Global aspects of homoclinic bifurcations of vector fields / Ale Jan Homburg.

Format:

Book

Author/Creator:

Homburg, Ale Jan, 1966- author.

Series:

Memoirs of the American Mathematical Society ; Volume 121, Number 578.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 121, Number 578

Language:

English

Subjects (All):

Bifurcation theory.

Vector fields.

Differentiable dynamical systems.

Physical Description:

1 online resource (143 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 1996.

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""1 Introduction""; ""2 Invariant manifolds and foliations""; ""2.1 Construction of foliations and invariant manifolds""; ""2.2 Saddle-saddle type homoclinic bifurcation""; ""2.3 Generalized homoclinic orbits""; ""3 Homoclinic intermittency""; ""3.1 Properties of the Poincare return map""; ""3.2 An orientable centre manifold""; ""3.3 A nonorientable centre manifold""; ""4 Suspended basic sets""; ""4.1 Properties of the Poincare return map""; ""4.2 The orientable suspended horseshoe""; ""4.3 The nonorientable suspended horseshoe""; ""4.4 Several generalized homoclinic orbits""

""Appendix A: Invariant foliations""""A.1 Linearisation on a centre manifold""; ""A.2 Smoothness of foliations""; ""Bibliography""

Notes:

"May 1996, volume 121, number 578 (second of 4 numbers)."

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-0163-0