Numerical dynamics of integrodifference equations: Hierarchies of invariant bundles in $L^p(\Omega)$

TL;DR

This paper investigates how invariant manifolds for nonautonomous integrodifference equations in $L^p()$ spaces behave under spatial discretization, providing quantitative bounds and a perturbation theorem for their approximation.

Contribution

It establishes the closeness of invariant manifolds under Galerkin-type discretizations, extending the theory to nonautonomous integrodifference equations with smoothing kernels.

Findings

Invariant manifolds can be approximated with $C^{m-1}$-closeness under discretization.

The results include a quantitative perturbation theorem for invariant bundles.

Discretizations preserve the convergence order of the numerical method.

Abstract

We study how the "full hierarchy" of invariant manifolds for nonautonomous integrodifference equations on the Banach spaces of $p$-integrable functions behaves under spatial discretization of Galerkin type. These manifolds include the classical stable, center-stable, center, center-unstable and unstable ones, and can be represented as graphs of $C^m$-functions. For kernels with a smoothing property, our main result establishes closeness of these graphs in the $C^{m-1}$-topology under numerical discretizations preserving the convergence order of the method. It is formulated in a quantitative fashion and technically results from a general perturbation theorem on the existence of invariant bundles (i.e.\ nonautonomous invariant manifolds).