Local bifurcations in a class of piecewise-smooth Filippov systems with a nonregular switching curve via a nonlinear double regularization process

We are interested in analyzing the preservation of bifurcations in a class of piecewise smooth vector fields with a nonregular switching set under a smoothing process that approximates them by smooth vector fields. We examine cases in which the codimension is either preserved or altered, as well as whether the generic nature of the bifurcation is maintained.

💡 Research Summary

The paper investigates how low‑codimension bifurcations of planar piecewise‑smooth Filippov systems behave when the discontinuous vector field is approximated by smooth ones through a nonlinear double‑regularization process. The authors focus on a class of systems whose switching set Σ is nonregular: Σ = { (x₁,x₂) ∈ ℝ² | x₁x₂ = 0 }, i.e., the union of two transverse lines intersecting at the origin. In this setting the usual Filippov convention does not apply at the intersection point, and the dynamics near Σ can be highly degenerate.

To regularize the system they introduce two small parameters ε and η, one for each coordinate direction, together with transition functions φ and ψ. These functions satisfy φ(s)=ψ(s)=sgn(s) for |s|≥1 and are smooth with positive derivative on (−1,1). Unlike most previous works, φ and ψ are allowed to be non‑monotone, and a further small nonlinear term ξ G(φ,ψ) is added, where G is a continuous map on