The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter \(\epsilon \rightarrow 0\). Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit \(\epsilon \rightarrow 0\), grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law.

在分段平滑系统中，平面可见折痕是简单的奇点。在本文中，我们将限制于此分段光滑分叉的奇异摄动系统视为奇异摄动参数\（\ epsilon \ rightarrow 0 \）。或者，可以将这些奇异摄动系统视为其分段对应物的正则化。本文的主要贡献是演示了在这种情况下连续爆破变换的使用，使我们能够在非常笼统的假设下获得有关折叠附近过渡图的详细信息。我们应用此信息来首次证明，在极限环为奇数极限\（\ epsilon \ rightarrow 0 \）的情况下，存在局​​部唯一的鞍形节点分叉，吃草不连续集。我们将此结果应用于由Stribeck型摩擦定律描述的运动皮带上的质量弹簧系统。