Overview

This book is based on a year long course taught by the author to graduate students at the University of Missouri over several years. The main objective is to provide, in the first semester, a student friendly account of some of the standard topics encountered in an introductory course of ordinary differential equations, and then, in the second semester, to expand on these ideas by introducing more advanced concepts and applications. A central theme in the book is the use of Implicit Function Theorem. A proof of the existence and uniqueness results for solutions of differential equations based on the Implicit Function Theorem in Banach spaces is presented. Then in the latter sections of the book, the basic ideas of perturbation theory are introduced as applications of the Implicit Function Theorem (with the Lyapunov-Schmidt reduction technique): continuation of subharmonics and the existence of periodic solutions via the averaging method. Finally, local bifurcations-saddle node and Hopf-are studied as applications of the Lyapunov-Schmidt reduction. The book also contains material that is different from standard treatments. For example, the Fiber Contraction Principle is introduced as a useful tool for proving the smoothness of functions that are obtained as fixed points of contractions. This idea is used to give an alternate proof of the smoothness of the flow of a differential equation. Later, the Fiber Contraction Principle appears in the nontrivial proof of the smoothness of invariant manifolds at a rest point. The proof for this existence and smoothness of the stable center manifolds is obtained as a corollary of a more general existence theorem for invariant manifolds at a rest point in the presence of a ""spectral gap"". The ideas introduced in this section can be extended to infinite dimensions.

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