This book examines qualitative properties of vector fields in the plane, in the spirit of Hilbert's Sixteenth Problem. Two principal topics explored are bifurcations of limit cycles of planar vector fields and desingularization of singular points for individual vector fields and for analytic families of such fields. In addition to presenting important new developments in this area, this book contains an introductory paper which outlines the general context and describes connections between the papers in the volume. The book will appeal to researchers and graduate students working in the qualitative theory of ordinary differential equations and dynamical systems.
Concerning the Hilbert sixteenth problem by Yu. Ilyashenko and S. Yakovenko Finite cyclicity of elementary polycycles in generic families by Yu. Ilyashenko and S. Yakovenko Desingularization in families of analytic differential equations by S. Trifonov Order of the topologically sufficient jet of a smooth vector field on the real plane at a singular point of finite multiplicity by O. Kleban On few-parameter generic families of vector fields on the two-dimensional sphere by A. Kotova and V. Stanzo A geometric proof of the Bautin theorem by S. Yakovenko.