### Abstract

An asymptotic and numerical study is made of the singularity structure, in the complex t-plane, of the Duffing oscillator. The presence of logarithmic terms in the local psi-series expansion, of the form t^{4} ln t, leads to a multisheeted singularity structure of great complexity. This structure is built recursively from an elemental pattern which takes the form of four-armed 'stars' of singularities. This construction is deduced analytically from the properties of the mapping z=t^{4} ln t and is confirmed quite accurately numerically. A systematic resummation of the psi series, in terms of Lame functions, is developed. This series exhibits the same analytic structure at all orders and provides a 'semi-local' analytical representation of the solution which is apparently valid even in the chaotic regime.

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

Article number | 014 |

Pages (from-to) | 33-54 |

Number of pages | 22 |

Journal | Journal of Physics A: General Physics |

Volume | 21 |

Issue number | 1 |

DOIs | |

State | Published - 1988 |

Externally published | Yes |

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### ASJC Scopus subject areas

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

### Cite this

*Journal of Physics A: General Physics*,

*21*(1), 33-54. [014]. https://doi.org/10.1088/0305-4470/21/1/014

**Singularity clustering in the Duffing oscillator.** / Fournier, J. D.; Levine, G.; Tabor, Michael.

Research output: Contribution to journal › Article

*Journal of Physics A: General Physics*, vol. 21, no. 1, 014, pp. 33-54. https://doi.org/10.1088/0305-4470/21/1/014

}

TY - JOUR

T1 - Singularity clustering in the Duffing oscillator

AU - Fournier, J. D.

AU - Levine, G.

AU - Tabor, Michael

PY - 1988

Y1 - 1988

N2 - An asymptotic and numerical study is made of the singularity structure, in the complex t-plane, of the Duffing oscillator. The presence of logarithmic terms in the local psi-series expansion, of the form t4 ln t, leads to a multisheeted singularity structure of great complexity. This structure is built recursively from an elemental pattern which takes the form of four-armed 'stars' of singularities. This construction is deduced analytically from the properties of the mapping z=t4 ln t and is confirmed quite accurately numerically. A systematic resummation of the psi series, in terms of Lame functions, is developed. This series exhibits the same analytic structure at all orders and provides a 'semi-local' analytical representation of the solution which is apparently valid even in the chaotic regime.

AB - An asymptotic and numerical study is made of the singularity structure, in the complex t-plane, of the Duffing oscillator. The presence of logarithmic terms in the local psi-series expansion, of the form t4 ln t, leads to a multisheeted singularity structure of great complexity. This structure is built recursively from an elemental pattern which takes the form of four-armed 'stars' of singularities. This construction is deduced analytically from the properties of the mapping z=t4 ln t and is confirmed quite accurately numerically. A systematic resummation of the psi series, in terms of Lame functions, is developed. This series exhibits the same analytic structure at all orders and provides a 'semi-local' analytical representation of the solution which is apparently valid even in the chaotic regime.

UR - http://www.scopus.com/inward/record.url?scp=0002799998&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002799998&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/21/1/014

DO - 10.1088/0305-4470/21/1/014

M3 - Article

AN - SCOPUS:0002799998

VL - 21

SP - 33

EP - 54

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 1

M1 - 014

ER -