# Validated computations for connecting orbits in polynomial vector fields

## TL;DR

This paper introduces a computer-assisted method to rigorously prove the existence of connecting orbits in polynomial vector fields, combining Taylor approximations, Chebyshev series, and fixed-point analysis.

## Contribution

It develops a novel, rigorous computational framework for establishing the existence and transversality of heteroclinic orbits in polynomial vector fields.

## Key findings

- Successfully proves existence of heteroclinic orbits
- Provides explicit bounds on approximation errors
- Establishes transversality of stable and unstable manifolds

## Abstract

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.