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.

中文翻译：

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多项式向量场中连接轨道的验证计算

在本文中，我们提出了一种计算机辅助程序，用于证明连接多项式向量场的双曲平衡的横向异斜轨道的存在。这个想法是通过使用参数化方法计算（不稳定）流形上局部图的高阶泰勒近似，并使用切比雪夫级数来参数化之间的轨道，从而解决边界值问题。然后可以通过设置适合计算机辅助分析的适当不动点问题来确定异宿轨道的存在。不动点问题同时解决了局部（不稳定）流形和连接这些流形的轨道。我们对数值近似和异宿轨道之间的距离进行了明确的严格控制。