The interest in this topic comes from my research work on an NSF project "Symbolic Stability and Bifurcation Analysis of Time-Periodic Differential-Delay Equations: Applications to High-Speed Machining Models". You can find more information about this project on a page of my former advisor, Ed Bueler.

DDEs are used to describe many phenomena in science - economics, mechanics, population dynamics, physics, medicine, chemistry, engineering, and so on. In fact, a time delay occurs naturally in just about any interaction of the real world, hence almost any model without delays is an approximation at best. In many applications it is assumed that the future state of the system is independent of the past and is determined solely by the present. Models based on this assumption involve ordinary and partial differential equations. However, in many cases a more realistic model of the system must include some of the past history of the system. For example, in turning, a machine process in which the tool cuts the surface formed in the previous cut, self-excited vibrations may result from the regenerative nature of the process. In order to model this regenerative effect we need to include a time-delay into a mathematical model of the process, where the delay period is the time required for one rotation of the machined part.

Explicit forms of the solutions to the mathematical models stated by DDEs can be found for linear constant coefficient cases, but introduction of time-dependent coefficients virtually always prohibits exhibiting explicit solutions. Due to this fact relatively little is achieved in the existing theory of DDEs on qualitative analysis of the solutions such as their asymptotic behavior, stability properties, and the possibility of bifurcation on varying a parameter contained in the equations. Techniques for determining analytical stability boundaries of system of constant coefficient delay differential equations as a function of the system of parameters are well-known. The symbolic computation (using computer algebra systems) of stability boundaries for system of time-periodic ordinary differential equations has also been recently demonstrated. However, there are no general analytical methods for systems with both time-periodic coefficients and time-delay.

The initial aim of our work was to develop methods for the symbolic computation of linear stability boundaries for linear system with parameters of time-periodic DDEs with the period being equal to the delay. We did develop such a method, however it turned out that the symbolic computations needed to obtain the results are enormously complicated and cannot be handled effectively in a modern version of any computer algebra system.

At the moment we concentrate our efforts on developing an effective approach to finding numerical stability of DDEs using collocation approximation with Chebyshev polynomials. We also have broadened our goal from DDEs with a delay being equal to the period of coefficients to DDEs with multiple delays which are integer multiples of the period.