Stability of Periodic Linear Delay–differential Equations and the Chebyshev Approximation of Fundamental Solutions

These notes are a general introduction and exposition. They contain more theory and more appendices than are likely to appear in any paper but components will be published. We start with relatively well-known theory including an important variation–of– parameters formula whose (period)=(delay) form is equation (7). Next we review the Sinha–Wu [SW] method of approximation of fundamental solutions to ODEs. We give additional results on the accuracy of Picard iteration and Chebyshev approximation. We extend the method to the approximation of the “infinite–dimensional Floquet transition matrix” U in (7). The stability of the DDE is approximately determined by the eigenvalues of the approximating matrix to U . Examples are given in [A] which show the effectiveness of the resulting approximation. V. Averina’s thesis [A] should be regarded as part II of these notes. See [BM] for numerical results using the ideas of these notes.