We study bi-Lyapunov stable homoclinic classes for a C¹ generic flow on a closed Riemannian manifold and prove that such a homoclinic class contains no singularity. This enables a parallel study of bi-Lyapunov stable dynamics for flows and for diffeomorphisms. For example, we can then show that a bi-Lyapunov stable homoclinic class for a C¹ generic flow is hyperbolic if and only if all periodic orbits in the class have the same stable index.

中文翻译：

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C1 泛型流的双李雅普诺夫稳定同宿类

我们研究了闭黎曼流形上C ¹泛型流的双李雅普诺夫稳定同宿类，并证明这样的同宿类不包含奇点。这使得能够并行研究流动和微分同胚的双李雅普诺夫稳定动力学。例如，我们可以证明C ¹泛型流的双李雅普诺夫稳定同宿类是双曲线的，当且仅当该类中的所有周期轨道都具有相同的稳定指数。