Electronic ISBN:  9781470401634 
Product Code:  MEMO/121/578.E 
List Price:  $46.00 
MAA Member Price:  $41.40 
AMS Member Price:  $27.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 121; 1996; 128 ppMSC: 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 saddlesaddle type, characterized by the existence of welldefined 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.ReadershipGraduate students and research mathematicians interested in differential equations.

Table of Contents

Chapters

1. Introduction

2. Invariant manifolds and foliations

3. Homoclinic intermittency

4. Suspended basic sets

Appendix A. Invariant foliations


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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 saddlesaddle type, characterized by the existence of welldefined 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