  The Hypothalamic-Pituitary-Adrenal (HPA) axis is a major neuroendocrine
system, and its dysregulation is implicated in various diseases. This system
also presents interesting mathematical challenges for modeling. We consider a
nonlinear delay differential equation model and calculate pseudospectra of
three different linearizations: a time-dependent Jacobian, linearization around
the limit cycle, and dynamic mode decomposition (DMD) analysis of Koopman
operators (global linearization). The time-dependent Jacobian provided insight
into experimental phenomena, explaining why rats respond differently to
perturbations during corticosterone secretion's upward versus downward slopes.
We developed new mathematical techniques for the other two linearizations to
calculate pseudospectra on Banach spaces and apply DMD to delay differential
equations, respectively. These methods helped establish local and global limit
cycle stability and study transients. Additionally, we discuss using
pseudospectra to substantiate the model in experimental contexts and establish
bio-variability via data-driven methods. This work is the first to utilize
pseudospectra to explore the HPA axis.
