Bifurcations in Hamiltonian systems. Computing singularities by Gröbner bases
type: Item
instance of: scholarly article
title: Bifurcations in Hamiltonian systems. Computing singularities by Gröbner bases
zbMATH Open document ID: 1029.37033
author: Igor Hoveijn
author: Gerton Lunter
author: Henk Broer
author: Gert Vegter
published in: Lecture Notes in Mathematics
publication date: 2003-04-09T00:00:00Z
full work available at URL: t1806.htm
review text: ``This book deals with nonlinear Hamiltonian systems, depending on parameters.\dots The general goal is to understand their dynamics in a qualitative, and if possible, also quantitative way. In many important cases, it is possible to reduce a skeleton of the dynamics to lower dimension, sometimes leading to a Hamiltonian system in one degree of freedom. Such reduced systems allow a singularity theory or catastrophe theory approach which gives raise to transparent, in a sense polynomial, normal forms. Moreover the whole process of arriving at these normal forms is algorithmic. The purpose of this book is to develop computer-algebraic tools for the implementation of these algorithms. This set-up allows for many applications concerning resonances in coupled or driven oscillators, the \(n\)-body problem, the dynamics of the rigid body \dots''. This description opens the book, and conveniently summarizes its scope. Given a Hamiltonian system (around an equilibrium point) described by a Hamiltonian \(H\) and the associated Hamiltonian vector field \(X\), the latter can be taken to Birkhoff-Gustavsson normal form up to some degree \(k\) by means of a sequence of canonical transformations; denoting the resulting canonical transformation as \(\Phi\), one has thus mapped \(X\) into the vector field \(\Phi_* (X) = N + R\), where \(N\) is the normalized part, and \(R\) a (small) remainder. Thus, the flow under the normalized part \(N\) provides an approximation of the ``true'' flow under \(\Phi_*(X)\), hence (upon use of \(\Phi^{-1}\)) also under \(X\). This is the ``skeleton'' of the dynamics mentioned above. In many interesting cases, the vector field \(N\) in normal form exhibits a large degree of symmetry, and can thus be reduced to a lower-dimensional (Hamiltonian) system \(N_0\) by more and less standard methods. As mentioned above, it is not so rare that the lower-dimensional system is actually a Hamiltonian system in one degree of freedom, thus amenable to complete anal…
reviewed by: Giuseppe Gaeta
Mathematics Subject Classification ID: 37J15
Mathematics Subject Classification ID: 37G05
Mathematics Subject Classification ID: 37J20
Mathematics Subject Classification ID: 58K99
Mathematics Subject Classification ID: 37M20
Mathematics Subject Classification ID: 70F10
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 70E99
zbMATH DE Number: 1896085
zbMATH Keywords: singularities
zbMATH Keywords: bifurcation
zbMATH Keywords: normal forms
zbMATH Keywords: Gröbner bases
zbMATH Keywords: standard bases
zbMATH Keywords: numerical simulations
zbMATH Keywords: symmetry reduction
zbMATH Keywords: discrete symmetries
MaRDI profile type: Publication
OpenAlex ID: W2481544634
DOI: 10.1007/B10414
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description: scientific article; zbMATH DE number 1896085