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Global Aspects of Homoclinic Bifurcations of Vector Fields
Ale Jan Homburg, University of Groningen, Netherlands
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Memoirs of the American Mathematical Society
1996; 128 pp; softcover
Volume: 121
ISBN-10: 0-8218-0441-3
ISBN-13: 978-0-8218-0441-4
List Price: US$43 Individual Members: US$25.80
Institutional Members: US\$34.40
Order Code: MEMO/121/578

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.