POINCARÉ CODE: A package of open-source implements for normalization and computer algebra reduction near equilibria of coupled ordinary differential equations

Abstract

The Poincaré code is a Maple project package that aims to gather significant computer algebra normal form (and subsequent reduction) methods for handling nonlinear ordinary differential equations. As a first version, a set of fourteen easy-to-use Maple commands is introduced for symbolic creation of (improved variants of Poincaré’s) normal forms as well as their associated normalizing transformations. The software is the implementation by the authors of carefully studied and followed up selected normal form procedures from the literature, including some authors’ contributions to the subject. As can be seen, joint-normal-form programs involving Lie-point symmetries are of special interest and are published in CPC Program Library for the first time, Hamiltonian variants being also very useful as they lead to encouraging results when applied, for example, to models from computational physics like Hénon–Heiles. Program summaryProgram title: POINCARÉCatalogue identifier: AEPJ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEPJ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 7694No. of bytes in distributed program, including test data, etc.: 597943Distribution format: tar.gzProgramming language: Maple V11.Computer: See specifications for running Maple V11 or above.Operating system: MS Windows.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 the structure-preserving normal forms in the sense that all properties that can be formulated in terms of graded Lie algebras are preserved.Restrictions: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 that 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 matrix (semisimple or not), type of resonances, order of normalization and required options. Instantaneous for the provided examples.