Equations with Infinite Delay: Pseudospectral Discretization for Numerical Stability and Bifurcation in an Abstract Framework

We consider nonlinear delay differential and renewal equations with infinitedelay. We extend the work of Gyllenberg et al, Appl. Math. Comput. (2018) byintroducing a unifying abstract framework, and derive a finite-dimensionalapproximating system via pseudospectral discretization. For renewal equations,we consider a reformulation in the space of absolutely continuous functions viaintegration. We prove the one-to-one correspondence of equilibria between theoriginal equation and its approximation, and that linearization anddiscretization commute. Our most important result is the proof of convergenceof the characteristic roots of the pseudospectral approximation of thelinear(ized) equations when the collocation nodes are chosen as the family ofscaled zeros or extrema of Laguerre polynomials. This ensures that thefinite-dimensional system correctly reproduces the stability properties of theoriginal linear equation if the dimension of the approximation is large enough.The result is illustrated with several numerical tests, which also demonstratethe effectiveness of the approach for the bifurcation analysis of equilibria ofnonlinear equations. The new approach used to prove convergence also providesthe exact location of the spectrum of the differentiation matrices for theLaguerre zeros and extrema, adding new insights into properties that areimportant in the numerical solution of differential equations by pseudospectralmethods.