Multiscale time averaging, Reloaded

TL;DR

This paper introduces a rigorous multi-time scale averaging method for finite-dimensional conservative dynamical systems with slow frequencies, addressing small divisor issues through iterative scaling and normal form transformations.

Contribution

It presents a novel controlled averaging technique that operates on finite intervals and scales to arbitrary order, improving analysis of quasiperiodic systems.

Findings

Effective handling of small divisor problems in quasiperiodic systems

Development of a scalable multi-time averaging framework

Enhanced perturbative analysis through normal form transformations

Abstract

We develop a rigorously controlled multi-time scale averaging technique; the averaging is done on a finite time interval, properly chosen, and then, via iterations and normal form transformations, the time intervals are scaled to arbitrary order. Here, we consider as an example the problem of finite dimensional conservative dynamical system, which is quasiperiodic and dominated by slow frequencies, leading to small divisor problems in perturbative schemes.