tipping by rate.

Key results

The three types (§1, p. 3; Fig. 2, p. 5)

• “‘B-tipping’ where the output from an open system changes abruptly or qualitatively due to a bifurcation of a quasi-static attractor.”
• “‘N-tipping’ where noisy fluctuations result in the system departing from a neighbourhood of a quasi-static attractor.”
• “‘R-tipping’ where the system fails to track a continuously changing quasi-static attractor.”
• “each mechanism on its own can produce a tipping response. Furthermore, any open system may exhibit tipping phenomena that result from a combination of several of the above” (p. 3).
• Background (p. 2): Thompson and Sieber’s time-series analysis of B-tipping relies on the separation κ_drift ≪ κ_crit ≪ κ_stab, eq. (3) (parameter drift, slowest decay of the QSE, faster decays); the authors add that κ_drift is hard to define coordinate-free and “there is no a priori reason for inequality (3) to hold for a given system”.

The linear model (§2, pp. 5–6)

• Tracking and tipping: if |x(t) − x̃(λ(t))| < R the solution “tracks (or adiabatically follows)” the QSE; if |x(t₀) − x̃(λ(t₀))| = R “the solution tips (adiabatic approximation fails) at t₀” (p. 5). “System (4) shows only R-tipping - because M is fixed there is no bifurcation in the system and no noise is present.”
• Solution, eq. (6): x(t) − x̃(λ(t)) = −∫₀^∞ e^{Mu} (dx̃/dt)(λ(t − u)) du. Integrated again by parts it is L(t) + E(t), with the linear instantaneous lag L(t) = M⁻¹ r(t), eqs. (7)–(9), where the drift of the QSE is r(t) := dx̃/dt = (dx̃/dλ)(dλ/dt), eq. (8); E(t) is a history-dependent error in d²x̃/dt².

Criteria (§§2.1–2.2, pp. 7–8; the corrected text)

• Steady drift (r constant): E = 0 and |x − x̃| = |M⁻¹r|, eq. (10). Since ‖M‖⁻¹|r| ≤ |M⁻¹r| ≤ ‖M⁻¹‖|r|, R-tipping is avoided if ‖M⁻¹‖·|r| < R, eq. (11), and occurs if ‖M‖⁻¹·|r| > R. “In the intermediate case, the path of parameter variation will determine whether or not there is any R-tipping” (p. 7).
• Varying drift: with r_max(t) = sup_{s≤t} |dx̃/dt| and ‖e^{Mu}‖ ≤ c e^{−βu}, eq. (12), tipping is avoided by time t if (c/β) r_max(t) < R, eq. (13); for scalar M, ‖M⁻¹‖ r_max(t) < R, eq. (14). The approach “does not seem to easily give any sufficient condition to undergo R-tipping in terms of r_max(t)” (p. 8).
• Timescales (p. 8): a natural timescale for the QSE’s motion is R/|r(t)|; “For an R-tipping to occur, this natural timescale may be comparable to the slowest timescale (e.g. the reciprocal of the leading eigenvalue of M) of the parametrized system.” With |dx̃/dλ| ≈ 1 and R ≈ 1, R-tipping is expected when |dλ/dt| ≈ ‖M‖; “if |dx̃/dλ| ≈ 1/ε, then clearly we can have R-tipping even when |dλ/dt| ≈ ε‖M‖.”
• For a nonlinear system an “effective tipping radius” can be defined, but “There is however no exact analogy possible”, since it depends on the basin’s shape, the nonlinearities and “the exact path taken in parameter space” (p. 8).

Examples (§3, pp. 8–16)

• Saddle-node normal form, steady drift (§3.1): dx/dt = (x + λ)² − µ, dλ/dt = r, eqs. (15)–(16). For 0 < r < µ a repelling invariant line B(µ, r) is the tipping threshold; at r = µ it merges with the attracting line A and disappears. The critical rate depends on the initial condition: r_c = µ − (λ₀ + x₀)² if −x₀ < λ₀ < −x₀ + √µ, and r_c = µ if λ₀ ≤ −x₀, eq. (17), p. 10. The linear model with R = 2√µ overestimates r_c “because the linear attraction weakens on moving away from the stable QSE”; the effective radius is R_c = √µ/2. In the co-moving coordinate y = x + λ the problem is a saddle-node bifurcation at r = µ.
• Subcritical Hopf, steady drift (§3.2.1): dz/dt = F(z − λ), F(z) = (−1 + iω)z + |z|²z, eq. (18), one QSE at z̃ = λ(t). In the co-moving frame dw/dt = F(w) − r, eq. (19), the stable equilibrium that “represents the ability to track” disappears in a saddle-node when ω² < 1/4 and loses stability in a subcritical Hopf when ω² > 1/4; r_c(ω) = √((1 + 4ω²)/8) in the latter case, eq. (23); Bogdanov–Takens points at (±1/2, ±1/2) (Fig. 4, p. 12).
• Subcritical Hopf, unsteady drift (§3.2.2): λ moves from 0 to ∆ by dλ/dt = ρλ(∆ − λ), eq. (24), maximum rate ∆²ρ/4. “Observe that there is no change in stability or basin size of the QSE as t changes” (p. 12). For ∆ = 8, ρ_c = 4.78: at ρ = 4.76 the system tracks, at ρ = 4.8 it tips (Fig. 5, p. 13). Sieber (pers. comm.) suggests ρ_c is a heteroclinic connection in the extended system; there is no reduction to a bifurcation.
• Fast-slow system (§3.3): εẋ₁ = x₂ + λ + x₁(x₁ − 1), ẋ₂ = −Σ_{n=1}^N x₁ⁿ, eqs. (26)–(27), N odd. The equilibrium x̃(λ) = (0, −λ) is unique, asymptotically stable for every fixed λ “and globally asymptotically stable if N ≥ 5”. The fold of the critical manifold “defines a tipping threshold that is not associated with any basin boundary” (p. 13). With steady drift, r_c = Σ 2⁻ⁿ as ε → 0, eq. (33) (r_c = 1/2 for N = 1, Fig. 6a). With dλ/dt = ρe^{−λ}, eq. (34), ρ_c depends on the initial condition, eq. (37). “only when the parameter variation is very slow with respect to the slow variable and there are three timescales is tracking guaranteed. In this sense, the rate dependent tipping occurs when the slow and very slow timescales are no longer separable” (p. 16).

The climate model (§4, pp. 16–19; Tables 1–2, Fig. 7)

• Fraedrich’s zero-dimensional energy balance c dT/dt = R↓ − R↑, with ice-albedo α_p(T) = a₂ − b₂T², eqs. (38)–(40), and Sutera’s noisy form dT = f(T)dt + √ν dW, eq. (41). For µ above a critical 0 < µ_c < 1 there are two equilibria, T⁺ stable and T⁻ unstable, merging in a saddle-node at µ_c.
• R-tipping (Fig. 7a–b): parameters move smoothly between two steady settings by dλ/dt = ρλ(1 − λ), with a₂ chosen to hold b_µ² − 4d_µ (the discriminant of (40) as a quadratic in T²) constant (p. 17, Table 2); the paper does not say why that quantity is held. At ρ = 0.18 “the system returns to the QSE after a transient”; at ρ = 0.19 it “becomes unbounded”, so ρ_c ≈ 0.185 yr⁻¹.
• N-tipping (Fig. 7c): all parameters fixed, noise ν = 1.0 yr⁻¹.
• B-tipping (Fig. 7d): µ decreased from 1 at −0.0004 yr⁻¹ through the saddle-node.

Conclusions (§5, pp. 17–19; Fig. 8, p. 20)

• Mixed cases: B-tipping “may turn into R-tipping upon increasing the rate of change”, but also “B-tipping may be suppressed for fast enough variation of parameters”, or R-tipping may come first (p. 19; Fig. 8a–c).
• “We emphasise that neither N-tipping and R-tipping require any change of stability. Hence there is no reason to assume that the techniques of [4], based on a de-trended autoregressive model for B-tipping, should deliver useful predictions in such cases” (p. 19). R-tipping reducible to a co-moving bifurcation “may be expected to be predictable using similar methods”; others may be quantifiable by heteroclinic bifurcations of an extended system.
• Applications suggested, not shown: cell fate (Nene and Zaikin), “from mechanics and ecology to economics and social sciences” (p. 19).

Claims

id	claim	strength	support
C1	R-tipping occurs with no bifurcation and no noise: a system can leave an attractor that exists and keeps its stability at every frozen parameter	strong	the linear model with fixed M (§2, p. 5); the Hopf example with unchanged stability and basin (§3.2.2); the globally stable fast-slow example (§3.3)
C2	With steady drift, ‖M⁻¹‖·	r	< R suffices to avoid R-tipping and ‖M‖⁻¹·
C3	With steady drift, R-tipping is a bifurcation of the ability to track in a co-moving frame, not of the QSE	strong	§3.1 (y = x + λ) and §3.2.1 (eq. 19), worked exactly
C4	The critical rate depends on the initial condition and, outside steady drift, on the path	strong	eqs. (17) and (37); p. 7 on the intermediate case
C5	R-tipping can occur where the equilibrium is unique and globally asymptotically stable, at a threshold that is no basin boundary	strong	§3.3, N ≥ 5, in the singular limit ε → 0
C6	Each of B-, N- and R-tipping occurs alone in the Fraedrich–Sutera model	moderate	simulations, Fig. 7 and Table 2; ρ_c ≈ 0.185 yr⁻¹ read from two runs
C7	Critical-slowing early warnings have no reason to work for N- or R-tipping	moderate	argued from the absence of a change of stability (p. 19); no test is run
C8	Tracking is guaranteed only when parameter change is slow relative to the slowest dynamics; R-tipping is a failure of that separation	moderate	sufficient condition (13)–(14) in the linear model; stated for the fast-slow example (p. 16); for nonlinear systems “no exact analogy possible” (p. 8)
C9	The classification applies widely beyond climate	weak	suggested in §5, with no example outside climate and normal forms

Concepts

• Open system — one with inputs λ(t) that “can change the fate of system trajectories” (§1.1, p. 4); a closed one’s future depends only on its current state.
• Quasi-static attractor / QSE — the stable solution, here an equilibrium x̃(λ), of the system with λ held constant (p. 2).
• Tracking — staying within the tipping radius of the moving QSE, “adiabatically follows” (p. 5).
• Tipping radius R — the distance from the QSE at which the system is taken to have tipped; may but need not be a basin boundary (p. 5). Effective tipping radius — the radius that makes the linear criterion match a nonlinear system (p. 8; §3.1, §3.2.1).
• Drift of the QSE — r(t) = dx̃/dt = (dx̃/dλ)(dλ/dt), eq. (8); steady drift when r is constant.
• Linear instantaneous lag — L(t) = M⁻¹r(t), eq. (9).
• Critical rate r_c or ρ_c — the rate of change above which the system tips.
• B-, N-, R-tipping — as quoted under Key results (p. 3).

Connections

• Wieczorek, Ashwin, Luke and Cox 2010 (ref. [28], the compost-bomb instability, “Excitability in ramped systems”), not held, is where R-tipping was “identified … as an important tipping mechanism” (p. 3) and the source of the fast-slow theory used in §3.3.2.
• Thompson and Sieber (refs. [25], [26]), on bifurcation- and noise-induced tipping and the separation κ_drift ≪ κ_crit ≪ κ_stab, and their safe, explosive and dangerous bifurcations (p. 3).
• Dakos et al. 2008 and Ditlevsen and Johnsen 2010 (refs. [4], [5]) on early warning; Lenton et al. (refs. [16], [17]) on tipping elements; Kuehn 2011 (ref. [15]) on fast-slow critical transitions; none held.
• Scheffer and Carpenter (LIT-726, read in NOTE-571), the record’s standard of evidence for alternative stable states (THEORY-136), is not cited by the paper.
• Falmagne, Stephenson and Levin (NOTE-731, THEORY-214) use the B/S/R classification in a social system and find rate-induced tipping preceded by speeding up, not slowing: the empirical counterpart of C7.

Bearing on the record

• QUESTION-281: A14’s gloss is right in its first half and stronger than the paper in its second. “Not because its equilibrium ceases to exist” matches: R-tipping is the failure to track “a continuously changing quasi-static attractor” (p. 3); in the linear model M is fixed and “there is no bifurcation” (p. 5); in §3.2.2 “there is no change in stability or basin size” (p. 12). “Cannot reorganize fast enough” needs three corrections. (i) The paper’s quantity is relaxation, not reorganization: the lag M⁻¹r against the radius R (§2.1), so the distance to the threshold counts as much as the rate. (ii) “Destroyed” is stronger than “tips”: in the climate model at ρ just below ρ_c the system “returns to the QSE after a transient” (Fig. 7a), and in §3.3 the equilibrium is unique and globally stable, so a tip is a large excursion whose permanence the paper does not state. A form is lost by R-tipping for good only if something lies past the threshold to catch it (the unbounded runs of §3.1 and Fig. 7b). (iii) Rate can also preserve: “B-tipping may be suppressed for fast enough variation of parameters” (p. 19, Fig. 8a). The question’s “pass away” route gets a precise form: what bifurcates is “‘the ability to track the QSE’, rather than the QSE itself” (p. 11). The paper says nothing on how forms arise.
• CLAIM-157: the duration clause, read on frozen dynamics, cannot see R-tipping. In the paper’s own model the dynamics matrix M is the same at every instant (eq. 4), so any spectral or gap verdict computed from it is constant across the tip. “It lasts as long as that block persists” then says the form lasts while the system has left it. The clause holds up only if “persists” is read on the trajectory, or in the co-moving frame where the tracking solution is what bifurcates (§3.1, §3.2.1). The dated note of 2026-10-27 (“which bears on ‘it lasts as long as that block persists’”) is confirmed and can be made exact: a block can keep clearing its threshold at every frozen parameter while the form tips by rate. The paper does not address individuation or decomposability.
• CLAIM-478: rate-induced tipping is the failure of the separation the claim relies on, and the paper says so. “only when the parameter variation is very slow with respect to the slow variable and there are three timescales is tracking guaranteed … rate dependent tipping occurs when the slow and very slow timescales are no longer separable” (p. 16). So the quasi-static attractor is a constraint only over the window between the drift time R/|r| and the slowest relaxation (p. 8), which is the claim’s “over the window between the two time scales” for the attractor itself. Two sharpenings. The relevant drift is the QSE’s in state space, not the parameter’s: with |dx̃/dλ| ≈ 1/ε, a parameter moving at ε‖M‖ still tips the system (p. 8). And R-tipping needs no slowing of modes (p. 19), so it is separation lost by the input speeding up, not by the system slowing near an instability, the route in the claim’s second half. Filed as THEORY-1052.
• THEORY-136: a third route to abrupt change without alternative states. THEORY-136 holds that jumps also arise from a stepwise driver or a simple threshold. §3.3 adds a smooth but fast driver acting on a system with one globally stable equilibrium, at a threshold that “is not associated with any basin boundary” (p. 13). That supports THEORY-136’s caution. Where the frozen system is bistable, as in the climate model, a shift that depends on the rate of an otherwise identical parameter path is a form of initial-state or history dependence, which THEORY-136 counts as near-diagnostic; the paper does not discuss evidence.
• THEORY-214 and CASE-111. The paper’s argument that early warnings have “no reason” to work for R-tipping (p. 19) is the theoretical counterpart of THEORY-214’s finding that rate-induced tipping in r/place is preceded by speeding up, and it supports CASE-111’s caveat that “rate-dependent effects” make early-warning signals unreliable.
• Filed: THEORY-1079 (Active), THEORY-1052 (Proposed). No instruction for machine-learning practice is carried.

Limitations

• The journal’s §§2.1–2.2 were corrected after publication, and I read only the corrected preprint; I cannot say what the original criteria were or whether any later citation relies on them.
• R-tipping is defined by a tipping radius chosen by the analyst, and for nonlinear systems “no exact analogy possible” (p. 8). The definition of when a system has “tipped” is therefore model-relative; the paper gives no general definition of R-tipping for nonlinear systems.
• No sufficient condition for R-tipping with varying drift (p. 8).
• The climate example is a zero-dimensional toy, and ρ_c ≈ 0.185 yr⁻¹ is bracketed by two runs; no claim about a real climate subsystem is made.
• Whether a tipped system is lost for good is not treated: Fig. 7b is “unbounded”, and §3.3’s divergence “in the fast x₁ direction” in a globally stable system is left without its sequel.
• Small slips in the text: §2.2 cites “the elegant estimates of Godunov [27, Eq.(13)]” where reference [27] is Veselić; §5 names “(15) and (24)” for a heteroclinic computation that §3.2.2 attaches to (18) and (24). Neither affects the results. The extraction flattens super- and subscripts (b_µ² prints as “b2µ”; the initial λ in §4 prints as “10 … 6”, read as 10⁻⁶).

Open questions

• A general theory of R-tipping for nonlinear systems, giving the critical rate from the system and the path; the paper expects “significant mathematical development” (p. 19).
• Predictive techniques for R-tipping that does not reduce to a co-moving bifurcation (p. 19).
• How the three mechanisms combine in realistic models (p. 19).