On the average principle for one-frequency systems

TL;DR

This paper provides a fully quantitative estimate for the error in the averaging approximation of one-frequency perturbed integrable systems, improving understanding of the approximation's accuracy over various time scales.

Contribution

It introduces a new, fully quantitative error estimate for the averaging approximation, applicable even in resonant cases, and demonstrates its computational efficiency.

Findings

Estimator closely matches the actual oscillating distance in many cases

Good error estimates are obtained even in resonant scenarios

Estimator computation is faster than direct numerical simulation

Abstract

We consider a perturbed integrable system with one frequency, and the approximate dynamics for the actions given by averaging over the angle. The classical theory grants that, for a perturbation of order epsilon, the error of this approximation is O(epsilon) on a time scale O(1/epsilon), for epsilon -> 0. We replace this qualitative statement with a fully quantitative estimate; in certain cases, our approach also gives a reliable error estimate on time scales larger than 1/epsilon. A number of examples are presented; in many cases our estimator practically coincides with the envelope of the rapidly oscillating distance between the actions of the perturbed and of the averaged systems. Fairly good results are also obtained in some "resonant" cases, where the angular frequency is small along the trajectory of the system. Even though our estimates are proved theoretically, their computation in specific applications typically requires the numerical solution of a system of differential equations. However, the time scale for this system is smaller by a factor epsilon than the time scale for the perturbed system. For this reason, computation of our estimator is faster than the direct numerical solution of the perturbed system; the estimator is rapidly found also in cases when the time scale makes impossible (within reasonable CPU times) or unreliable the direct solution of the perturbed system.