Ordinary Differential Equations with Applications by Carmen Chicone (Springer Texts in Applied Mathematics, vol. 34, second edition; Springer, 2006; ISBN-13: 978-0387-30769-5). We expect to cover much of the material in Chapters 1-3 of the book
Prerequisites: An undergraduate course on ordinary differential equations, linear algebra, advanced calculus and some basic results from analysis, e.g., the definition of a Banach space. An attempt will be made to keep the course self-contained. Thus, while we shall use the implicit function theorem in Banach spaces, a complete proof will be given.
Description: This will be an introduction to the theory of ordinary differential equations. We will discuss existence and uniqueness theorems for the initial value problem. linearization and linear theory, stability (Lyapunov functions), omega limit sets, Poincare-Bendixson theory and invariant manifolds. If time permits, we may also discuss the Hopf bifurcation theorem and some topics from the theory of order-preserving dynamical systems.