Integrating the nonintegrable: Analytic structure of the Lorenz system revisited

G. Levine, M. Tabor

Research output: Contribution to journalArticlepeer-review

46 Scopus citations


A study of the complex time analytic structure of the Lorenz system in nonintegrable parameter regimes reveals the special sets of parameter values for which one (time-dependent) integral of motion exists. Furthermore, the analysis yields the exact form of the part of the integral with the highest homogeneous weight and a method to construct the rest of the integral. Recursive clustering of singularities in the chaotic regimes of the system is observed in computer studies and explained by a simple analytic argument. The analytic techniques used in these studies, a systematic resummation of a logarithmic psi-series, appears to be quite general and can provide explicit representations of a solution-even in the chaotic regimes-in the neighborhood of a given movable singularity. Furthermore, we suggest that this technique provides a type of renormalization program to study a wide class of nonintegrable systems.

Original languageEnglish (US)
Pages (from-to)189-210
Number of pages22
JournalPhysica D: Nonlinear Phenomena
Issue number1-3
StatePublished - 1988
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


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