Application of the Liapunov-Floquet transformation to differential equations with time delay and periodic coefficients

In this paper, the Liapunov-Floquet transformation (LFT) is applied to a time-periodic delay differential equation (DDE) discretized by the Chebyshev spectral continuous time approximation. The proposed combined approach allows for the stability and time-response analysis of a constant non-delayed analog of the original periodic DDE by applying the LFT to an equivalent large-order system of time-periodic ordinary differential equations. The implementation issues are analyzed for the time-delayed Mathieu's equation which is used as an example. It is shown that an order reduction procedure in which only the dominant modes of the infinite-dimensional DDE are retained in the LFT is necessary. The application of the proposed technique is studied in the presence of delay and parametric resonances for the delayed Mathieu's equation, as well as for a double inverted pendulum subjected to a time-periodic retarded follower force.