Averaging and computing normal forms with word series algorithms

Abstract

In the first part of the present work we consider periodically or quasiperiodically forced systems of the form $(d/dt)x = εf(x,t ω)$, where $ε\ll 1$, $ω\in\mathbb{R}^d$ is a nonresonant vector of frequencies and $f(x,θ)$ is $2π$-periodic in each of the $d$ components of $θ$ (i.e.\ $θ\in\mathbb{T}^d$). We describe in detail a technique for explicitly finding a change of variables $x = u(X,θ;ε)$ and an (autonomous) averaged system $(d/dt) X = εF(X;ε)$ 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 $x(t) = u(X(t),tω;ε)$. 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.