*Phil. Trans. R. Soc. A* 370, 1166–1184 (Published 30 January 2012) (doi:10.1098/rsta.2011.0306)

In this note, we correct an error in the first two paragraphs of §2*b* of the paper by Ashwin *et al*. [1]. This section attempts to generalize sufficient conditions for R-tipping in the linear model [1, equation (2.1)] with steady drift in §2*a* to the case of time-varying rates *r*(*t*). Starting with [1, equation (2.3)] and noting that |e^{Mu}*v*|≤∥e^{Mu}∥|*v*|, we have the upper bound
If *M* is stable, then
1.1for some real *c*,*β*>0 [2], and so
Hence, one can guarantee that [1, equation (2.1)] avoids R-tipping by time *t* if
1.2If *M* is scalar, then we can choose *c*=1, *β*=−*M*, and (1.2) reduces to [1, equation (2.9)]. On the other hand, if *M* is a matrix, then we need a good choice of *c* and *β* in (1.1) to make the tipping condition (1.2) optimal, but this depends on the matrix structure and not simply the norm; see, for example, the text by Hinrichsen & Pritchard [2] and the elegant estimates of Godunov [3, equation (13)]. Incidentally, we remark that within the unnumbered equation between [1, equation (2.1)] and [1, equation (2.2)] there should be a minus sign before the integral, though this is corrected in the rest of the paper.

The converse condition [1, equation (2.10)] is not correct, but can be corrected as follows. From the formula between [1, equation (2.3)] and [1, equation (2.4)] recall
where we define
Note that in the case of constant drift, while in the more general case, the expression for includes an additional term depending on *M* and the history of the rate of change of drift. Because , one can guarantee that an R-tipping occurs before time *t* if

## Acknowledgements

We thank Jan Sieber, Stuart Townley and Rowen Learoyd for conversations that helped us to clarify these points.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.