Integral Operators and Delay Differential Equations

Abstract

The monodromy operator of a linear delay differential equation with periodic coefficients is formulated as an integral operator. The kernel of this operator includes a factor formed from the fundamental solution of the linear delay differential equation. Although the properties of the fundamental solutions are known, in general there is no closed form for the fundamental solution. This paper describes a collocation procedure to approximate the fundamental solution before the integral operator is discretized. Using arguments on collectively compact operators, the eigenvalues of the discretized monodromy operator are shown to converge to the eigenvalues of the monodromy operator in integral form. The eigenvalues of the monodromy operator tell the stability of the linear delay differential equation. An application to several cases of the Van der Pol oscillator with delay will be given. AMS (MOS) Subject Classification. 34K11, 34K13, 34K28, 45P05, 47A75, 47B38, 47G10.