A folded saddle singularity for equation (4.9) (figure 5d) is a sufficient condition for the existence of a critical ramping rate in system (4.2), not a necessary and sufficient condition as we say in the abstract, the second paragraph of §1a, the title of §4a and the second paragraph of §6.
Another sufficient condition for the existence of a critical ramping rate in system (4.2) is a transition between the phase portrait in figure 5b and any of the phase portraits in figure 5c,g–k as the ramping rate v is varied. Such a transition defines a critical ramping rate that, unlike the one defined by a folded saddle singularity in equation (4.12), is independent of the initial condition within Sa. An example is given in Ashwin et al. (2011, §3.3.1). Note that system (4.2) may not have a unique critical ramping rate. For example, as v is varied, there can be multiple transitions between the phase portraits in figure 5b–k, giving rise to more than one v-interval where the system ‘tips’ (produces an excitable response).
Finally, there is an error in equation (4.8). This equation should read and the scaling below equation (5.3) should read
- Received April 6, 2011.
- Accepted April 12, 2011.
- This journal is © 2011 The Royal Society