Barreira Luis, Valls Claudia. Ordinary Differential Equations. Qualitative Theory

American Mathematical Society, 2012. — 265 p.The main objective of this book is to give a comprehensive introduction to the qualitative theory of ordinary differential equations. In particular,
among other topics, we study the existence and uniqueness of solutions, phase portraits, linear equations and their perturbations, stability and Lya­punov functions, hyperbolicity, and equations in the plane.The book is also intended to serve as a bridge to important topics that are often left out o f a second course of ordinary differential equations. Examples include the smooth dependence o f solutions on the initial conditions, the existence of topological and differentiable conjugacies between linear systems, and the Holder continuity of the conjugacies in the Grobman-Hartman theorem. We also give a brief introduction to bifurcation theory,
center manifolds, normal forms, and Hamiltonian systems.We describe mainly notions, results and methods that allow one to dis­cuss the qualitative properties o f the solutions of an equation without solving
it explicitly. This can be considered the main aim of the qualitative theory of ordinary differential equations.The book can be used as a basis for a second course of ordinary differen­tial equations. Nevertheless, it has more material than the standard courses,
and so, in fact, it can be used in several different ways and at various levels.