Galerkin Projections for Delay Differential Equations

Abstract

We present a Galerkin projection technique by which finite-dimensional ordinary differential equation (ODE) approximations for delay differential equations (DDEs) can be obtained in a straightforward fashion. The technique requires neither the system to be near a bifurcation point, nor the delayed terms to have any specific restrictive form, or even the delay, nonlinearities, and/or forcing to be small. We show through several numerical examples that the systems of ODEs obtained using this procedure can accurately capture the dynamics of the DDEs under study, and that the accuracy of solutions increases with increasing numbers of shape functions used in the Galerkin projection. Examples studied here include a linear constant coefficient DDE as well as forced nonlinear DDEs with one or more delays and possibly nonlinear delayed terms. Parameter studies, with associated bifurcation diagrams, show that the qualitative dynamics of the DDEs can be captured satisfactorily with a modest number of shape functions in the Galerkin projection.