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The theory for \u2018oscillatory\u2019 multiple time-scale systems which possess a limit cycle manifold instead of (or in addition to) a critical manifold is less developed, particularly in the non-normally hyperbolic regime. We use the blow-up method to analyse the global oscillatory transition near a regular folded limit cycle manifold in a class of three time-scale \u2018semi-oscillatory\u2019 systems with two small parameters. The systems considered behave like oscillatory systems as the smallest perturbation parameter tends to zero, and stationary systems as both perturbation parameters tend to zero. The additional time-scale structure is crucial for the applicability of the blow-up method, which cannot be applied directly to the two time-scale oscillatory counterpart of the problem. Our methods allow us to describe the asymptotics and strong contractivity of all solutions which traverse a neighbourhood of the global singularity. Our main results cover a range of different cases with respect to the relative time-scale of the angular dynamics and the parameter drift. We demonstrate the applicability of our results for systems with periodic forcing in the slow equation, in particular for a class of Li\u00e9nard equations. Finally, we consider a toy model used to study tipping phenomena in climate systems with periodic forcing in the fast equation, which violates the conditions of our main results, in order to demonstrate the applicability of classical (two time-scale) theory for problems of this kind.\n                  <\/jats:p>","DOI":"10.1007\/s00332-023-09987-x","type":"journal-article","created":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T13:01:51Z","timestamp":1700139711000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Geometric Blow-Up for Folded Limit Cycle Manifolds in Three Time-Scale Systems"],"prefix":"10.1007","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8539-320X","authenticated-orcid":false,"given":"S.","family":"Jelbart","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7063-6173","authenticated-orcid":false,"given":"C.","family":"Kuehn","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-4611-9742","authenticated-orcid":false,"given":"S.-V.","family":"Kuntz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,11,16]]},"reference":[{"key":"9987_CR1","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1023\/A:1021739205527","volume":"5","author":"OD Anosova","year":"1999","unstructured":"Anosova, O.D.: On invariant manifolds in singularly perturbed systems. 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