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.

Readership

Researchers and graduate students working in the qualitative theory of ordinary differential equations.

Table of Contents

• Yu. Ilyashenko and S. Yakovenko -- Concerning the Hilbert sixteenth problem
• Yu. Ilyashenko and S. Yakovenko -- Finite cyclicity of elementary polycycles in generic families
• S. Trifonov -- Desingularization in families of analytic differential equations
• O. Kleban -- Order of the topologically sufficient jet of a smooth vector field on the real plane at a singular point of finite multiplicity
• A. Kotova and V. Stanzo -- On few-parameter generic families of vector fields on the two-dimensional sphere
• S. Yakovenko -- A geometric proof of the Bautin theorem