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

TL;DR

This work develops a unified abstract framework for nonlinear infinite-delay equations (iDEs) by recasting iDDEs and iREs as semilinear ADEs in exponentially weighted spaces. It then applies pseudospectral discretization (PSD) with Laguerre-based collocation to obtain finite-dimensional ODEs that preserve equilibria and linearization, and proves convergence of the discrete spectra to the true spectra for the linearized problem when collocation nodes are scaled Laguerre zeros or extrema. Theoretical results include explicit error bounds and exact spectral information for the differentiation matrices, complemented by numerical experiments on linear iDDEs/iREs and nonlinear iDEs to demonstrate exponential convergence and reliable bifurcation analysis. The approach provides a practically effective tool for stability assessment and bifurcation analysis of nonlinear iDEs, with insights into the spectral properties of Laguerre-based differentiation operators and potential extensions to resolvent convergence and node optimization.

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.

Figures (4)

• Figure 1: Linear DDE \ref{['linear-DDE']}: log-log plot of $|\lambda-\lambda_N|$ (solid) and $\|w(\psi_\lambda-\psi_{\lambda,N})\|_{\infty,N}$ (dotted), for parameters specified in \ref{['t:linear-DE']}. See main text for further details.
• Figure 2: Linear RE \ref{['linear-RE']}: log-log plot of $|\lambda-\lambda_N|$ (solid) and $\|\psi_\lambda-\psi_{\lambda,N}\|_{1,N}$ (dotted), for parameter values specified in \ref{['t:linear-DE']}.
• Figure 3: Top-left: Hopf bifurcation curve of \ref{['beretta-breda']} in $(\tau,m)$ computed with MatCont, for $\delta_A=0.5,\,\delta_J=1,\,a=7,\,b=350$, and $\rho=(\delta_J+m/\tau)/4$, for $N=5,10,20$ (stable positive equilibrium below the curve). Note that the curves for $N=10,\,20$ are indistinguishable. The other panels show the error in the two Hopf bifurcation points and the critical characteristic root at Hopf, for $m=6.5,\,7,\,7.5$, with MatCont tolerance $10^{-10}$. Reference Hopf values are computed from the equivalent ODE formulation GSV18.
• Figure 4: Left: Hopf bifurcation curve of \ref{['Nich-RE']} in $(\mu,\gamma/\mu)$, for $\eta:=\beta_0 \mathrm{e}^{-\mu}/\mu$, computed using MatCont tolerance and $10^{-6}$, $\rho=\mu/2$. The positive equilibrium exists above the line $\eta=1$ ('branching point', BP). Gray lines are computed with DDE-BIFTOOL on the equivalent DDE formulation. Right: error in BP and Hopf for $\mu=2$, with MatCont tolerance $10^{-10}$. Reference values are computed with $N=50$.

Theorems & Definitions (16)

• Theorem 2.1
• Theorem 2.2
• Remark 3.1
• Proposition 4.2
• Proof 1
• Theorem 4.3: One-to-one correspondence of equilibria
• Proof 2
• Proposition 4.4
• Theorem 4.5
• Proof 3