Multiple time scales method for a two dimensional system with distributed delays

Building on the simplified mathematical model derived from the dynamics of the hypothalamic-pituitary-adrenal (HPA) axis, which describes the interactions between ACTH and cortisol through a system of differential equations with distributed delays, this paper focuses on the criticality analysis of the Hopf bifurcation. This analysis is conducted using the method of multiple time scales, which is applicable to systems with arbitrary distributed delays and indicates whether the Hopf bifurcation leads to a stable or unstable limit cycle. Furthermore, the approach presented in this paper can be applied to a broader class of systems that exhibit similar structural properties, extending its relevance beyond the specific biological context considered here.