Global Aspects of Homoclinic Bifurcations of Vector Fields

Know about the book -

Global Aspects of Homoclinic Bifurcations of Vector Fields is written by Ale Jan Homburg and published by American Mathematical Soc.. It's available with International Standard Book Number or ISBN identification 0821804413 (ISBN 10) and 9780821804414 (ISBN 13).

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.