# Floquet theory for periodic delay differential equations: current numerical routines

I would like to determine the stability of a system of periodic delay differential equations (a seasonal host-parasite model). I've tried to implement the method described in Lemma 2.5 in this paper:

Lemma 2.5 Assume that (E, E+) is an ordered Banach space with E+ being normal and Int(E+) $$\neq$$ ∅, which is equipped with the norm ||·||E . Let L be a positive bounded linear operator. Choose v0 ∈ Int(E+) and define an = ||Lvn-1||E , vn = $$\frac{Lv_{n-1}}{ > a_{n}}$$, ∀n ≥ 1. If $$\lim_{(n→+∞)}a_n$$ exists, then r(L) = $$\lim_{n→+∞}a_n$$.

But I'm not sure it's working properly (I've probably made a mistake somewhere in the implementation). The problem is it doesn't always seem to correspond to equilibrium points I find when simply simulating the model for a very long time. I looked into different ways of approximating Floquet multipliers/monodromy matrices etc. for delay equations, just so I could double check my work, and there seem to be many different approaches (e.g., semi-discretization, finite element analysis, Chebyshev polynomials, etc.)

My question, for anyone much more familiar with this literature than I am, is: Is there a currently preferred method? Are there any that are easy and quick to implement (I'm not a very sophisticated programmer or mathematician)? I would be especially interested in an example with published code. Some advice would be greatly appreciated.