Bifurcations of Planar Vector Fields: Nilpotent Singularities and Abelian Integrals

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This monograph 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 cases are studied in detail. The results presented intend to 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 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.