Planar Dynamical Systems - Selected Classical Problems

(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."