This book provides an introduction to and a comprehensive study of the qualitative theory of ordinary differential equations. It begins with fundamental theorems on existence, uniqueness, and initial conditions, and discusses basic principles in dynamical systems and PoincaréBendixson theory. The authors present a careful analysis of solutions near critical points of linear and nonlinear planar systems and discuss indices of planar critical points. A very thorough study of limit cycles is given, including many results on quadratic systems and recent developments in China. Other topics included are: the critical point at infinity, harmonic solutions for periodic differential equations, systems of ordinary differential equations on the torus, and structural stability for systems on twodimensional manifolds. This books is accessible to graduate students and advanced undergraduates and is also of interest to researchers in this area. Exercises are included at the end of each chapter. Readership Graduate students and research mathematicians interested in differential equations. Reviews "This book attests to the depth, the breadth and the sweeping success of the new methods introduced by the Chinese in the study of nonlinear systems of differential equations. One leafs through the pages of this book with a feeling of awe that one seldom gets from a math book, and one salutes the bravery, the grit and the genius that have produced so much mathematics of the very first rank."  The Bulletin of Mathematics Books and Computer Software "The book could serve both as an excellent introduction to the field for the beginner, and as a valuable reference for the specialist, especially in presenting work of Chinese mathematicians."  Mathematical Reviews "An excellent blend of abstract theorems and valuable concrete examples ... written interestingly ... useful for students, teachers and researchers."  Zentralblatt MATH Table of Contents  Fundamental theorems
 Critical points on the plane
 Indices of planar critical points
 Limit cycles
 Critical points at infinity
 Harmonic solutions for twodimensional periodic systems
 Systems of ordinary differential equations on the torus
 Structural stability
 Exercises
 References
