Averaging and Computing Normal Forms with Word Series Algorithms

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 267)


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.


Averaging High-order averaging Quasi-stroboscopic averaging Hamiltonian problems Near-integrable systems Normal forms 

Mathematics Subject Classification (2010)

34C20 34C29 70H05 



A. Murua and J. M. Sanz-Serna have been supported by projects MTM2013-46553-C3-2-P and MTM2013-46553-C3-1-P from Ministerio de Economía y Comercio, Spain. Additionally A. Murua has been partially supported by the Basque Government (Consolidated Research Group IT649-13).


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Konputazio Zientziak eta A. A. Saila, Informatika FakultateaUPV/EHUDonostia–San SebastiánSpain
  2. 2.Departamento de MatemáticasUniversidad Carlos III de MadridLeganés, MadridSpain

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