The aim of this article is twofold. Firstly, we study the existence of limit cycles in a family of piecewise smooth vector fields corresponding to an unfolding of an invisible fold–fold singularity. More precisely, given a positive integer k, we prove that this family has exactly k hyperbolic crossing limit cycles in a suitable neighborhood of this singularity. Secondly, we provide a complete study relating the existence and stability of these crossing limit cycles with the limit cycles of the family of smooth vector fields obtained by the regularization method. This relationship is obtained by studying the equivalence between the signs of the Lyapunov coefficients of the family of piecewise smooth vector fields and the ones of its regularization.

本文的目的是双重的。首先，我们研究了一系列分段光滑向量场中极限环的存在，这些极限场对应于不可见的折叠-折叠奇点的展开。更精确地，给定正整数k，我们证明该族在这个奇点的合适邻域中恰好具有k个双曲穿越极限环。其次，我们提供了一个完整的研究，将这些交叉极限环的存在和稳定性与通过正则化方法获得的光滑向量场族的极限环相关。通过研究分段光滑矢量场族的Lyapunov系数的正负号与其正则化的等价性之间的等价关系。