Asymptotic behaviour of coupled systems in discrete and continuous time

TL;DR

This paper analyzes the long-term behavior of solutions to infinite coupled systems in discrete and continuous time, providing convergence criteria, optimal rates, and applications to specific examples.

Contribution

It offers a new characterization of initial conditions leading to convergence and improves convergence rate estimates by linking discrete and continuous systems.

Findings

Characterization of initial values for convergence

Optimal convergence rate estimates

Application to concrete examples in both settings

Abstract

This paper investigates the asymptotic behaviour of solutions to certain infinite systems of coupled recurrence relations. In particular, we obtain a characterisation of those initial values which lead to a convergent solution, and for initial values satisfying a slightly stronger condition we obtain an optimal estimate on the rate of convergence. By establishing a connection with a related problem in continuous time, we are able to use this optimal estimate to improve the rate of convergence in the continuous setting obtained by the authors in a previous paper. We illustrate the power of the general approach by using it to study several concrete examples, both in continuous and in discrete time.