The regularized visible fold revisited

Kristian Uldall Kristiansen

REVIEW 3 major objections 6 minor 51 references

The regularized visible fold revisited

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For every sufficiently small $\epsilon>0$, a limit cycle grazing the discontinuity set in the singular limit unfolds into a locally unique saddle-node bifurcation of limit cycles at $\alpha=\epsilon^{2k/(2k+1)}\alpha_2(\epsilon)$, with…

desk verdict A serious blowup analysis that proves the first rigorous version of the grazing saddle-node bifurcation for general regularizations, but the quantitative derivative estimate at the heart of the uniqueness argument is underproved and needs referee scrutiny. read the letter →

arxiv 1908.06781 v5 pith:EYHIV4NT submitted 2019-08-19 math.DS

Kristian Uldall Kristiansen This is my paper · ORCID

classification math.DS MSC 34E1534C2337G15

keywords visiblefoldpiecewisesmoothsystemsregularizationsingularperturbationblowupmethodsaddle-nodebifurcationgrazingfrictionoscillator

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a precise statement about what happens when a smooth ordinary differential equation approaches a piecewise smooth system with a visible fold. The main result is that if a repelling limit cycle of the upper vector field grazes the discontinuity line in the singular limit, then for every sufficiently small regularization parameter $\epsilon>0$ the regularized system undergoes a locally unique saddle-node bifurcation of limit cycles, at a parameter value that differs from the grazing value by order $\epsilon^{2k/(2k+1)}$, where $k$ is the decay rate of the regularization function above the fold. The proof relies on two consecutive blowup transformations that resolve the fold into hyperbolic and center-manifold building blocks, yielding the first detailed description of the transition map in a full neighborhood of the fold. As a consequence, the paper settles a claim left open in earlier treatments and applies the result to a mass-spring system with Stribeck friction, where numerical continuation confirms the predicted scaling.

What carries the argument

The machinery is the double blowup: first blow up the discontinuity line $y=\epsilon=0$ to a cylinder, then blow up the nonhyperbolic point $T$, the imprint of the visible fold, to a sphere with weights $(2k,k,1)$. In the charts $(\bar r=1)_1$ and $(\bar\epsilon=1)_2$, this desingularizes the fold: the critical manifold ends at a nonhyperbolic saddle $p_a$, while the grazing orbit enters and leaves through hyperbolic points $p_L$ and $p_R$. Partial linearizations near $p_L$ and $p_R$ are combined with a global analysis of the Chini equation $v'=2u+v^{-k}$, the reduced equation for the middle transition map, to obtain the S-shaped derivative profile in Lemmas 3.14 and 3.17. The saddle-node then follows by writing the Poincar\'e map as $Q\circ R^{-1}$ and applying the implicit function theorem in the scaled variables $x=\epsilon^{2k/(2k+1)}x_2$, $\alpha=\epsilon^{2k/(2k+1)}\alpha_2$.

What would settle it

A direct numerical evaluation of the derivative $X'_{C,0}(x_1)$ of the middle transition map, defined through the Chini equation $v'=2u+v^{-k}$, for small $\nu$ and $x_1\in[-1-\zeta,-1+\zeta]$, would settle the key estimate: it must lie strictly between $-1$ and $0$ with negative second derivative; observing any value outside that interval for some small $\nu$ would contradict Lemma 3.14 and with it the uniqueness part of Theorem 1.7.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.7. Under assumptions (A0)-(A2) on the system and regularization, and (B1)-(B3) on a repelling limit cycle with a single quadratic tangency to the switching manifold, there exists, for all sufficiently small $\epsilon>0$, a locally unique saddle-node bifurcation of limit cycles at $\alpha=\epsilon^{2k/(2k+1)}\alpha_2(\epsilon)$ with $\alpha_2$ continuous. Locally, limit cycles exist precisely for $\alpha\leq\epsilon^{2k/(2k+1)}\alpha_2(\epsilon)$: two for smaller $\alpha$, one at the bifurcation value, and the saddle-node periodic orbit converges in Hausdorff distance to the grazing limit cycle $\Gamma_0$ as $\epsilon\to 0$. This is obtained not from the discontinuous return map but from the regularized transition map itself: Theorem 1.3(d) shows that on an $\epsilon^{2k/(2k+1)}$-neighborhood of the grazing point the map's derivative is monotone, crosses every value in $(-1,0)$ exactly once, and has second derivative uniformly negative, which is exactly the profile needed to separate two fixed points before the bifurcation and none after.

Load-bearing premise

The whole proof hinges on the claimed derivative profile of the middle transition map near the fold—slope strictly between -1 and 0, with decreasing slope—which the paper supports only by a calculation sketch rather than a full proof.

Editorial extensions

If this is right

For any regularization function satisfying (A1)-(A2), the grazing bifurcation in the regularized system is a genuine saddle-node of limit cycles at the $\epsilon^{2k/(2k+1)}$-scale, where $k$ is the decay rate of the regularization above the fold.
Below the bifurcation value there are locally exactly two periodic orbits: one attracting whose $\epsilon\to 0$ limit has a sliding segment, and one repelling that converges to the repelling grazing cycle; above the value there are none.
The saddle-node periodic orbit converges in Hausdorff distance to the grazing limit cycle $\Gamma_0$ as $\epsilon\to 0$, so the bifurcation is a true regularized unfolding of the piecewise-smooth grazing bifurcation.
In the mass-spring-on-belt example with a Stribeck friction law and a subcritical Hopf bifurcation, the theorem predicts a locally unique saddle-node of limit cycles near the grazing parameter $\alpha_*$ for every small $\epsilon$; the paper's AUTO computation gives a numerically observed slope of about 0.8024, matching the predicted exponent $4/5$ for $k=2$.
The transition-map description in Theorem 1.3(d) is sharper than previous treatments: in the middle region the map has a single point with any prescribed slope in $(-1,0)$, and the uniform curvature bound rules out additional saddle-nodes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The transition-map argument is essentially local in the fold, so a natural extension is to higher-dimensional systems whose grazing is a planar quadratic tangency plus strong contraction; one expects the same $\epsilon^{2k/(2k+1)}$ scaling for the saddle-node in that setting.
If the regularization function is non-monotone and produces a fold in the critical manifold, the same consecutive-blowup framework suggests that the unstable limit cycle can be continued past the tangency to a canard-like bifurcation, a case the paper explicitly leaves open.
The Chini equation $v'=2u+v^{-k}$ is the scalar model for the middle transition; a computer-assisted or analytic bound on its variational equation could replace the sketched proof of Lemma 3.14 and make the uniqueness proof fully checkable.
The power law $\alpha_{SN}(\epsilon)-\alpha_* \sim \epsilon^{2k/(2k+1)}$ is a fingerprint of the regularization's decay rate, so measuring the saddle-node shift for different smoothing functions with known $k$ would directly test the theorem's quantitative claim.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper does something real: it gives the first rigorous proof that a grazing limit cycle in a piecewise smooth system becomes a locally unique saddle-node bifurcation under a broad class of regularizations, with a precise scaling exponent epsilon^{2k/(2k+1)}. That goes beyond Bonet-Reves and Seara's work, which had the same conclusion only for Sotomayor-Teixeira functions and without a proof. The blowup construction is coherent, the transition map estimates in Theorem 1.3(d) are genuinely new, and the numerical exponent 0.8024 matches the theoretical 4/5. The paper is honest about its limitations and does not overclaim. The citations look appropriate; the self-citations are to prior blowup methods that are directly relevant.

The soft spot is exactly the one you flagged. Lemma 3.14 is the quantitative core: it claims the derivative of the middle transition map X'_C,0 lies in (-1+c,-c) with X''_C,0<0 on the whole interval, and this is what makes the S-shaped graph and the unique saddle-node work. But the proof at the endpoints is only sketched: "a simple calculation" for one end, "following the flow" for the other, and the uniform negativity of the second derivative is asserted rather than shown. This is not a matter of taste. If that derivative profile fails, the nondegeneracy condition (4.7) can fail and the implicit function theorem argument in Lemma 4.5 collapses. The stress-test note is fair, and I would not soften it. That said, the gap looks fillable; the rest of the proof is detailed and the claim is plausible. A referee should ask for a complete calculation, not reject the paper on suspicion.

Minor points: the paper says "smooth" means C^l with l sufficiently large without being precise, and the numerical artifacts behind Figure 5 are not documented. Neither is serious for the main argument.

The paper is for people working in geometric singular perturbation theory and regularized piecewise smooth systems, especially those studying grazing and boundary bifurcations. It is a solid piece of work that deserves a serious referee. I would send it out with a request for a rigorous proof of Lemma 3.14, and I would not be surprised if the paper came back accepted after that.

Referee Report

3 major / 6 minor

Summary. The paper studies singularly perturbed planar systems that limit, as ε→0, to piecewise smooth systems with a visible fold singularity. The main technical result, Theorem 1.3, gives a detailed description of the transition map near the fold under general assumptions (A0)-(A2) on the regularization function, including contraction away from the grazing point, a derivative estimate in an O(ε^{2k/(2k+1)}) neighborhood of the grazing point, and a description of the derivative profile there. Theorem 1.7 then uses this transition map to prove, for a grazing repelling limit cycle satisfying (B1)-(B3), the existence of a locally unique saddle-node bifurcation of limit cycles at α = ε^{2k/(2k+1)} α_2(ε), with exactly two limit cycles below the bifurcation value and one at the value, and with Hausdorff convergence of the saddle-node orbit to the grazing cycle. The proof is built on two consecutive blowups, a partial linearization near the hyperbolic points p_L and p_R, an analysis of a Chini-type equation, and an implicit function argument. Section 2 applies the result to a mass-spring system on a moving belt with a Stribeck friction law and provides AUTO-based numerical evidence, including the predicted exponent 4/5 for k=2.

Significance. If the quantitative estimates in Section 3.5 are supplied in full, the paper would constitute the first rigorous proof of local uniqueness of the saddle-node bifurcation in the regularized grazing bifurcation for the visible fold, extending earlier work by Bonet-Revés and Seara from Sotomayor-Teixeira regularization functions to the class of asymptotic regularization functions satisfying (A1)-(A2). The consecutive-blowup framework is appropriate and general, and the paper gives explicit, falsifiable predictions: the scaling exponent ε^{2k/(2k+1)} in Theorem 1.7 is verified numerically, and the transition-map derivative profile in Theorem 1.3(d) is a concrete quantitative statement. The assumptions (A0)-(A2) and (B1)-(B3) do not appear to encode the conclusion; there is no circular parameter fitting. The application to the Stribeck friction oscillator is a genuine and well-motivated addition. However, the central derivative profile of the middle transition map, Lemma 3.14, is only sketched, and the chain-rule estimate that converts it into the uniform negativity bound in Lemma 3.17 is asserted rather than demonstrated.

major comments (3)

[Section 3.5, Lemma 3.14 and Lemma 3.17] The proof of Lemma 3.14 is only a sketch, and this is the quantitative core of Theorem 1.3(d)(ii). For the endpoint x1 = -1+ζ the argument invokes 'a simple calculation' together with an odd variational equation to conclude X'_C,0 approaches -1; for x1 = -1-ζ it asserts convergence to 0^- by 'following the flow ... up close to the center manifold of p_a' without any estimate. The uniform negativity of X''_C,0 on the whole interval is inferred from Lemma 3.13, but Lemma 3.13 only gives sign information, not a positive lower bound for |X''_C,0|. These facts are load-bearing: Lemma 3.17 needs X''_ε < -c^{-1} after the rescaling, and Lemma 4.5's nondegeneracy condition (4.7) plus the uniqueness of the saddle-node in Theorem 1.7 depend on that bound. An independent, detailed verification of both endpoint asymptotics and of a uniform bound on X''_C,0 is required.
[Section 3.5, Lemma 3.16, Eq. (3.47)] The chain-rule estimate in Eq. (3.47) is not justified by the displayed argument. Using the asymptotics (3.44)-(3.45), the contributions to X''_ε coming from the second derivatives of X_R,ε and ~X_L,ε are of order ε^{-2k/(2k+1)} or larger, not O(1); they can be absorbed into the leading term only if those second derivatives are uniformly O(c) and (X_C,ε)'' is bounded away from zero. The proof says 'simple calculation' but does not track these terms. Since Eq. (3.47) is the bridge from Lemma 3.14 to Lemma 3.17 and hence to the nondegeneracy condition (4.7), a complete derivation with all remainders accounted for is needed.
[Section 3.5, proof of Theorem 1.3(d)(i), and Section 4, Lemma 4.1] The proof of the contraction estimate in Theorem 1.3(d)(i) is only sketched. The first part of the passage from Σ_out,L_L to Σ_in,C_R is described as 'standard and left out of this manuscript completely', and the second part is asserted to be contracting because of exponential contraction toward the center manifold of p_a. This estimate is used in Lemma 4.1 to prove that for α<0 there are exactly two fixed points and for α>0 there are none, which is part of the full statement of Theorem 1.7. Please supply the missing estimates or state and prove the needed contraction bound explicitly.

minor comments (6)

[Section 3.2, Eqs. (3.19) and (3.23)] In the displayed forms of Ψ_L and Ψ_R, the function ~R_L is written as O(ρ^k_1), but the coordinate change should have the form 1 + O(ρ^k_1) to be invertible near the identity; please clarify the intended expansion.
[Section 3.2, Eq. (3.26)] There is a typo in the line 'XR(0, 1, 0, ) = 0': an extra comma appears after the third argument.
[Lemma 3.6] The phrase 'C2 O(ε_1^{1/(2k+1)})-close' is awkward; it should be rephrased as 'C^2 and O(ε_1^{1/(2k+1)})-close' to avoid ambiguity.
[Section 1.4 and throughout] The paper says it will leave to the reader what 'sufficiently large' smoothness is, but the main theorems rely on C^2 estimates and an implicit function theorem argument. Please state a specific finite order of smoothness for the main results, or at least specify that C^3 or C^4 suffices for Theorem 1.7.
[Section 2, Figure 5 and surrounding text] The subfigure references are inconsistent: the text refers to 'Fig. 5(d)' for the log-log plot of α*−α, while the caption labels that plot as '(b)', and the caption's '(d)' is the phase portrait. Please align the in-text references with the caption.
[Section 5, paragraph 'Comparison with previous results'] There is a typo in the phrase '[26, Thoerem 3.3]'; it should read 'Theorem'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the saddle-node result is derived from independent transition-map estimates and standard blowup/Fenichel theory, not encoded in the assumptions.

full rationale

The derivation chain in Theorem 1.7 is not circular. The assumptions (A0)-(A2) and (B1)-(B3) are structural hypotheses about the regularized vector field and the grazing limit cycle; they do not contain the conclusion of a locally unique saddle-node bifurcation. The normal form (1.11) is imported from the external reference [3], not from the author's own prior work. The blowup framework is stated as following [32], which is a self-citation, but the load-bearing analysis is carried out in the manuscript itself: Appendix A gives a proof of Theorem 1.3(a), and Sections 3.2--3.5 derive the transition-map estimates (Lemmas 3.6, 3.8, 3.13-3.17) from direct variational calculations on the blown-up equations and the Chini equation. Lemma 3.13, which supplies U'(u) in (-1,0) and U''(u)<0, is proved in detail. Lemma 3.14 uses this lemma together with stated endpoint asymptotics; those endpoint claims are only sketched ('a simple calculation', 'following the flow ... up close to the center manifold of pa'), but a sketch is a rigor or correctness concern, not circularity, because the claims are not assumed as inputs. The final saddle-node argument applies the implicit function theorem to the fixed-point equation Q2 = R2^{-1}, with nondegeneracy following from the derivative profile and from R's hyperbolic derivative estimates in Lemma 1.6; these ingredients are independent of the advertised uniqueness. The paper also provides an external numerical check (AUTO continuation in Section 2) consistent with the derived scaling. No fitted parameter is renamed as a prediction, and no load-bearing uniqueness theorem is imported from the author's own prior work. Hence the correct circularity finding is no significant circularity, even though the proof of Lemma 3.14 would benefit from a fuller exposition.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

free_parameters: none. The numbers k, α, ε are inputs from the stated assumptions, and δ, ξ, ν, θ, η are auxiliary constants chosen in the proofs, not fitted to data. The axioms are either explicit model hypotheses or standard analytic background. The normal form (1.11) is the only nontrivial imported result. No new physical or mathematical entities are postulated.

assumptions (5)

domain assumption A0-A2: affine dependence in p, smooth monotone asymptotically flat regularization φ, finite algebraic decay rates k± in the tails (1.14)-(1.15).
The entire paper is conditional on these hypotheses; A2 in particular fixes the integer k in the scaling exponents.
domain assumption B1-B3: Z+ has a hyperbolic repelling limit cycle Γ0 tangent to Σ, the tangent point moves across Σ with positive speed in α, and Z− is transverse at the tangency.
Defines the grazing-sliding scenario that Theorem 1.7 addresses.
domain assumption Normal form (1.11) from [3, Prop. 14]: a smooth coordinate change puts the visible fold into Z+=(1+f, 2x+yg), Z−=(0,1).
Imported from prior literature and not reproved; all chart calculations start from this form.
standard math Fenichel theory, center manifold theory and partial linearization theorems apply as used in Lemmas 3.3-3.9 and Appendix A.
Standard background in geometric singular perturbation theory; used to construct S_epsilon, M1, and the local diffeomorphisms.
standard math Implicit function theorem and regular perturbation theory for finite-time flow maps apply to the global return map R and to the Poincaré map P.
Used in Lemmas 1.6 and 4.5 to pass from estimates at ε=0 to small ε and to solve (4.5).

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Pith. "Pith review of The regularized visible fold revisited." pith.science (2026). https://pith.science/paper/EYHIV4NT

@misc{pith2026190806781,
  author       = {Pith},
  title        = {Pith review of: The regularized visible fold revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EYHIV4NT}},
  note         = {Machine review of arXiv:1908.06781}
}

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.

Figures

Figures reproduced from arXiv: 1908.06781 by the authors.

**Figure 1.** Geometric construction of the Filippov sliding vectorfield Zsl as the convex combination of Z± such that Zsl is tangent to Σ. We consider the case illustrated in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** The visible fold. Z+ has a quadratic tangency with Σ at (x, y) = (0, 0) while Z−(0, 0) is transverse. Along the sliding region, Σsl = {x < 0}, the Filippov vector-field gives ˙x > 0. Every point in ΣL with x < γL therefore reaches ΣR at x = γR by following the Filippov flow. See also Remark 1.5. by the first intersection (Q(x, ), δ) ∈ ΣR of the forward flow, defined by (1.2), with Z± as in (1.11), of the point (x, … view at source ↗

Figures from the paper (8 more)

**Figure 3.** Figure 3: A grazing limit cycle for α = = 0. Assumption (B1) is so that Γ0 is repelling for Z+ for α = 0. By (B2), Γα<0 (in blue) locally intersects Σ twice near the fold. Black orbits are (backwards) transients for α = 0, demonstrating the repelling nature of Γ0. of Z+, Qe is…

**Figure 4.** Figure 4: In (a): The mass-spring system on a moving belt. In (b): A Stribeck friction law with a minimum at y = y0. square we obtain a slope ≈ 0.8024 which is also in agreement with Theorem 1.7 for k = 2; notice 2k/(2k+1) = 4/5 = 0.8 for this value of k. In [PITH_FULL_IM…

**Figure 5.** Figure 5: In (a): Bifurcation diagram of limit cycles using min y as a measure of the amplitude for varying values of : in cyan: = 5 × 10−3 , magenta: = 2.5 × 10−3 , blue: = 10−3 , red: = 5 × 10−4 , and finally in green: = 10−4 . In (b): α∗ − α along the saddle-node b…

**Figure 6.** Figure 6: Illustration of the cylindrical blowup of the visible fold. The blown down version is on the left whereas the blowup picture on the right. Our viewpoint is from > 0, this axis coming out of the diagram. Since ≥ 0 only the part of the cylinder with ¯ ≥ 0 is relevan…

**Figure 7.** Figure 7: Illustration of the subsequent blowup (3.2) of the nonhyperbolic point (r1, x1, 1) = (0, 0, 0), corresponding to T, in the chart (¯y = 1)1. The weights in (3.2) are so that the critical manifold and the nonhyperbolic fiber gets separated into two poins pa and pf on …

**Figure 8.** Figure 8: Illustration of the dynamics in chart (¯r = 1)1, see (3.3). pL and pR are hyperbolic and we use partial, smooth linearizations, see Lemma 3.4 and Lemma 3.5, near this points to describe the local transition maps between the various sections shown. Notice ρ1 = 0 corresp…

**Figure 9.** Figure 9: Dynamics within the (u, v)-plane. We describe the transition map from ΣL and ΣR using a simple analysis of the variational equations, which in the (u, v)-variables, take a simple form. Lemma 3.16. Consider any c > 0, then there exist constants η, θ ∈ (0, η), δ1, and ν1…

**Figure 10.** Figure 10: Graphs of Q (thick curve in different colours) and R−1 (thick dashed line), following Theorem 1.3(d) and Lemma 1.6. The different colours on the graph of Q, represents the different domains in Theorem 1.3(d): Green for the domain in (i), purple for the domain in (ii),…

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