Monodromy operator approximation for periodic solutions of differential-difference equations

The following nonlinear equation with delay is considered:
\begin{equation} x'(t)= f(x(t),x(t-1)). \tag{1} \end{equation}
It is supposed that a periodic solution $\tilde x$ of this equation with period $T$ is known. The Floquet theory is known to be valid for Eq. [1], which permits to describe the behavior of the solutions in the field the of periodical solution $\tilde x$ in terms of monodromy operator eigenvalues associated with the $\tilde x$ solution.
Unlike the case of ordinary differential equations, yet there is no a universal efficient method to find Floquet multipliers for the casual Eq. (1) and solution $\tilde x$ (at least approximately). In [2–4], there are most sufficient results for such an approach. If the period is not commensurable with delay, then considerable difficulties arise. One of the ways to solve this problem is based on the approximation of an original problem by means of a sequence of auxiliary problems. The problem of building of auxiliary model problems has not been solved yet and still there is no evaluation method to calculate the error within such an approach to the finding of Floquet multipliers.
A new way of approximation of the original problem is introduced in this paper. A number of examples are provided.