Bifurcations of Planar Vector Fields: Nilpotent Singularities and Abelian Integrals

Freddy Dumortier, Robert Roussarie, Jorge Sotomayor, Henryk Żaładek (auth.)3540545212, 9783540545217, 0387545212

The book reports on recent work by the authors on the bifurcation structure of singular points of planar vector fields whose linear parts are nilpotent. The bifurcation diagrams of the most important codimension-three cases are studied in detail. The results presented reach the limits of what is currently known on the bifurcation theory of planar vector fields. While the treatment is geometric, special analytical tools using abelian integrals are needed, and are explicitly developed. The rescaling and normalization methods are improved for application here. The reader is assumed to be familiar with the elements of Bifurcation and Dynamical Systems Theory. The book is addressed to researchers and graduate students working in Ordinary Differential Equations and Dynamical Systems, as well as anyone modelling complex multiparametric phenomena.

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Table of contents :
Introduction….Pages 1-18
Definitions and notations….Pages 19-21
Transformation into normal form….Pages 22-27
Bifurcations of codimension 1 and 2….Pages 28-56
Elementary properties….Pages 57-84
The central rescaling….Pages 85-134
Conclusions and discussion of remaining problems….Pages 135-164
Abelian integrals in unfoldings of codimension 3 singular planar vector fields….Pages 165-224