Transversal connecting orbits from shadowing

Brian A. Coomes, Hüseyin Koçak, Kenneth J. Palmer

Research output: Contribution to journalArticle

11 Scopus citations

Abstract

A rigorous numerical method for establishing the existence of a transversal connecting orbit from one hyperbolic periodic orbit to another of a differential equation in ℝ is presented. As the first component of this method, a general shadowing theorem that guarantees the existence of such a connecting orbit near a suitable pseudo connection orbit given the invertibility of a certain operator is proved. The second component consists of a refinement procedure for numerically computing a pseudo connecting orbit between two pseudo periodic orbits with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component consists of a numerical procedure to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse. Using this method, existence of chaos is demonstrated on examples with transversal homoclinic orbits, and with cycles of transversal heteroclinic orbits.

Original languageEnglish (US)
Pages (from-to)427-469
Number of pages43
JournalNumerische Mathematik
Volume106
Issue number3
DOIs
StatePublished - May 1 2007

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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