Bifurcations of mixed-mode oscillations in three-timescale systems: an extended prototypical example

The paper’s Proposition 5 classifies singular cycles in the double singular limit into remote, aligned, and connected cases and constructs them as concatenations of fast, intermediate, and slow segments; two-scale cycles occur on constant-z planes in the remote/aligned cases, and a unique three-scale cycle occurs in the connected case on P− ∪ P+ ∪ Za∓. The candidate solution reproduces exactly this geometric construction and orientation logic (fast jump maps J±, intermediate flow toward the folds, slow drift along Za∓ with a µ-interval fixing its sign). The only gaps are minor: the solution implicitly uses the outer/middle branch structure of M2 (requiring two fold points) and identifies Z± by x < 0 and x > xq+, whereas the paper defines Z± via the fold points p± of M2; monotonicity/uniqueness of intersections on Z± is also used but not stated. Overall, both arguments agree in substance and conclusion, with the paper offering a qualitative proof sketch rather than a full proof.