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首页> 外文期刊>Journal of dynamics and differential equations >Computer Assisted Proof of Transverse Saddle-to-Saddle Connecting Orbits for First Order Vector Fields
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Computer Assisted Proof of Transverse Saddle-to-Saddle Connecting Orbits for First Order Vector Fields

机译：一阶矢量场的横向鞍到鞍连接轨道的计算机辅助证明

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摘要

In this paper we introduce a computational method for proving the existence of generic saddle-to-saddle connections between equilibria of first order vector fields. The first step consists of rigorously computing high order parametrizations of the local stable and unstable manifolds. If the local manifolds intersect, the Newton-Kantorovich theorem is applied to validate the existence of a so-called short connecting orbit. If the local manifolds do not intersect, a boundary value problem with boundary values in the local manifolds is rigorously solved by a contraction mapping argument on a ball centered at the numerical solution, yielding the existence of a so-called long connecting orbit. In both cases our argument yields transversality of the corresponding intersection of the manifolds. The method is applied to the Lorenz equations, where a study of a pitchfork bifurcation with saddle-to-saddle stability is done and where several proofs of existence of short and long connections are obtained.
机译：在本文中，我们介绍了一种计算方法，用于证明一阶向量场的平衡之间存在一般的鞍对鞍连接。第一步包括严格计算局部稳定和不稳定歧管的高阶参数。如果局部流形相交，则应用牛顿-坎托罗维奇定理来验证所谓短连接轨道的存在。如果局部流形不相交，则通过以数值解为中心的球上的收缩映射参数来严格解决具有局部流形中的边界值的边值问题，从而产生所谓的长连接轨道。在这两种情况下，我们的论点都产生了歧管相应交点的横向性。将该方法应用于Lorenz方程，在此研究了具有鞍到鞍稳定性的干草叉分叉，并获得了存在短连接和长连接的几种证明。

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