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Global Aspects of Homoclinic Bifurcations of Vector Fields

Ale Jan Homburg University of Groningen, Netherlands
Available Formats:
Electronic ISBN: 978-1-4704-0163-4
Product Code: MEMO/121/578.E
List Price: $46.00 MAA Member Price:$41.40
AMS Member Price: $27.60 Click above image for expanded view Global Aspects of Homoclinic Bifurcations of Vector Fields Ale Jan Homburg University of Groningen, Netherlands Available Formats:  Electronic ISBN: 978-1-4704-0163-4 Product Code: MEMO/121/578.E  List Price:$46.00 MAA Member Price: $41.40 AMS Member Price:$27.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1211996; 128 pp
MSC: Primary 34; Secondary 58;

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.

• Chapters
• 1. Introduction
• 2. Invariant manifolds and foliations
• 3. Homoclinic intermittency
• 4. Suspended basic sets
• Appendix A. Invariant foliations
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Volume: 1211996; 128 pp
MSC: Primary 34; Secondary 58;

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.