Nonsymmetric Lorenz attractors from a homoclinic bifurcation

Clark Robinson*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

21 Scopus citations

Abstract

We consider a bifurcation of a flow in three dimensions from a double homoclinic connection to a fixed point satisfying a resonance condition between the eigenvalues. For correctly chosen parameters in the unfolding, we prove that there is a transitive attractor of Lorenz type. In particular we show the existence of a bifurcation to an attractor of Lorenz type which is semiorientable, i.e., orientable on one half and nonorientable on the other half. We do not assume any symmetry condition, so we need to discuss nonsymmetric one-dimensional Poincaré maps with one discontinuity and absolute value of the derivative always greater than one. We also apply these results to a specific set of degree four polynomial differential equations. The results do not apply to the actual Lorenz equations because they do not have enough parameters to adjust to make them satisfy the hypothesis.

Original languageEnglish (US)
Pages (from-to)119-141
Number of pages23
JournalSIAM Journal on Mathematical Analysis
Volume32
Issue number1
DOIs
StatePublished - 2000

Keywords

  • Attractors
  • Homoclinic bifurcation
  • Lorenz

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

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