Grazing-sliding bifurcations in planar Z2-symmetric Filippov systems

The paper’s main theorem (Theorem 3.3) proves that, near a Z2-symmetric figure-eight loop formed by two stable limit cycles grazing a visible fold-fold, there is a two-parameter reparameterization (β1,β2) and five bifurcation curves β2=ψi(β1) that are quadratically tangent to β2=0, with the ordering ψ1>ψ2>ψ4>0 for β1<0 and ψ3<ψ5<0 for β1>0; it also classifies all phase portraits across these boundaries. This is stated explicitly and in detail in the paper, including the full list of subcases (1)–(4) and the ordering (3.5) with λ(0)<1, and is supported by a displacement-map construction and asymptotic expansions for ψi(β1) (e.g., explicit ψ4, ψ5 coefficients) . The model’s solution reproduces the same reparameterization idea, the contraction-based fixed-point analysis of 1D return maps, the quadratic (parabolic) local thresholds across a visible fold, and the same ordering and phase-portrait transitions, relying on standard Filippov notions and a canonical local fold normal form; this aligns with the paper’s Section 2 definitions and the proof strategy using return/displacement maps and the nondegeneracy condition (3.3) for the (β1,β2) change of parameters . Hence both are consistent and essentially follow the same proof architecture, with the model presenting a concise, high-level sketch and the paper providing full technical details.