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Uniqueness and non-uniqueness of normal forms for vector fields

Published online by Cambridge University Press:  14 November 2011

Alberto Baider  and
Richard Churchill
Alberto Baider
Affiliation:
Department of Mathematical Sciences, Hunter College, 695 Park Avenue, New York, New York 10021, U.S.A.
Richard Churchill
Affiliation:
Department of Mathematical Sciences, Hunter College, 695 Park Avenue, New York, New York 10021, U.S.A.
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Synopsis

The use of normal forms in the study of equilibria of vector fields and Hamiltonian systems is a well-established practice and is described in standard references (e.g. [1], [7] or [10]). Also well known is the fact that such normal forms are not unique, and the relationship between distinct normal forms of the same vector field has also been investigated, in particular by M. Kummer [8] and A. Brjuno [2,3] (also see [12]). In this paper we use this relationship to extract invariants of the vector field directly from an arbitrary normal form. The treatment is sufficiently general to handle the vector field and Hamiltonian cases simultaneously, and applications in these contexts are presented.

The formulation of our main result (Theorem 1.1) is reminiscent of, and was heavily influenced by, work of Shi Songling on planar vector fields [11]. Additional inspiration was provided by M. Kummer's contributions to the 1:1 resonance problem in [9]. The authors are grateful to Richard Cushman for comments on an earlier version of this paper.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 108 , Issue 1-2 , 1988 , pp. 27 - 33
Copyright
Copyright © Royal Society of Edinburgh 1988

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