In 2008 November 23-28 the workshop of Classical Problems on Planar Polynomial Vector Fields
was held in the Banff International Research Station Canada. Called classical problems it was
concerned with the following:(1) Problems on integrability of planar polynomial vector fields.
(2) The problem of the center stated by Poincaré for real polynomial differential systems
which asks us to recognize when a planar vector field defined by polynomials of degree at most
n possesses a singularity which is a center.(3) Global geometry of specific classes of planar
polynomial vector fields.(4) Hilbert s 16th problem.These problems had been posed more than 110
years ago. Therefore they are called classical problems in the studies of the theory of
dynamical systems. The qualitative theory and stability theory of differential equations
created by Poincaré and Lyapunov at the end of the 19th century had major developments as two
branches of the theory of dynamical systems during the 20th century. As a part of the basic
theory of nonlinear science it is one of the very active areas in the new millennium. This
book presents in an elementary way the recent significant developments in the qualitative
theory of planar dynamical systems. The subjects are covered as follows: the studies of center
and isochronous center problems multiple Hopf bifurcations and local and global bifurcations
of the equivariant planar vector fields which concern with Hilbert s 16th problem. The book is
intended for graduate students post-doctors and researchers in dynamical systems. For all
engineers who are interested in the theory of dynamical systems it is also a reasonable
reference. It requires a minimum background of a one-year course on nonlinear differential
equations.
