Validated computations for connecting orbits in polynomial vector fields

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 Parameteri...

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Bibliographic Details
Published in:Indagationes mathematicae 2020-03, Vol.31 (2), p.310-373
Main Authors: van den Berg, Jan Bouwe, Sheombarsing, Ray
Format: Article
Language:English
Subjects:
Boundary value problems
Chebyshev approximation
Computer-assisted proof
Fields (mathematics)
Heteroclinic orbits
Manifolds
Mathematical analysis
Orbital stability
Orbits
Parameterization
Polynomials
Validated computations
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Summary: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.
ISSN:0019-3577
1872-6100
DOI:10.1016/j.indag.2020.01.007