Ordinary Differential Equations and Dynamical Systems

✍ Scribed by Thomas C. Sideris (auth.)

• Publisher: Atlantis Press
• Year: 2013
• Tongue: English
• Leaves: 230
• Series: Atlantis Studies in Differential Equations 2
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xi
Introduction....Pages 1-3
Linear Systems....Pages 5-19
Existence Theory....Pages 21-51
Nonautonomous Linear Systems....Pages 53-72
Results from Functional Analysis....Pages 73-87
Dependence on Initial Conditions and Parameters....Pages 89-94
Linearization and Invariant Manifolds....Pages 95-118
Periodic Solutions....Pages 119-154
Center Manifolds and Bifurcation Theory....Pages 155-198
The Birkhoff Smale Homoclinic Theorem....Pages 199-217
Back Matter....Pages 219-225

✦ Subjects

Ordinary Differential Equations