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Geometric singular perturbation theory provides a powerful mathematicalframework for the analysis of 'stationary' multiple time-scale systems whichpossess a critical manifold, i.e. a smooth manifold of steady states for thelimiting fast subsystem, particularly when combined with a method ofdesingularization known as blow-up. The theory for 'oscillatory' multipletime-scale systems which possess a limit cycle manifold instead of (or inaddition to) a critical manifold is less developed, particularly in thenon-normally hyperbolic regime. We use the blow-up method to analyse the globaloscillatory transition near a regular folded limit cycle manifold in a class ofthree time-scale 'semi-oscillatory' systems with two small parameters. Thesystems considered behave like oscillatory systems as the smallest perturbationparameter tends to zero, and stationary systems as both perturbation parameterstend to zero. The additional time-scale structure is crucial for theapplicability of the blow-up method, which cannot be applied directly to thetwo time-scale oscillatory counterpart of the problem. Our methods allow us todescribe the asymptotics and strong contractivity of all solutions whichtraverse a neighbourhood of the global singularity. Our main results cover arange of different cases with respect to the relative time-scale of the angulardynamics and the parameter drift. We demonstrate the applicability of ourresults for systems with periodic forcing in the slow equation, in particularfor a class of Li\'enard equations. Finally, we consider a toy model used tostudy tipping phenomena in climate systems with periodic forcing in the fastequation, which violates the conditions of our main results, in order todemonstrate the applicability of classical (two time-scale) theory for problemsof this kind.

Geometric Blow-Up for Folded Limit Cycle Manifolds in Three Time-Scale Systems