Singularly Perturbed Boundary-Focus Bifurcations

The paper rigorously proves, for BF3 regularisations with algebraic tails, the existence of a continuous family of stable cycles for µ in (K ε^{k/(k+1)}, µ+) and Hausdorff proximity O(ε^{2k/(2k+1)}) to the PWS cycle, via multi-stage blow-up in an extended space (Theorems 3.8 and 3.11). The candidate solution establishes the same conclusions by a different route: a u=φ(y/ε) slow–fast reduction, Fenichel away from the corner, a weighted blow-up at the end of sliding, and a perturbative return map. Apart from a mild unstated global-return assumption in Step 1, the model’s argument is consistent with, and reaches the same results as, the paper.