Source
Ashwin, Wieczorek, Vitolo and Cox (2012), LIT-1654, §1 (p. 2), §2.2 (p. 8), §3.3.2 (p. 16) and §5 (p. 19), arXiv v5; read in NOTE-1215.
What was actually shown
In the linear model, tracking is a rate comparison. With ‖e^{Mu}‖ ≤ c e^{−βu}, the system avoids tipping by time t if (c/β)·r_max(t) < R, eq. (13), where r_max is the largest drift |dx̃/dt| so far; for scalar M this is ‖M⁻¹‖·r_max(t) < R, eq. (14). The paper reads it as time scales: the QSE’s motion has a natural time R/|r(t)|, and “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” (p. 8).
The drift that counts is the attractor’s, not the parameter’s. r = (dx̃/dλ)(dλ/dt), eq. (8). “if |dx̃/dλ| ≈ 1/ε, then clearly we can have R-tipping even when |dλ/dt| ≈ ε‖M‖” (p. 8). A parameter that moves slowly against the system’s own rates can still tip it if the attractor’s position is sensitive to the parameter.
In the fast-slow example the separation is named. “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).
No slowing is needed. “neither N-tipping and R-tipping require any change of stability” (p. 19). In the linear model M is fixed, so no mode slows; the separation is lost because the input speeds up. This is set against the separation κ_drift ≪ κ_crit ≪ κ_stab that time-series analysis of bifurcation-induced tipping relies on, for which “there is no a priori reason … to hold for a given system” (p. 2).
What this does not say
- That the comparison decides for nonlinear systems. The tipping-radius model has “no exact analogy” there: an effective radius “may depend on the shape of the local basin of attraction, the nonlinearities present and the exact path taken in parameter space” (p. 8). The linear condition is sufficient for tracking, not necessary, and no sufficient condition for tipping under varying drift is given (p. 8).
- That separation lost always means tipping. Between the two steady-drift bounds the path decides (p. 7), and the starting state moves the critical rate (eqs. 17, 37).
- That this is the only way separation fails. Near a bifurcation the slowest relaxation goes to zero and the same comparison fails from the other side; that is B-tipping with slow passage effects, which the paper names (p. 2) and does not analyse.
- Anything about constraints or individuation. The reading of the quasi-static attractor as a constraint that holds over a window between time scales is CLAIM-478’s, not the paper’s.