Dynamical systems with time delay describe many phenomena in science - engineering, physics, biology, to name a few. In many applications inclusion of the past history of the system is not only desirable but is necessary for obtaining practical results.

The stability of the delay-differential equations (DDEs) with linear constant coefficients has been thoroughly studied. However, there are no general analytical methods for DDE systems with time-dependent coefficients. The importance of this area is apparent in engineering fields such as machine tool vibrations and optimal control, among others. We propose a numerical method to study the parameter-dependent stability of this kind of systems with the period of coefficients being rationally related to the delay.

It has been shown that an infinite-dimensional version of Floquet theory can be applied to periodic DDEs, thus the stability of the system can be determined by infinitely many eigenvalues. We construct an approximation of the ''infinite-dimensional Floquet transition matrix'' by considering differentiation and coefficient multiplication as operators on space of Chebyshev polynomials. We show the stability boundaries of some well-known examples of DDEs in mathematics and mechanics. We also consider application of the proposed method to the problem of air-to-fuel ratio regulation in internal combustion engines.