# The regularized visible fold revisited

## TL;DR

This paper uses advanced mathematical techniques to analyze how regularized systems behave near a visible fold, revealing a saddle-node bifurcation when a limit cycle grazes a discontinuity, with applications to friction models.

## Contribution

It introduces a novel application of consecutive blowup transformations to study transition maps near visible folds in regularized systems, proving the existence of a saddle-node bifurcation in this context.

## Key findings

- Demonstrated detailed transition map analysis near the fold.
- Proved the existence of a saddle-node bifurcation in regularized systems.
- Applied results to a mass-spring system with Stribeck friction.

## Abstract

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.