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Manuscript Title: POINCARÉ CODE : a Package of Open-source Implements for Normalization and Computer Algebra Reduction near Equilibria of Coupled Ordinary Differential Equations
Authors: J. Mikram, F. Zinoun, A. El Abdllaoui
Program title: POINCARÉ
Catalogue identifier: AEPJ_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 184(2013)2204
Programming language: Maple V11.
Computer: See specifications for running Maple V11 or above.
Operating system: MS Windows.
Keywords: Maple package, Ordinary differential equations, Poincaré - Dulac normal form, Birkhoff - Gustavson normal form, Joint normal form, Lie transform.
PACS: 02.70.Wz.
Classification: 4.3, 5, 16.9.

Nature of problem:
Computing structure-preserving normal forms near the origin for nonlinear vector fields.

Solution method:
14 Maple commands are designed to compute various normal forms as well as their generating functions and associated normalizing transformations for nonlinear systems of ordinary differential equations, including Hamiltonian ones. Further reduction of these normal forms - in the presence of Lie-point symmetries - is also considered. All algorithms are based on Lie transform, thus leading to structure-preserving normal forms in the sense that all properties that can be formulated in terms of graded Lie algebras are preserved.

The semisimple part of the leading matrix about the origin (resp. the quadratic part in the Hamiltonian case) is assumed to be already taken into diagonal form (resp. canonical form) via a linear change of variables. In other words, the eigenvalues must be explicitly known.

Unusual features:
Further reduction of Poincaré-Dulac normal form via the joint normal form commands, taking profitably Lie-point symmetries, is perhaps the main feature of the software. Besides, to the best of our knowledge, and except the Hamiltonian case, it seems there is no CPC programs for computation of (structure-preserving) normal forms in the general case.

Running time:
Depends on the input data, mainly on system size, type of the leading matrice (semisimple or not), type of resonances, order of normalization and required options. Instantaneous for the provided examples.