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 $(d/dt)x = \varepsilon f(x,t \omega )$, where $\varepsilon \ll 1$, $\omega \in \mathbb {R}^d$ 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 $x = u(X,\theta ;\varepsilon )$ and an (autonomous) averaged system $(d/dt) X = \varepsilon F(X;\varepsilon )$ 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 $x(t) = u(X(t),t\omega ;\varepsilon )$. 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 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 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.