On the Chebyshev spectral continuous time approximation for constant and periodic delay differential equations

In this paper, the approximation technique proposed in Breda et al. (2005) [1] for converting a linear system of constant-coefficient delay differential equations (DDEs) into a system of ordinary differential equations (ODEs) using pseudospectral differencing is extended to linear and nonlinear systems of DDEs with time-periodic coefficients. The Chebyshev spectral continuous time approximation (ChSCTA) technique is used to study the stability of first and second-order constant coefficient DDEs, a delayed system with a cubic nonlinearity and parametric sinusoidal excitation, the delayed Mathieu’s equation, and delayed systems with two fixed delays. In all the examples, the stability and time response obtained from ChSCTA show good agreement with either analytical results, or the results obtained before by other reliable approximation methods. The “spectral accuracy” convergence behavior of Chebyshev spectral collocation shown in Trefethen (2000) [2] which the proposed technique possesses is compared to the convergence properties of finite difference-based continuous time approximation for constant-coefficient DDEs proposed recently in Sun (2009) [3] and Sun and Song (2009) [4].