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International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 590856, 11 pages
Research Article

The Return Map for a Planar Vector Field with Nilpotent Linear Part: A Direct and Explicit Derivation

Ohio State University, Columbus, OH 43210, USA

Received 18 May 2009; Revised 4 July 2009; Accepted 24 August 2009

Academic Editor: Manfred H. Moller

Copyright © 2009 Rodica D. Costin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Using a direct approach the return map near a focus of a planar vector field with nilpotent linear part is found as a convergent power series which is a perturbation of the identity and whose terms can be calculated iteratively. The first nontrivial coefficient is the value of an Abelian integral, and the following ones are explicitly given as iterated integrals.