Source
Ashwin, Wieczorek, Vitolo and Cox (2012), LIT-1654, §§1–3 (pp. 3–16 of arXiv v5, whose §§2.1–2.2 are the corrected text); read in NOTE-1215.
What was actually shown
The definition. For an open system dx/dt = f(x, λ(t)), “‘R-tipping’ where the system fails to track a continuously changing quasi-static attractor” (p. 3), set beside B-tipping (a bifurcation of that attractor) and N-tipping (noise carries the system out of its neighbourhood).
The linear model (§2). dx/dt = M(x − x̃(λ)) inside a tipping radius R, with M “a fixed stable linear operator”. “because M is fixed there is no bifurcation in the system and no noise is present” (p. 5). The state lags the moving equilibrium by x − x̃ = −∫₀^∞ e^{Mu}(dx̃/dt) du, eq. (6), whose leading term is M⁻¹r with r = (dx̃/dλ)(dλ/dt). With steady drift the lag is exactly M⁻¹r: tipping is avoided if ‖M⁻¹‖·|r| < R, eq. (11), and occurs if ‖M‖⁻¹·|r| > R (p. 7). So the outcome compares how fast the equilibrium moves with how fast the system relaxes, scaled by how far the threshold is.
The examples (§3), each of which could have failed to tip:
- Subcritical Hopf with a smooth shift of the equilibrium: “there is no change in stability or basin size of the QSE as t changes”, yet for ∆ = 8 the system tracks at ρ = 4.76 and tips at ρ = 4.8 (p. 12, Fig. 5).
- A fast-slow system whose equilibrium is unique and “globally asymptotically stable if N ≥ 5”, where the fold of the slow manifold “defines a tipping threshold that is not associated with any basin boundary” (p. 13); r_c = Σ 2⁻ⁿ with steady drift, eq. (33).
- The critical rate depends on where the system starts: eq. (17) for the saddle-node normal form, eq. (37) for the fast-slow system with unsteady drift; in the linear model, between the two bounds “the path of parameter variation will determine whether or not there is any R-tipping” (p. 7).
What bifurcates. With steady drift, a change to co-moving coordinates turns R-tipping into a bifurcation problem: a saddle-node at r = µ in y = x + λ (§3.1), a saddle-node or subcritical Hopf of the co-moving equilibrium of dw/dt = F(w) − r (§3.2.1). “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” (p. 11). With unsteady drift there is no such reduction (§3.2.2).
Rate cuts both ways. “B-tipping may be suppressed for fast enough variation of parameters” (p. 19, Fig. 8a): fast change can carry a system past a bifurcation it would have tipped at under slow change.
What this does not say
- That a system that tips is lost for good. Tipping is reaching the radius R, past which the linear model is “unphysical” (p. 5). In the climate model just below ρ_c “the system returns to the QSE after a transient” (Fig. 7a), and in the globally stable fast-slow system the paper does not say what follows the divergence. Loss is permanent only if the region past the threshold leads elsewhere, as in §3.1 and Fig. 7b, where solutions are unbounded.
- That the threshold is a basin boundary. It may be one (§§3.1–3.2) or not (§3.3); in the linear model it is a modelling choice.
- That the rate alone decides. The critical rate depends on the relaxation rate, the distance to the threshold, the starting state and, away from steady drift, the path.
- A general nonlinear theory. For nonlinear systems “There is however no exact analogy possible” with the tipping-radius model (p. 8), and no sufficient condition for tipping under varying drift is given.
- That any real system has R-tipped. The climate example is a zero-dimensional energy balance model; applications beyond climate are suggested, not shown (p. 19).