This is the first book to systematically state the fundamental theory of
integrability and its development of ordinary differential equations
with emphasis on the Darboux theory of integrability and local
integrability together with their applications. It summarizes the
classical results of Darboux integrability and its modern development
together with their related Darboux polynomials and their applications
in the reduction of Liouville and elementary integrabilty and in the
center--focus problem, the weakened Hilbert 16th problem on algebraic
limit cycles and the global dynamical analysis of some realistic models
in fields such as physics, mechanics and biology.
Although it can be used as a textbook for graduate students in dynamical
systems, it is intended as supplementary reading for graduate students
from mathematics, physics, mechanics and engineering in courses related
to the qualitative theory, bifurcation theory and the theory of
integrability of dynamical systems.