Averaging and computing normal forms with word series algorithms

TL;DR

This paper introduces a series-based method using word series algorithms for averaging and transforming periodically or quasiperiodically forced systems into simpler normal forms, enabling explicit solution relations and invariants.

Contribution

It develops a universal, recursive series technique for averaging and normal form transformations applicable to forced and perturbed systems, with explicit formulas and invariants.

Findings

Explicit series formulas for averaging transformations.

Universal coefficients independent of specific functions.

Construction of formal invariants in Hamiltonian systems.

Abstract

In the first part of the present work we consider periodically or quasiperiodically forced systems of the form $(d/dt)x = \epsilon f(x,t \omega )$, where $\epsilon\ll 1$, $\omega\in\mathbb{R}^d$ is a nonresonant vector of frequencies and $f(x,\theta)$ is $2\pi$-periodic in each of the $d$ components of $\theta$ (i.e.\ $\theta\in\mathbb{T}^d$). We describe in detail a technique for explicitly finding a change of variables $x = u(X,\theta;\epsilon)$ and an (autonomous) averaged system $(d/dt) X = \epsilon F(X;\epsilon)$ so that, formally, the solutions of the given system may be expressed in terms of the solutions of the averaged system by means of the relation $x(t) = u(X(t),t\omega;\epsilon)$. Here $u$ and $F$ are found as series whose terms consist of vector-valued maps weighted by suitable scalar coefficients. The maps are easily written down by combining the Fourier coefficients of $f$ and the coefficients are found with the help of simple recursions. Furthermore these coefficients are {\em universal} in the sense that they do not depend on the particular $f$ under consideration. In the second part of the contribution, we study problems of the form $(d/dt) x = g(x)+f(x)$, where one knows how to integrate the "unperturbed" problem $(d/dt)x = g(x)$ and $f$ is a perturbation satisfying appropriate hypotheses. It is shown how to explicitly rewrite the system in the "normal form" $(d/dt) x = \bar g(x)+\bar f(x)$, where $\bar g$ and $\bar f$ are {\em commuting} vector fields and the flow of $(d/dt) x = \bar g(x)$ is conjugate to that of the unperturbed $(d/dt)x = g(x)$. In Hamiltonian problems the normal form directly leads to the explicit construction of formal invariants of motion. Again, $\bar g$, $\bar f$ and the invariants are written as series consisting of known vector-valued maps and universal scalar coefficients that may be found recursively.