Global aspects of homoclinic bifurcations of vector fields
About this Title
Ale Jan Homburg
Publication: Memoirs of the American Mathematical Society
Publication Year 1996: Volume 121, Number 578
ISBNs: 978-0-8218-0441-4 (print); 978-1-4704-0163-4 (online)
MathSciNet review: 1327210
MSC: Primary 58F14; Secondary 34C23, 34C37, 58F15
In this book, the author investigates a class of smooth one parameter families of vector fields on some $n$-dimensional manifold, exhibiting a homoclinic bifurcation. That is, he considers generic families $x_\mu$, where $x_0$ has a distinguished hyperbolic singularity $p$ and a homoclinic orbit; an orbit converging to $p$ both for positive and negative time. It is assumed that this homoclinic orbit is of saddle-saddle type, characterized by the existence of well-defined directions along which it converges to the singularity $p$.
The study is not confined to a small neighborhood of the homoclinic orbit. Instead, the position of the stable and unstable set of the homoclinic orbit is incorporated and it is shown that homoclinic bifurcations can lead to complicated bifurcations and dynamics, including phenomena like intermittency and annihilation of suspended horseshoes.
Graduate students and research mathematicians interested in differential equations.
Table of Contents
- 1. Introduction
- 2. Invariant manifolds and foliations
- 3. Homoclinic intermittency
- 4. Suspended basic sets
- Appendix A. Invariant foliations