Transversal connecting orbits from shadowing

Brian A Coomes, Huseyin Kocak, Kenneth J. Palmer

Research output: Contribution to journalArticle

7 Citations (Scopus)

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 2007

Fingerprint

Connecting Orbits
Shadowing
Orbits
Invertibility
Periodic Orbits
Heteroclinic Orbit
Homoclinic Orbit
Numerical Procedure
Operator
Theorem
Chaos
Refinement
Orbit
Numerical Methods
Differential equation
Verify
Upper bound
Norm
Cycle
Computing

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics
  • Computational Mathematics

Cite this

Transversal connecting orbits from shadowing. / Coomes, Brian A; Kocak, Huseyin; Palmer, Kenneth J.

In: Numerische Mathematik, Vol. 106, No. 3, 05.2007, p. 427-469.

Research output: Contribution to journalArticle

Coomes, Brian A ; Kocak, Huseyin ; Palmer, Kenneth J. / Transversal connecting orbits from shadowing. In: Numerische Mathematik. 2007 ; Vol. 106, No. 3. pp. 427-469.
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