Ordinary Differential Equations and Dynamical Systems
About this Title
Publication: Graduate Studies in Mathematics
Publication Year: 2012; Volume 140
ISBNs: 978-0-8218-8328-0 (print); 978-0-8218-9104-9 (online)
MathSciNet review: MR2961944
MSC: Primary 34-01; Secondary 37-01, 39-01
This book provides a self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate students.
The first part begins with some simple examples of explicitly solvable equations and a first glance at qualitative methods. Then the fundamental results concerning the initial value problem are proved: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore, linear equations are considered, including the Floquet theorem, and some perturbation results. As somewhat independent topics, the Frobenius method for linear equations in the complex domain is established and Sturm–Liouville boundary value problems, including oscillation theory, are investigated.
The second part introduces the concept of a dynamical system. The Poincaré–Bendixson theorem is proved, and several examples of planar systems from classical mechanics, ecology, and electrical engineering are investigated. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed. Finally, stability is studied, including the stable manifold and the Hartman–Grobman theorem for both continuous and discrete systems.
The third part introduces chaos, beginning with the basics for iterated interval maps and ending with the Smale–Birkhoff theorem and the Melnikov method for homoclinic orbits.
The text contains almost three hundred exercises. Additionally, the use of mathematical software systems is incorporated throughout, showing how they can help in the study of differential equations.
Graduate students interested in ordinary differential equations and dynamical systems.
Table of Contents
Part 1. Classical theory
- Chapter 1. Introduction
- Chapter 2. Initial value problems
- Chapter 3. Linear equations
- Chapter 4. Differential equations in the complex domain
- Chapter 5. Boundary value problems
Part 2. Dynamical systems
- Chapter 6. Dynamical systems
- Chapter 7. Planar dynamical systems
- Chapter 8. Higher dimensional dynamical systems
- Chapter 9. Local behavior near fixed points
Part 3. Chaos
- Chapter 10. Discrete dynamical systems
- Chapter 11. Discrete dynamical systems in one dimension
- Chapter 12. Periodic solutions
- Chapter 13. Chaos in higher dimensional systems