The generic failure of persistence through the focus-node bifurcation

Let $t^{**}$ be the anti-bump and let $t^*_\mu$ be the corresponding bump, with these quantities defined by

$$R_\lambda(t_c)=R_\mu(t_c)=R_\lambda(t^{**})=R_\mu(t^{**})=R_\lambda(t^*_\lambda)=R_\mu(t^*_\mu).$$

Assume hypotheses HFN1 through HFN5. The focus-node persistence conjecture. The following proposition is generically wrong: if a trajectory of the system goes along the slow curve at least on $]t^{**},a[$, then it goes along the slow curve at least on $[t^{**},t^*_\mu]$. This conjecture concerns the case in which the computed bump occurs after the focus-node bifurcation; the author states that the expected results are not proved and presents numerical evidence supporting this failure.