Nonlocal branches of cycles, bistability, and topologically persistent mixed mode oscillations.

A possible mechanism for generating mixed mode oscillations is based on an appropriate S-shaped structure, which graphs the relation between the parameter and the collection of periodic oscillations existing for a particular parameter value in the product of parameter and phase spaces. This natural scenario should be supplemented by simple and constructive criteria of existence, and methods of localization, of such S-shaped structures. These criteria are the main focus of the paper.