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.
