Stability of linear time-periodic delay-differential equations via Chebyshev polynomials

This paper presents a new technique for studying the stability properties of dynamic systems modeled by delay-differential equations (DDEs) with time-periodic parameters. By employing a shifted Chebyshev polynomial approximation in each time interval with length equal to the delay and parametric excitation period, the dynamic system can be reduced to a set of linear difference equations for the Chebyshev expansion coefficients of the state vector in the previous and current intervals. This defines a linear map which is the 'infinite-dimensional Floquet transition matrix U′. Two different formulas for the computation of the approximate U. whose size is determined by the number of polynomials employed, are given. The first one uses the direct integral form of the original system in state space form while the second uses a convolution integral (variation of parameters) formulation. Additionally, a variation on the former method for direct application to second-order systems is also shown. An error analysis is presented which allows the number of polynomials employed in the approximation to be selected in advance for a desired tolerance. An extension of the method to the case where the delay and parametric periods are commensurate is also shown. Stability charts are produced for several examples of time-periodic DDEs, including the delayed Mathieu equation and a model for regenerative chatter in impedance-modulated turning. The results indicate that this method is an effective way to study the stability of time-periodic DDEs.