Encyclopedia of Complexity and Systems Science

2009 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Diagrammatic Methods in Classical Perturbation Theory

  • Guido Gentile
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_122

Definition of the Subject

Recursive equations naturally arise whenever a dynamical system is considered in the regime of perturbation theory; for an introductory articleon perturbation theory see  Perturbation Theory. A classical example is provided by Celestial Mechanics, where perturbation series, known asLindstedt series , are widely used; see Gallavotti [21] and  Perturbation Theory in Celestial Mechanics.

A typical problem in Celestial Mechanics is to study formal solutions of given ordinary differential equations in the form of expansions ina suitable small parameter, the perturbation parameter. In the case of quasi-periodic solutions, the study of the series, in particular of itsconvergence, is made difficult by the presence of the small divisors – which will be defined later on. Under some non-resonance condition onthe frequency vector, one can show that the series are well-defined to any order. The first proof of such a property was given byPoincaré [53], even if the...

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Guido Gentile
    • 1
  1. 1.Dipartimento di MatematicaUniversità di Roma TreRomaItaly