A Simple Criterion for Identifying Slow–Fast Relaxation Oscillations

Simple conditions have been developed in [Zhang, W., Wahl, L. & Yu, P. [2014] “Viral blips may not need a trigger: How transient viremia can arise in deterministic in-host models,” SIAM Rev. 56, 127–155; Yu, P. & Wang, X. [2019] “Analysis on recurrence behavior in oscillating networks of biologically relevant organic reactions,” Math. Biosci. Eng. 16, 5263–5286], which are used to identify the existence of slow–fast relaxation oscillations that appear in differential systems, where the geometric singular perturbation theory is not applicable. In this study, we present seven models which fail to satisfy these conditions, yet still show recurrence phenomena, that motivate us to generalize these conditions to establish new simple hypothetical conditions for identifying slow–fast relaxation oscillations and providing arguments toward the development of a mathematical theory.