### Abstract

The authors consider stationary bifurcations with Z_{4}.Z _{2}^{4} symmetry. For an open set of cubic coefficients in the normal form, we prove the existence of a limit cycle with frequency approximately mod lambda mod ^{-1} as lambda to 0. It is shown that there are two types of structurally stable heteroclinic cycles in this example, one of which is of a new type. They find the stability of all of the zeros and heteroclinic cycles which branch from the origin at the bifurcation. Novel techniques are needed for the calculation of stability for the new type of heteroclinic cycle, and the proof of existence of the limit cycles.

Original language | English (US) |
---|---|

Article number | 001 |

Pages (from-to) | 1001-1043 |

Number of pages | 43 |

Journal | Nonlinearity |

Volume | 4 |

Issue number | 4 |

DOIs | |

State | Published - 1991 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematical Physics
- Statistical and Nonlinear Physics
- Applied Mathematics
- Mathematics(all)

### Cite this

*Nonlinearity*,

*4*(4), 1001-1043. [001]. https://doi.org/10.1088/0951-7715/4/4/001

**Stationary bifurcation to limit cycles and heteroclinic cycles.** / Field, M.; Swift, James W.

Research output: Contribution to journal › Article

*Nonlinearity*, vol. 4, no. 4, 001, pp. 1001-1043. https://doi.org/10.1088/0951-7715/4/4/001

}

TY - JOUR

T1 - Stationary bifurcation to limit cycles and heteroclinic cycles

AU - Field, M.

AU - Swift, James W

PY - 1991

Y1 - 1991

N2 - The authors consider stationary bifurcations with Z4.Z 24 symmetry. For an open set of cubic coefficients in the normal form, we prove the existence of a limit cycle with frequency approximately mod lambda mod -1 as lambda to 0. It is shown that there are two types of structurally stable heteroclinic cycles in this example, one of which is of a new type. They find the stability of all of the zeros and heteroclinic cycles which branch from the origin at the bifurcation. Novel techniques are needed for the calculation of stability for the new type of heteroclinic cycle, and the proof of existence of the limit cycles.

AB - The authors consider stationary bifurcations with Z4.Z 24 symmetry. For an open set of cubic coefficients in the normal form, we prove the existence of a limit cycle with frequency approximately mod lambda mod -1 as lambda to 0. It is shown that there are two types of structurally stable heteroclinic cycles in this example, one of which is of a new type. They find the stability of all of the zeros and heteroclinic cycles which branch from the origin at the bifurcation. Novel techniques are needed for the calculation of stability for the new type of heteroclinic cycle, and the proof of existence of the limit cycles.

UR - http://www.scopus.com/inward/record.url?scp=0039898994&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0039898994&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/4/4/001

DO - 10.1088/0951-7715/4/4/001

M3 - Article

VL - 4

SP - 1001

EP - 1043

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 4

M1 - 001

ER -