Classification of convergence rates of solutions of perturbed ordinary differential equations with regularly varying nonlinearity

TL;DR

This paper classifies the convergence rates of solutions to perturbed scalar ODEs with regularly varying nonlinearities, showing how different decay rates of perturbations affect the speed of convergence to equilibrium.

Contribution

It provides a comprehensive classification of convergence rates for solutions of perturbed ODEs with regularly varying nonlinearities, including new rates at critical perturbation decay.

Findings

Convergence rate is preserved under rapid decay of perturbation.

At critical decay rate, convergence is slightly slower.

Slower decay of perturbation results in strictly slower convergence.

Abstract

In this paper we consider the rate of convergence of solutions of a scalar ordinary differential equation which is a perturbed version of an autonomous equation with a globally stable equilibrium. Under weak assumptions on the nonlinear mean reverting force, we demonstrate that the convergence rate is preserved when the perturbation decays more rapidly than a critical rate. At the critical rate, the convergence to equilibrium is slightly slower than the unperturbed equation, and when the perturbation decays more slowly than the critical rate, the convergence to equilibrium is strictly slower than that seen in the unperturbed equation. In the last case, under strengthened assumptions, a new convergence rate is recorded which depends on the convergence rate of the perturbation. The latter result relies on the function being regularly varying at the equilibrium with index greater than unity; therefore, for this class of regularly varying problems, a classification of the convergence rates is obtained.