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Normal Forms in Perturbation Theory

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Nonlinear dynamical systems are notoriously hard to tackle by analytic means. One of the few approaches that has been effective for the last coupleof centuries, is Perturbation Theory. Here systems are studied, which in an appropriate sense, can be seen as perturbations of a given system with‘well-known’ dynamical properties. Such ‘well-known’ systems usually are systems with a great amount of symmetry(like integrable Hamiltonian systems [1]) or very low-dimensional systems. The methods ofPerturbation Theory then try to extend the ‘well-known’ dynamical properties to the perturbed system. Methods to do this are often basedon the Implicit Function Theorem, on normal hyperbolicity [85,87] or on Kolmogorov–Arnold–Moser theory [1,10,15,18,38].

To obtain a perturbation theory set-up, normal form theory is a vital tool. In its most elementary form it amounts to‘simplifying’ the Taylor series of a dynamical system at an equilibrium point by successive changes of...

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Abbreviations

Normal form procedure:

This is the stepwise ‘simplification’ by changes of coordinates, of the Taylor series at an equilibrium point, or of similar series at periodic or quasi-periodic solutions.

Preservation of structure:

The normal form procedure is set up in such a way that all coordinate changes preserve a certain appropriate structure. This applies to the class of Hamiltonian or volume preserving systems, as well as to systems that are equivariant or reversible with respect to a symmetry group. In all cases the systems may also depend on parameters.

Symmetry reduction:

The truncated normal form often exhibits a toroidal symmetry that can be factored out, thereby leading to a lower dimensional reduction.

Perturbation theory:

The attempt to extend properties of the (possibly reduced) normal form truncation, to the full system.

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Authors and Affiliations

  1. University of Groningen, Groningen, The Netherlands

    Henk W. Broer

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  1. Henk W. Broer
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Broer, H.W. (2009). Normal Forms in Perturbation Theory. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_372

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