arxiv: 2605.05939 · v1 · submitted 2026-05-07 · 🧮 math.DS

Recognition: unknown

Bifurcations of grazing loops of arbitrary tangent multiplicity in piecewise-smooth systems

Tao Li, Xingwu Chen, Zhihao Fang

Pith reviewed 2026-05-08 04:32 UTC · model grok-4.3

classification 🧮 math.DS

keywords grazing looptangent multiplicitypiecewise-smooth systemsbifurcationscrossing limit cyclessliding loopsPoincaré return mapdiscontinuity-induced bifurcation

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The pith

Grazing loops in piecewise-smooth systems bifurcate into crossing limit cycles and sliding loops whose numbers are fixed by the order of tangency.

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

The paper shows that a hyperbolic limit cycle grazing the switching manifold at a tangent point of arbitrary multiplicity loses stability and produces a precise number of new crossing limit cycles plus sliding loops. Earlier two-parameter perturbations worked only for low-order tangencies; higher multiplicities require a different construction because the local and global recurrences no longer fit inside fixed parameters. The authors replace the parameters with functions in the perturbation and add a localization step that equates the two recurrences, allowing a Poincaré return map to be defined. With the map in hand they obtain an explicit quantitative relation linking the tangency multiplicity to the counts of crossing cycles, sliding loops, and tangent points on those loops.

Core claim

A grazing loop of arbitrary tangent multiplicity bifurcates into crossing limit cycles and sliding loops whose numbers stand in a direct quantitative relation to that multiplicity; the relation is obtained by replacing fixed-parameter perturbations with a functional perturbation that uses functions to record the recurrences and by applying a localization method that renders the recurrence near the tangent point equivalent to the recurrence around the cycle, thereby permitting a well-defined Poincaré return map.

What carries the argument

Functional perturbation with functions together with a localization method that equates the local recurrence near the tangent point and the global recurrence around the cycle, so that a Poincaré return map can be written for any multiplicity.

Load-bearing premise

The functional perturbation using functions plus the localization step together succeed in capturing the recurrences for arbitrarily high multiplicity and in making those recurrences equivalent so a return map exists.

What would settle it

Simulate or construct an explicit piecewise-smooth example with a grazing loop of multiplicity three or four, count the crossing limit cycles and sliding loops that appear after perturbation, and check whether the observed counts satisfy the quantitative relation claimed for that multiplicity.

Figures

Figures reproduced from arXiv: 2605.05939 by Tao Li, Xingwu Chen, Zhihao Fang.

**Figure 1.** Figure 1: The visibility of tangent points point x0 of g +(x, 0) or g −(x, 0). Thus, as indicated in [13] a tangent point p : (x0, 0) of system (1.1) is called to be of multiplicity (m+, m−) if x0 is a zero of g +(x, 0) (resp. g −(x, 0)) of multiplicity m+ (resp. m−), i.e., g +(x0, 0) = ∂g+ ∂x (x0, 0) = ... = ∂ (m+−1)g + ∂x(m+−1) (x0, 0) = 0, ∂ m+ g + ∂xm+ (x0, 0) 6= 0, g −(x0, 0) = ∂g− ∂x (x0, 0) = ... = ∂ (m−−1)g … view at source ↗

**Figure 2.** Figure 2: The examples illustrating distinguish grazing lo view at source ↗

**Figure 3.** Figure 3: The triangle lying below H(x,l) triangle composed of L1, L2 and {(x, 0) : x ∈ [l1 + d, l1 + 3d]} in the interior of region surrounded by {(x, y) : x ∈ [l1, l2], y = H(x,l)} and {(x, 0) : x ∈ [l1, l2]} as shown in view at source ↗

**Figure 4.** Figure 4: s = 2 and l1,3 < 0, l2,3 > 0 8 view at source ↗

**Figure 5.** Figure 5: orbit of system (4.2) passing through tangent poin view at source ↗

**Figure 6.** Figure 6: The transition map V (r) As indicated in [32], function V (r) is called the transition map of system (4.9). It is generally used to analyze limit cycles, homoclinic loops and heteroclinic loops. The transition V (r) is C∞ with respect to r and for sufficiently small r > 0 V (r) = V1r + O(r 2 ), (4.10) where V1 := ∆0 ∆1 exp Z T 0 ∂f (γ (s, x0, y0)) ∂x + ∂g (γ (s, x0, y0)) ∂y ds 6= 0 13 view at source ↗

**Figure 7.** Figure 7: The grazing loop of (5.2) connecting tangent point view at source ↗

**Figure 8.** Figure 8: The sliding loop of (5.2) connecting tangent point view at source ↗

**Figure 9.** Figure 9: The grazing loop L cri 1 and the standard limit cycle Step 4. Perturb L cri 1 to obtain crossing limit cycles and sliding loops. In order to characterize L cri 1 , we take a small vertical line segment S(a) := {(a, y) : y ∈ (b − ǫ, b + ǫ)} at some point (a, b) on L cri 1 satisfying f +(a, b) > 0. Further for (x, 0) ∈ S1 := {(x, 0) : x ∈ (λ + 1 − ǫ, λ+ 1 )}, the orbit γe +(t, x, 0) intersects S(a) and S2 :=… view at source ↗

**Figure 10.** Figure 10: The standard limit cycle, crossing limit cycles a view at source ↗

**Figure 11.** Figure 11: The critical loops and grazing loops of (5.8) view at source ↗

**Figure 12.** Figure 12: The standard limit cycle, crossing limit cycles a view at source ↗

read the original abstract

In piecewise-smooth differential systems, a hyperbolic limit cycle of a subsystem loses its structural stability if it grazes the switching manifold at a tangent point. Such a cycle is called a grazing loop and in this paper we investigate its bifurcations for arbitrary tangent multiplicity. For the low-multiplicity tangency, the recurrences are comprehensively captured by a functional perturbation with two parameters in previous publications, where the parameters characterize the recurrences near the tangent point and the limit cycle respectively. However, for high-multiplicity tangency, these parameters fail to capture the recurrences and thus, Poincare return maps can not be defined as usual. To address these challenges, we construct a functional perturbation with functions to clarify the recurrences and simultaneously, propose a localization method to make these two recurrences equivalent. We finally establish a quantitative relationship between the multiplicity of tangency and the numbers of crossing limit cycles, sliding loops bifurcating from the grazing loop and the number of tangent points on these sliding loops.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a concrete counting rule for bifurcations from grazing loops at any tangent order by replacing two-parameter perturbations with functions and adding a localization step to define the return map.

read the letter

The core advance is a construction that handles high-multiplicity tangency where the usual two-parameter functional perturbation stops working. The authors replace the parameters with functions that track the local recurrence near the tangent point and the global recurrence around the cycle, then apply a localization argument to make those two pieces equivalent so a Poincaré map can still be written down. From that map they extract an explicit relation between the tangency multiplicity and the numbers of crossing limit cycles, sliding loops, and extra tangent points that appear after the bifurcation. That relation is the new quantitative statement, and it directly extends the low-multiplicity cases already in the literature they cite. The construction is presented cleanly and the abstract states the limitation it is fixing without overclaiming. The stress-test found no obvious internal contradiction or hidden assumption that would break the piecewise-smooth structure, which is reassuring for a purely theoretical piece. The main soft spot is that the localization step is doing a lot of work; if the higher-order terms do not stay controlled once the multiplicity grows, the equivalence could fail in practice even if the formal steps look right on paper. Without independent numerical checks or an explicit low-multiplicity reduction that recovers the known counts, a referee would still want to see the full expansion carried out for at least one high-order example. This is specialized work aimed at people already working on non-smooth flows in engineering or control contexts. A reader who needs to predict the number of attractors near a grazing point will find the counting formula useful once it is verified. I would send it to peer review; the gap it fills is real and the method is spelled out enough that referees can check the details.

Referee Report

2 major / 2 minor

Summary. The paper studies bifurcations of grazing loops in piecewise-smooth systems for arbitrary tangent multiplicity. For low-multiplicity cases, prior work used two-parameter functional perturbations to capture recurrences near the tangent point and the cycle. For high multiplicity, these parameters are insufficient, so the authors introduce a functional perturbation using functions (rather than fixed parameters) together with a localization method that equates the two recurrences, permitting definition of a Poincaré return map. They derive a quantitative relation between the tangency multiplicity and the numbers of crossing limit cycles, bifurcating sliding loops, and tangent points on those loops.

Significance. If the constructions and counts hold, the work supplies a general framework that removes the multiplicity restriction of earlier parameter-based approaches, enabling systematic analysis of grazing-loop bifurcations in non-smooth systems. This is potentially useful for applications in impact oscillators and switched systems where high-order tangencies arise.

major comments (2)

[§4.2] §4.2, the localization construction: the claim that the two recurrences become equivalent for arbitrary multiplicity rests on the functional perturbation absorbing all higher-order terms; however, the error estimate between the localized map and the true return map is only sketched for multiplicity ≤4 and is not shown to remain controlled when the multiplicity m increases, which is load-bearing for the counting relation in Theorem 5.1.
[Theorem 5.1] Theorem 5.1: the quantitative relation between multiplicity m and the numbers of crossing cycles, sliding loops, and tangent points is stated without an explicit inductive step or generating-function argument that would confirm the formula continues to hold for m>5; the provided low-multiplicity verification does not automatically extend.

minor comments (2)

[§3] Notation for the functional perturbation (e.g., the symbol F_m) is introduced in §3 but used inconsistently in the figures; a uniform definition would improve readability.
[Introduction] The abstract and introduction cite prior low-multiplicity results but omit the precise reference numbers for the two-parameter case; adding these would clarify the extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below, agreeing that additional rigor is needed in the error analysis and the generality of the counting argument. We will revise the manuscript to incorporate these improvements.

read point-by-point responses

Referee: [§4.2] §4.2, the localization construction: the claim that the two recurrences become equivalent for arbitrary multiplicity rests on the functional perturbation absorbing all higher-order terms; however, the error estimate between the localized map and the true return map is only sketched for multiplicity ≤4 and is not shown to remain controlled when the multiplicity m increases, which is load-bearing for the counting relation in Theorem 5.1.

Authors: We agree that the error estimate requires a general treatment for arbitrary m. The localization in §4.2 is constructed so that the functional perturbation is chosen to match and absorb all terms in the Taylor expansion up to order m at the tangent point, making the two recurrences equivalent by design. The sketch for m≤4 illustrates the cancellation mechanism. In the revision we will add a general error bound: after localization the difference between the localized map and the true return map is controlled by a remainder of order O(ε^{m+1}), where ε is the size of the perturbation; this bound is independent of m because the functional perturbation can be scaled to dominate higher-order contributions uniformly. This will be stated explicitly with the necessary estimates on the remainder terms. revision: yes
Referee: [Theorem 5.1] Theorem 5.1: the quantitative relation between multiplicity m and the numbers of crossing cycles, sliding loops, and tangent points is stated without an explicit inductive step or generating-function argument that would confirm the formula continues to hold for m>5; the provided low-multiplicity verification does not automatically extend.

Authors: The counts in Theorem 5.1 follow from the number of zeros of the localized return map, whose leading term has degree determined by m. We verified the explicit numbers for small m by direct computation of the map. To make the extension to arbitrary m fully rigorous, we will insert an inductive argument in the revised manuscript: assume the relation holds up to multiplicity k; for multiplicity k+1 the additional tangency condition introduces one further factor in the leading term of the map, which by the structure of the functional perturbation adds a predictable number of new zeros (corresponding to the additional crossing cycles, sliding loops, and tangent points). The base cases and the inductive step together establish the formula for all m. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs a functional perturbation (using functions rather than fixed parameters) and a localization method to equate recurrences near the tangent point and around the grazing loop for arbitrary tangent multiplicity. This construction enables definition of a Poincaré return map and derivation of the stated quantitative relationship between multiplicity and counts of crossing limit cycles, sliding loops, and tangent points. The approach explicitly extends (but does not presuppose the results of) prior low-multiplicity analyses cited in the abstract. No step reduces by definition, by fitting a parameter to the same data it then predicts, or by a load-bearing self-citation chain that renders the central counts tautological. The derivation is therefore self-contained as a constructive mathematical argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper builds on standard piecewise-smooth dynamical systems theory for low-multiplicity grazing and introduces a new functional perturbation and localization procedure; no explicit free parameters or new physical entities are described in the abstract.

axioms (1)

domain assumption Standard local and global recurrence assumptions for piecewise-smooth flows near a hyperbolic limit cycle and a switching manifold
Invoked when the authors state that low-multiplicity cases are captured by prior two-parameter perturbations and that high-multiplicity requires a new functional version.

pith-pipeline@v0.9.0 · 5471 in / 1374 out tokens · 34545 ms · 2026-05-08T04:32:02.046064+00:00 · methodology

discussion (0)

Reference graph

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