### 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) |
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Article number | 001 |

Pages (from-to) | 1001-1043 |

Number of pages | 43 |

Journal | Nonlinearity |

Volume | 4 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 1991 |

Externally published | Yes |

### ASJC Scopus subject areas

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

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## Cite this

Field, M., & Swift, J. W. (1991). Stationary bifurcation to limit cycles and heteroclinic cycles.

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