The transverse homoclinic dynamics and their bifurcations at nonhyperbolic fixed points

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

The complete description of the dynamics of diffeomorphisms in a neighborhood of a transverse homoclinic orbit to a hyperbolic fixed point is obtained. It is topologically conjugate to a non-Bernoulli shift called (Σ,σ). We also obtain a more or less complete picture, referred to as the net weaving bifurcation, when the fixed point of such a system is undergoing the generic saddle-node bifurcation. The idea of homotopy conjugacy is naturally introduced to show that systems whose fixed points undergo the pitchfork, transcritical, periodic doubling, and Hopf bifurcations are all homotopically conjugate to our shift dynamics (Σ,σ) in a neighborhood of a transverse homoclinic orbit. These bifurcations are also examined in the context of the spectral decomposition with respect to the maximal indecomposable nonwandering sets.

Original languageEnglish (US)
Pages (from-to)15-53
Number of pages39
JournalTransactions of the American Mathematical Society
Volume331
Issue number1
DOIs
StatePublished - May 1992

Fingerprint

Homoclinic
Orbits
Transverse
Bifurcation
Fixed point
Homoclinic Orbit
Hopf bifurcation
Saddle-node Bifurcation
Spectral Decomposition
Conjugacy
Doubling
Diffeomorphisms
Decomposition
Hopf Bifurcation
Homotopy

Keywords

  • Saddle-node bifurcation
  • Symbolic system
  • Topological conjugacy
  • Transverse homoclinic point

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

The transverse homoclinic dynamics and their bifurcations at nonhyperbolic fixed points. / Deng, Bo.

In: Transactions of the American Mathematical Society, Vol. 331, No. 1, 05.1992, p. 15-53.

Research output: Contribution to journalArticle

@article{c43117a816a940b78436d6bc79fa91d4,
title = "The transverse homoclinic dynamics and their bifurcations at nonhyperbolic fixed points",
abstract = "The complete description of the dynamics of diffeomorphisms in a neighborhood of a transverse homoclinic orbit to a hyperbolic fixed point is obtained. It is topologically conjugate to a non-Bernoulli shift called (Σ,σ). We also obtain a more or less complete picture, referred to as the net weaving bifurcation, when the fixed point of such a system is undergoing the generic saddle-node bifurcation. The idea of homotopy conjugacy is naturally introduced to show that systems whose fixed points undergo the pitchfork, transcritical, periodic doubling, and Hopf bifurcations are all homotopically conjugate to our shift dynamics (Σ,σ) in a neighborhood of a transverse homoclinic orbit. These bifurcations are also examined in the context of the spectral decomposition with respect to the maximal indecomposable nonwandering sets.",
keywords = "Saddle-node bifurcation, Symbolic system, Topological conjugacy, Transverse homoclinic point",
author = "Bo Deng",
year = "1992",
month = "5",
doi = "10.1090/S0002-9947-1992-1024768-9",
language = "English (US)",
volume = "331",
pages = "15--53",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "1",

}

TY - JOUR

T1 - The transverse homoclinic dynamics and their bifurcations at nonhyperbolic fixed points

AU - Deng, Bo

PY - 1992/5

Y1 - 1992/5

N2 - The complete description of the dynamics of diffeomorphisms in a neighborhood of a transverse homoclinic orbit to a hyperbolic fixed point is obtained. It is topologically conjugate to a non-Bernoulli shift called (Σ,σ). We also obtain a more or less complete picture, referred to as the net weaving bifurcation, when the fixed point of such a system is undergoing the generic saddle-node bifurcation. The idea of homotopy conjugacy is naturally introduced to show that systems whose fixed points undergo the pitchfork, transcritical, periodic doubling, and Hopf bifurcations are all homotopically conjugate to our shift dynamics (Σ,σ) in a neighborhood of a transverse homoclinic orbit. These bifurcations are also examined in the context of the spectral decomposition with respect to the maximal indecomposable nonwandering sets.

AB - The complete description of the dynamics of diffeomorphisms in a neighborhood of a transverse homoclinic orbit to a hyperbolic fixed point is obtained. It is topologically conjugate to a non-Bernoulli shift called (Σ,σ). We also obtain a more or less complete picture, referred to as the net weaving bifurcation, when the fixed point of such a system is undergoing the generic saddle-node bifurcation. The idea of homotopy conjugacy is naturally introduced to show that systems whose fixed points undergo the pitchfork, transcritical, periodic doubling, and Hopf bifurcations are all homotopically conjugate to our shift dynamics (Σ,σ) in a neighborhood of a transverse homoclinic orbit. These bifurcations are also examined in the context of the spectral decomposition with respect to the maximal indecomposable nonwandering sets.

KW - Saddle-node bifurcation

KW - Symbolic system

KW - Topological conjugacy

KW - Transverse homoclinic point

UR - http://www.scopus.com/inward/record.url?scp=84966233578&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966233578&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1992-1024768-9

DO - 10.1090/S0002-9947-1992-1024768-9

M3 - Article

AN - SCOPUS:84966233578

VL - 331

SP - 15

EP - 53

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -