Ordinary Differential Equations and Dynamical Systems

The proofs presented in this volume include complete mathematical details as well as extensive use of the implicit function theorem as a unifying approach to perturbative analysis. A wealth of end-of-chapter exercises enables its use as a textbook.

This book is a mathematically rigorous introduction to the beautiful subject of ordinary differential equations for beginning graduate or advanced undergraduate students. Students should have a solid background in analysis and linear algebra. The presentation emphasizes commonly used techniques without necessarily striving for completeness or for the treatment of a large number of topics. The first half of the book is devoted to the development of the basic theory: linear systems, existence and uniqueness of solutions to the initial value problem, flows, stability, and smooth dependence of solutions upon initial conditions and parameters. Much of this theory also serves as the paradigm for evolutionary partial differential equations. The second half of the book is devoted to geometric theory: topological conjugacy, invariant manifolds, existence and stability of periodic solutions, bifurcations, normal forms, and the existence of transverse homoclinic points and their link to chaotic dynamics. A common thread throughout the second part is the use of the implicit function theorem in Banach space. Chapter 5, devoted to this topic, the serves as the bridge between the two halves of the book.

Proofs are presented with complete mathematical details Extensive use of the implicit function theorem as a unifying approach to perturbative analysis Exercises are included at the end of each chapter permit use as a textbook

Inhalt
Introduction.- Linear Systems.- Existence Theory.- Nonautomous Linear Systems.- Results from Functional Analysis.- Dependence on Initial Conditions and Parameters.- Linearization and Invariant Manifolds.- Periodic Solutions.- Center Manifolds and Bifurcation Theory.- The Birkhoff Smale Homoclinic Theorem.- Appendix: Results from Real Analysis.