Validated computations for connecting orbits in polynomial vector fields

Jan Bouwe van den Berg*, Ray Sheombarsing

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review


In this paper we present a computer-assisted procedure for proving the existence of transverse heteroclinic orbits connecting hyperbolic equilibria of polynomial vector fields. The idea is to compute high-order Taylor approximations of local charts on the (un)stable manifolds by using the Parameterization Method and to use Chebyshev series to parameterize the orbit in between, which solves a boundary value problem. The existence of a heteroclinic orbit can then be established by setting up an appropriate fixed-point problem amenable to computer-assisted analysis. The fixed point problem simultaneously solves for the local (un)stable manifolds and the orbit which connects these. We obtain explicit rigorous control on the distance between the numerical approximation and the heteroclinic orbit. Transversality of the stable and unstable manifolds is also proven.

Original languageEnglish
Pages (from-to)310-373
Number of pages64
JournalIndagationes Mathematicae
Issue number2
Early online date31 Jan 2020
Publication statusPublished - Mar 2020


  • Computer-assisted proof
  • Heteroclinic orbits
  • Validated computations


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