This work covers the fundamentals of the theory of ordinary differential equations (ODEs). It includes an extensive discussion of the integration of differential inequalities, on which this theory relies heavily. In addition, the text illustrates techniques involving simple topological arguments, fixed point theorems and basic facts of functional analysis. The theory of ODEs is presented in a general way within this text, enabling it to act as a useful reference. This edition covers invariant manifolds, perturbations and dichotomies, making the text relevent to current studies of geometrical theory of differential equations and dynamical systems. In particular, it includes the proof of the Hartman-Grobman theorem on the equivalence of a nonlinear to a linear flow in the neighbourhood of a hyperbolic stationary point, as well as theorems on smooth equivalences, the smoothness of invariant methods and the reduction of problems on ODEs to those on "maps" (Poincare). "Ordinary Differential Equations" is based on the author's lecture notes from courses on ODEs taught to advanced undergraduate and graduate students in mathematics, physics and engineering. The book includes a selection of exercises varying in difficulty from routine examples to more challenging problems. Readers should have knowledge of matrix theory and the ability to deal with functions of real variables.
Philip Hartman is Professor Emeritus of The Johns Hopkins University, where he taught in the Department of Mathematics from 1946 to 1980. He was the recipient of a Guggenheim Fellowship in 1950-51. During his active career, Professor Hartman has been a visiting professor or fellow at UCLA, NYU, Warwick University (England), and University of Pisa (Italy), and he has also served on the editorial boards for the American Journal of Mathematics and Nonlinear Analysis: Theory, Methods and Applications.
Foreword to the Classics Edition; Preface to the First Edition; Preface to the Second Edition; Errata; I: Preliminaries; II: Existence; III: Differential In qualities and Uniqueness; IV: Linear Differential Equations; V: Dependence on Initial Conditions and Parameters; VI: Total and Partial Differential Equations; VII: The Poincare-Bendixson Theory; VIII: Plane Stationary Points; IX: Invariant Manifolds and Linearizations; X: Perturbed Linear Systems; XI: Linear Second Order Equations; XII: Use of Implicity Function and Fixed Point Theorems; XIII: Dichotomies for Solutions of Linear Equations; XIV: Miscellany on Monotomy; Hints for Exercises; References; Index.
