On the eigenvalue spectrum for time-delayed Floquet problems

A linear homogeneous scalar differential–difference equation with harmonic time dependence is investigated. The associated eigenvalue problem is solved in terms of a continued fraction expansion for the characteristic equation. The dependence of the largest eigenvalue on the system parameters, being relevant for stability of periodic states in delay systems, is discussed in detail. The competition between the two timescales, the delay and the external period cause intricate structures. The result suggests features to improve control of chaos by time-delayed feedback schemes with time-dependent control amplitudes.