### 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 language | English (US) |
---|---|

Pages (from-to) | 15-53 |

Number of pages | 39 |

Journal | Transactions of the American Mathematical Society |

Volume | 331 |

Issue number | 1 |

DOIs | |

State | Published - May 1992 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

}

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 -