Equations with infinite delay: pseudospectral discretization for numerical stability and bifurcation in an abstract framework

TL;DR

This paper develops a pseudospectral discretization method for nonlinear delay differential and renewal equations with infinite delay, ensuring numerical stability and accurate bifurcation analysis through convergence of characteristic roots.

Contribution

It introduces a unifying abstract framework and proves convergence of pseudospectral approximations, enabling reliable stability and bifurcation analysis of infinite delay equations.

Findings

Convergence of characteristic roots for Laguerre-based pseudospectral discretizations.

Equilibrium correspondence between original and approximated equations.

Effective numerical tests demonstrating bifurcation analysis capabilities.

Abstract

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