Singularity clustering in the Duffing oscillator

J. D. Fournier, G. Levine, M. Tabor

Research output: Contribution to journalArticlepeer-review

48 Scopus citations


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.

Original languageEnglish (US)
Article number014
Pages (from-to)33-54
Number of pages22
JournalJournal of Physics A: General Physics
Issue number1
StatePublished - 1988
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • General Physics and Astronomy
  • Mathematical Physics


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