TY - JOUR

T1 - Validated computations for connecting orbits in polynomial vector fields

AU - van den Berg, Jan Bouwe

AU - Sheombarsing, Ray

PY - 2020/3

Y1 - 2020/3

N2 - 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.

AB - 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.

KW - Computer-assisted proof

KW - Heteroclinic orbits

KW - Validated computations

UR - http://www.scopus.com/inward/record.url?scp=85079556239&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85079556239&partnerID=8YFLogxK

U2 - 10.1016/j.indag.2020.01.007

DO - 10.1016/j.indag.2020.01.007

M3 - Article

AN - SCOPUS:85079556239

VL - 31

SP - 310

EP - 373

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

SN - 0019-3577

IS - 2

ER -