Analysis und Zahlentheorie

Prof. Dr. A. Krieg

S. Mayer, RWTH Aachen, J. Scheurle, TU München, S. Walcher, RWTH Aachen

The Poincaré-Dulac normal form of a vector field at a stationary point is widely used, and is particularly useful in the analysis of non-hyperbolic stationary points. The special advantage of this type of normal form is a built-in linear symmetry group, thus there is a path to systematic reduction. On the other hand, the computation of a Poincaré-Dulac normal form is problematic, since most algorithms require the exact knowledge of the eigenvalues of the linearization. For computations coming from applied problems, this assumption is not realistic.

This page contains various programs (Maple 7) and example computations for Poincaré-Dulac normal forms and corresponding reduced vector fields. No explicit knowledge of the eigenvalues of the linearization is required, and only rational operations are needed in the normal form computations.

The programs and examples are based on two recent papers:

- J. Scheurle, S. Walcher: On normal form computations. In: P. K. Newton, P. Holmes, A. Weinstein (eds.): Geometry, Dynamics and Mechanics, Springer-Verlag, New York (2002), pp. 309-325.
- S. Mayer, J. Scheurle, S. Walcher: Practical normal form computations for vector fields, Preprint, 11 pp. (2002)

A short presentation (mostly focussed on solving the homological equation and determining invariants for arbitrary linear maps) can be downloaded here (pdf), while the html-version can be found here.

Readers are free to download the programs on this page, use them for their own computations or develop them further.

**Disclaimer:** While there is no indication of any problems with the algorithms and programs, the authors cannot guarantee the correctness of any of these programs.

S. Mayer

Lehrstuhl A für Mathematik

RWTH Aachen

D-52056 Aachen

Germany

Email: mayer@mathA.rwth-aachen.de

J. Scheurle

Zentrum Mathematik

der Technische Universität München

D-85747 Garching bei München

Germany

Email: scheurle@mathematik.tu-muenchen.de

S. Walcher

Lehrstuhl A für Mathematik

RWTH Aachen

D-52056 Aachen

Germany

Email: walcher@mathA.rwth-aachen.de

- Requisite programs
- Determining annihilating polynomials
- Homological equation at fixed degree
- Higher-order normalization
- Invariants at fixed degree
- Reduced vector field
- Complete program