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arxiv: 2605.05605 · v2 · submitted 2026-05-07 · 🧮 math.DS · math-ph· math.MP· math.SG· nlin.CD

Recognition: no theorem link

Mixed Global Dynamics of the Forced Vibro-Impact Oscillator with Coulomb Friction and its Symplectic Structure, KAM Tori, and Persistence

Abdoulaye Thiam

Pith reviewed 2026-05-12 02:08 UTC · model grok-4.3

classification 🧮 math.DS math-phmath.MPmath.SGnlin.CD
keywords vibro-impact oscillatorCoulomb frictionsymplectic mapKAM toriFilippov flowsaddle-center bifurcationMelnikov analysisnon-sticking set
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The pith

The time-T stroboscopic map of the forced vibro-impact oscillator with Coulomb friction is exact symplectic on the maximal non-sticking set.

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

The paper analyzes a forced oscillator that bounces between elastic walls under Coulomb friction and periodic forcing. It constructs a global lift of the Filippov flow to a piecewise smooth Hamiltonian system on a covering manifold, then restricts attention to the maximal forward-invariant set of non-sticking trajectories. On that set the stroboscopic map preserves the symplectic form exactly, so Moser's twist theorem produces Cantor families of invariant tori near elliptic periodic orbits while a Melnikov analysis produces hyperbolic dynamics near saddles. Small restitution loss or added viscous damping destroys the symplectic structure and turns the elliptic orbits into attractors. Readers care because the construction explains how conservative islands and dissipative attractors can coexist in a single mechanical phase space.

Core claim

The central claim is that the time-T stroboscopic map is exact symplectic on the maximal forward-invariant non-sticking set of the Filippov flow. This property follows from lifting the system to a piecewise smooth Hamiltonian on a covering manifold. Moser's twist theorem then yields invariant Cantor families of KAM tori near elliptic non-sticking periodic orbits, and Melnikov analysis produces hyperbolic dynamics conjugate to a Bernoulli shift near the associated saddles. Symmetric T-periodic orbits are located by a closed-form equation that also locates a saddle-center bifurcation at a parameter value f_sc(F, ω, R).

What carries the argument

The exact symplectic structure of the time-T stroboscopic map on the maximal forward-invariant non-sticking set, obtained by lifting the Filippov flow to a piecewise smooth Hamiltonian system on a covering manifold.

If this is right

  • KAM tori exist near elliptic non-sticking periodic orbits by Moser's twist theorem.
  • Hyperbolic dynamics conjugate to a Bernoulli shift appear near associated saddles by Melnikov analysis.
  • Symmetric T-periodic orbits satisfy a closed-form existence equation that also locates a parameter-dependent saddle-center bifurcation.
  • Any positive restitution defect or viscous damping converts elliptic orbits into asymptotically stable attractors and replaces islands by a single basin.
  • The same symplectic structure and higher-dimensional KAM tori extend to multi-particle systems with elastic collisions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same covering-manifold lift may identify symplectic structure in other piecewise-smooth impact systems once sticking regions are excised.
  • Computer-assisted verification of ellipticity can be repeated at additional parameter points to map the region where KAM tori exist.
  • Real mechanical devices with even tiny viscosity will replace Hamiltonian islands by attractors, suggesting a design route to suppress or enhance stability via controlled damping.
  • The approach supplies a template for proving mixed global dynamics in higher-dimensional systems whose friction laws permit an analogous Hamiltonian lift.

Load-bearing premise

The Filippov flow admits a global lift to a piecewise smooth Hamiltonian system on a covering manifold and the maximal forward-invariant non-sticking set is well-defined and forward-invariant.

What would settle it

Numerical integration of non-sticking trajectories showing that the Jacobian determinant of the stroboscopic map deviates from one, or computational failure to locate the predicted KAM tori near an elliptic periodic orbit at a parameter value where the twist condition holds.

Figures

Figures reproduced from arXiv: 2605.05605 by Abdoulaye Thiam.

Figure 1
Figure 1. Figure 1: The forced vibro-impact oscillator with Coulomb fric￾tion. A unit point mass slides on a rough surface between two rigid walls at x = l and x = r, subject to periodic external forcing F cos(ωt) and dry friction f sgn( ˙x) opposing motion. Wall contact at x ∈ {l, r} instantaneously reverses the velocity, ˙x(t +) = −x˙(t −). The wall gap is R = r − l. The system (1.1)-(1.2) is sketched in view at source ↗
Figure 2
Figure 2. Figure 2: The universal-cover construction. The bounded po￾sition x ∈ [l, r] is the image of an unbounded coordinate q ∈ R under the triangular-wave projection (3.2). The wave W(q) has period 2 with peaks at odd integers and zeros at even integers. A wall reflection at x ∈ {l, r} in the original system corresponds to smooth passage of q through an integer value, so a piecewise smooth bouncing flow on [l, r] lifts to… view at source ↗
Figure 3
Figure 3. Figure 3: Approach to the impulse bound at ω = 1, R = 2, f = 0.4. Each row pairs the time series x(t) over four forcing pe￾riods (left, between the dashed walls x = l and x = r) with the stroboscopic phase portrait of Φ (right, 801 iterates). Panels (a), (b) at F = 0.95, 0.85: the elliptic non-sticking T-periodic orbit per￾sists, and the iterates trace a small smooth invariant curve around P∗ (the rotation numbers θ… view at source ↗
Figure 3
Figure 3. Figure 3: Approach to the impulse bound at ω = 1, R = 2, f = 0.4. 5. Birkhoff normal form and KAM tori This section establishes the existence of a positive-measure family of invariant Cantor tori around any elliptic non-sticking T-periodic orbit of (1.1)-(1.2) satisfying the standard non-resonance and twist conditions. Although the original system is piecewise smooth across the velocity zero set and the wall surface… view at source ↗
Figure 4
Figure 4. Figure 4: Near-equilibrium gallery at F = 1, ω = 1, R = 2, f = 0.4. Each row pairs a time series x(t) over four forcing periods (left) with the corresponding stroboscopic phase portrait of Φ (right), zoomed on the elliptic fixed point P∗. Rows (a)-(c) are non-sticking orbits in the elliptic island (the FP itself, a small KAM curve, and a larger KAM curve); the time series in all three rows have the same impact patte… view at source ↗
Figure 4
Figure 4. Figure 4: Near-equilibrium gallery at F = 1, ω = 1, R = 2, f = 0.4. The four panels are arranged so that the qualitative transition predicted by Theorem 5.6 can be read off the phase portraits at a glance: a single point at the fixed point, two nested closed curves at intermediate radii, and a two-dimensional cloud of stroboscopic iterates outside the KAM region. The time series alone do not distinguish the four cas… view at source ↗
Figure 5
Figure 5. Figure 5: Schematic of the state-space decomposition X = ΩNS∪ Ωdissip. Inside the non-sticking forward-invariant set ΩNS (white) the stroboscopic map Φ is exact symplectic by Theorem 3.3; nested Φ-invariant Cantor curves (KAM tori) surround each elliptic non￾sticking T-periodic orbit P∗ by Theorem 5.6, and a hyperbolic Cantor set sits in the homoclinic tangle of the bifurcating saddle S∗ by Theorem 6.4. The compleme… view at source ↗
Figure 5
Figure 5. Figure 5: Schematic of the state-space decomposition X = ΩNS ∪ Ωdissip. Inside the non-sticking forward-invariant set ΩNS (white) the stroboscopic map Φ is exact symplectic by Theorem 3.3; nested Φ-invariant Cantor curves (KAM tori) surround each elliptic non-sticking T-periodic orbit P∗ by Theorem 5.6, and a hyperbolic Cantor set sits in the homoclinic tangle of the bifurcating saddle S∗ by Theorem 6.4. The complem… view at source ↗
Figure 6
Figure 6. Figure 6: The N-particle vibro-impact system (9.1)-(9.2). Or￾dered point masses m1, m2, . . . , mN confined to [l, r], each sub￾ject to the same external forcing F cos(ωt) and Coulomb friction −f sgn( ˙xi). The leftmost particle reflects elastically at x = l and the rightmost at x = r; adjacent particles mi , mi+1 exchange mo￾mentum via the elastic-collision formula (9.2) when xi = xi+1. 9.2. The sign-preservation p… view at source ↗
Figure 6
Figure 6. Figure 6: The N-particle vibro-impact system (9.1)-(9.2). Lemma 9.1 (Algebra of binary collisions). The collision map (9.2) is a linear involution that preserves the total momentum mix˙ i+mi+1x˙ i+1 and the total kinetic energy (mix˙ 2 i + mi+1x˙ 2 i+1)/2. For equal masses mi = mi+1, it exchanges velocities. For unequal masses, the post-collision velocities are linear combinations of the pre￾collision velocities wit… view at source ↗
Figure 7
Figure 7. Figure 7: The elliptic non-sticking T-periodic orbit at f = 0.4 with the parameters (10.1). Left: x(t) over one period, show￾ing two transverse wall hits and no zero-velocity events. Right: phase-plane trajectory, with the stroboscopic fixed point (x∗, v∗) ≈ (0.1003, 0.5419) marked by a red star. The numerical Jacobian Φ′ (P∗), computed by central differences with step 10−6 , has (10.4) Φ′ (P∗) ≈  +1.12010470 −3.28… view at source ↗
Figure 7
Figure 7. Figure 7: The elliptic non-sticking T-periodic orbit at f = 0.4 with the parame￾ters (10.1). Left: x(t) over one period, showing two transverse wall hits and no zero-velocity events. Right: phase-plane trajectory, with the stroboscopic fixed point (x∗, v∗) ≈ (0.1003, 0.5419) marked by a red star. The numerical Jacobian Φ′ (P∗), computed by central differences with step 10−6 , has (10.4) Φ′ (P∗) ≈  +1.12010470 −3.28… view at source ↗
Figure 8
Figure 8. Figure 8: shows the stroboscopic phase portrait in a small neighborhood of P∗. Initial conditions on a polar grid of radius up to 0.05 are iterated under Φ for 300 periods. Closed invariant curves and resonant island chains are visible, providing a direct picture of the KAM Cantor family of Theorem 5.6 view at source ↗
Figure 8
Figure 8. Figure 8: shows the stroboscopic phase portrait in a small neighborhood of P∗. Initial conditions on a polar grid of radius up to 0.05 are iterated under Φ for 300 periods. Closed invariant curves and resonant island chains are visible, providing a direct picture of the KAM Cantor family of Theorem 5.6. Three concentric layers of structure organize the picture. At the center, the elliptic fixed point P∗ is surrounde… view at source ↗
Figure 9
Figure 9. Figure 9: Saddle-center branch of the elliptic non-sticking T￾periodic orbit of Subsection 10.2, traced from f = 0.4 up to its numerical fold at ffold ≈ 0.467 for (F, ω, R) = (1, 1, 2). Left: posi￾tion x∗ of the elliptic fixed point along the branch. Right: rotation number θ∗/(2π) along the same branch. The horizontal dotted lines mark the resonances 1/4, 1/5, 1/6. The vertical red dashed line marks f = ffold, where… view at source ↗
Figure 9
Figure 9. Figure 9: shows the result. The position x∗(f) along the branch is monotone increasing, while the rotation number θ∗/(2π) decreases monotonically and tends to zero as f → f − fold, in agreement with the local saddle-center scenario of Theo￾rem 4.8 (c). The two folds, at fsc ≈ 0.34790 and ffold ≈ 0.467, illustrate the prolif￾eration of non-sticking T-periodic orbits with distinct impact patterns recorded in Subsectio… view at source ↗
Figure 10
Figure 10. Figure 10: SALI map at f = 0.4, zoomed on the elliptic island around the fixed point of Subsection 10.2 (marked by a black star). Blue: regular orbits on KAM curves (slow power-law SALI decay); red/yellow: chaotic non-sticking orbits in the homoclinic web of Theorem 6.4 (exponential SALI decay); grey: orbits that eventu￾ally stick and lie in Ωdissip. The contiguous blue island reflects the Cantor family of invariant… view at source ↗
Figure 10
Figure 10. Figure 10: SALI map at f = 0.4, zoomed on the elliptic island around the fixed point of Subsection 10.1 (marked by a black star). 61 [PITH_FULL_IMAGE:figures/full_fig_p061_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Basin partition X = ΩNS ∪ Ωdissip at f = 0.4 (zoom on the elliptic island), computed on a 70 × 70 grid with 20 strobo￾scopic iterates per initial condition. Blue: orbits in the non-sticking invariant set ΩNS. Gold: orbits in the dissipative subset Ωdissip. Numerical basin entropies (box size 5): Sb ≈ 0.036, Sbb ≈ 0.269. The blue ΩNS structure consists of a compact island surrounding the elliptic fixed poi… view at source ↗
Figure 11
Figure 11. Figure 11: Basin partition X = ΩNS ∪Ωdissip at f = 0.4, zoomed tightly on the elliptic island around P∗ (red star), computed on a 60 × 60 grid in the box {(x, v) : |x − x∗| ≤ 0.10, |v − v∗| ≤ 0.07} with 25 stroboscopic iterates per initial condition. Blue: orbits in the non-sticking invariant set ΩNS (occupying 22.6% of the grid). Gold: orbits in the dissipative subset Ωdissip. The dominant connected blue com￾ponent… view at source ↗
Figure 12
Figure 12. Figure 12: Two-parameter map of the existence of non-sticking T-periodic orbits in (f, R). White: no non-sticking T-periodic orbit detected; blue: elliptic only; gold: saddle only; purple: coex￾istence. The dashed red curve fsc(R) = (8 + 4R − 4π)/π2 is the saddle-center bifurcation locus predicted by Theorem 4.8 for the symmetric branch with one wall hit and one turning point per half￾period. Numerically detected or… view at source ↗
Figure 12
Figure 12. Figure 12: Two-parameter map of the existence of non-sticking T-periodic orbits in (f, R) at F = ω = 1. Region shadings come from the analytical formulas of Theorem 4.8 on a 220×220 background grid. White: no symmetric-branch orbit predicted; blue: elliptic-only region (where fsc(R) ≤ 0, so only the elliptic non-sticking orbit exists below the im￾pulse bound); purple: coexistence region (where fsc(R) < f < fimp, so … view at source ↗
Figure 13
Figure 13. Figure 13: Two coexisting T-periodic non-sticking orbits at F = 3, f = 0.4, ω = 1, R = 2, found by grid-Newton on Φ. Top row: elliptic, (x∗, v∗) = (+0.4064, −0.6708), tr Φ′ = −0.968. Bot￾tom row: saddle, (+0.1992, −1.4002), tr Φ′ = +11.765. Each row pairs the time series x(t) over two forcing periods (left, walls at x = ±1 dashed) with the phase-plane trajectory (right, fixed point marked by a red star). Right-wall … view at source ↗
Figure 13
Figure 13. Figure 13: Two coexisting T-periodic orbits at F = 3, f = 0.4, ω = 1, R = 2, found by grid-Newton on Φ. Each row pairs the time series x(t) over two forcing periods (left, walls at x = ±1 dashed) with the phase-plane trajectory (right, fixed point marked by a red star). Top row (blue): stable focus, (x e ∗ , ve ∗ ) = (+0.4064, −0.6708), tr Φ′ = −0.968, det Φ′ = +0.578. Bottom row (purple): dissipative saddle, (x s ∗… view at source ↗
Figure 14
Figure 14. Figure 14: Asymptotic regime gallery at F ∈ {1.5, 3.0, 4.5, 6.0}, ω = 1, R = 2, f = 0.4. Each panel shows four forcing periods of x(t) in the asymptotic regime; the transient (when present) is dropped before the time origin of the panel. Top-left (F = 1.5) and bottom-right (F = 6.0): initial datum at the elliptic non￾sticking T-periodic fixed point of Φ, no transient dropped. Top￾right (F = 3.0) and bottom-left (F =… view at source ↗
Figure 14
Figure 14. Figure 14: Asymptotic regime gallery, part I: F ∈ {1.5, 3.0, 4.5}, ω = 1, R = 2, f = 0.4. Each panel shows four forcing periods of x(t) along the elliptic T-periodic orbit located by Newton iteration on Φ at the corresponding F, with the fixed point and the stability data tr Φ′ , det Φ′ recorded in the panel title. The forcing F cos ωt is overlaid (light blue, rescaled to fit the window); walls at x = ±1 are dashed;… view at source ↗
Figure 15
Figure 15. Figure 15: Proliferation of T-periodic non-sticking orbits in the forcing-amplitude sweep at ω = 1, R = 2, f = 0.4. Each bar reports the number of distinct T-periodic non-sticking orbits found by grid-Newton continuation of Φ at the corresponding value of F, stacked into elliptic (|tr Φ′ | < 2, blue) and saddle (|tr Φ′ | > 2, purple) sub-counts; the integer above each bar is the total. The dashed orange line marks t… view at source ↗
Figure 15
Figure 15. Figure 15: extends the gallery into the wall-bouncing regime at F = 6, 12, 50. At F = 6 the system still admits a periodic elliptic orbit at (x∗, v∗) = (+0.0267, −0.4816) with tr Φ′ = −0.665, det Φ′ = +0.538 (located by Newton iteration), and the time￾series and phase-plane panels show the periodic attractor. At F = 12 and F = 50 no T-periodic elliptic orbit was found by grid-Newton with the seed sets used here, and… view at source ↗
Figure 16
Figure 16. Figure 16: Discontinuity-induced regimes. Left: time series at F = 0.55, f = 0.4 (well below the universal impulse bound fimp = 2F/π ≈ 0.350 at F = 0.55, but with F small relative to f, so the orbit lacks energy to sustain wall-bouncing motion), IC (0, 1.5). The orbit makes one right-wall hit at t ≈ 0.5, one left-wall hit at t ≈ 2.4, then a long sequence of small turnings (green dots) as the trajectory decays toward… view at source ↗
Figure 16
Figure 16. Figure 16: Continuation in F of T-periodic orbits at ω = 1, R = 2, f = 0.4. Each bar reports the number of distinct T-periodic orbits found by grid-Newton on Φ at the corresponding value of F, stacked into elliptic (|tr Φ′ | < 2, blue) and saddle (|tr Φ′ | > 2, purple) sub-counts; the integer above each bar is the total. The dashed orange line marks the predicted saddle-center fold Fsc(f = 0.4) ≈ 0.887 obtained from… view at source ↗
Figure 17
Figure 17. Figure 17: Discontinuity-induced regimes at ω = 1, R = 2, f = 0.4. Panel (a): x(t) at F = 0.55 from initial condition (0, 1.5); one right-wall hit (▽), one left-wall hit (△), one transverse turning (◦), and three sticking-onset/release pairs (brown squares and diamonds) over t ∈ [0, 12]. The decaying-amplitude pat￾tern visible after t ≈ 3 is the colloquial “chattering toward sticking” regime, dis￾tinguished from gen… view at source ↗
read the original abstract

The forced vibro-impact oscillator with Amonton-Coulomb friction and elastic walls was shown by Gendelman et al. (2019) to exhibit a coexistence of Hamiltonian stability islands and dissipative attractors in a single phase space. We provide a complete mathematical analysis of this phenomenon. We prove global well-posedness of the associated Filippov flow and construct a global lift to a piecewise smooth Hamiltonian system on a covering manifold. On the maximal forward-invariant non-sticking set, we show that the time-$T$ stroboscopic map is exact symplectic, within the formalism of symplectic dynamics. We derive a closed-form existence equation for symmetric $T$-periodic orbits and establish a parameter-dependent saddle-center bifurcation at $f_{\rm sc}(F,\omega,R)$, correcting a universality claim in prior work. Using Moser's twist theorem, we prove the existence of invariant Cantor families (KAM tori) near elliptic non-sticking periodic orbits, while a Melnikov analysis yields hyperbolic dynamics conjugate to a Bernoulli shift near the associated saddle. We further show that any positive restitution defect or viscous damping destroys the conservative structure: elliptic periodic orbits persist but become asymptotically stable, replacing Hamiltonian islands by a single attracting basin. The approach extends to multi-particle systems with elastic collisions, where a symplectic structure and higher-dimensional KAM tori are obtained. A computer-assisted proof verifies the existence and ellipticity of a non-sticking periodic orbit at a specific parameter point.

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.

Referee Report

4 major / 3 minor

Summary. The paper provides a mathematical analysis of the forced vibro-impact oscillator with Amonton-Coulomb friction, proving global well-posedness of the Filippov flow, constructing a global lift to a piecewise smooth Hamiltonian system on a covering manifold, and showing that the time-T stroboscopic map is exact symplectic on the maximal forward-invariant non-sticking set. It derives a closed-form existence equation for symmetric T-periodic orbits, establishes a parameter-dependent saddle-center bifurcation correcting prior universality claims, applies Moser's twist theorem to prove KAM tori near elliptic non-sticking periodic orbits, uses Melnikov analysis for hyperbolic dynamics near saddles, shows that restitution defects or viscous damping destroy the conservative structure, extends the approach to multi-particle systems, and includes a computer-assisted proof of existence and ellipticity for a specific non-sticking periodic orbit.

Significance. If the lift to the covering manifold and the exact symplecticity of the stroboscopic map are rigorously established, this work would be significant for providing a symplectic framework that explains the coexistence of Hamiltonian stability islands and dissipative attractors in a single non-smooth system with Coulomb friction. The closed-form bifurcation equation, correction to prior work, computer-assisted verification, and extension to multi-particle systems with higher-dimensional KAM tori represent concrete advances. The approach bridges non-smooth dynamics with classical symplectic geometry tools.

major comments (4)
  1. [Global lift and covering manifold construction] The global lift to a piecewise smooth Hamiltonian system on the covering manifold (central to §2-3) is load-bearing for the exact symplecticity claim. The manuscript must explicitly verify that this construction preserves the Filippov convexification at v=0 and correctly excises grazing trajectories from the non-sticking set; without this, the pulled-back symplectic form is not guaranteed to be preserved.
  2. [Symplecticity of the stroboscopic map] The proof that the time-T stroboscopic map is exact symplectic on the maximal forward-invariant non-sticking set (following the lift construction) requires an explicit calculation showing preservation of the symplectic form under the piecewise flow; the current outline invokes the formalism but lacks the detailed pullback verification needed to apply Moser's twist theorem.
  3. [Computer-assisted verification] The computer-assisted proof of existence and ellipticity of a non-sticking periodic orbit (final section) reports no error estimates, interval arithmetic bounds, or the specific parameter values (F, ω, R) used. These are required to confirm the non-degeneracy condition for Moser's theorem and to make the verification reproducible.
  4. [Periodic orbits and bifurcation analysis] The closed-form existence equation for symmetric T-periodic orbits and the saddle-center bifurcation locus f_sc(F, ω, R) are derived from the system equations, but the manuscript should include a direct comparison showing how this corrects the universality claim in Gendelman et al. (2019) without hidden parameter restrictions.
minor comments (3)
  1. Clarify the notation for the restitution coefficient and the friction parameter R consistently from the introduction onward.
  2. Add explicit theorem citations (e.g., specific statement of Moser's twist theorem and the Melnikov version employed) in the relevant sections.
  3. Improve figure captions for phase portraits to indicate the non-sticking set boundaries and the location of the verified periodic orbit.

Simulated Author's Rebuttal

4 responses · 0 unresolved

We thank the referee for the careful and constructive report. We address each major comment point by point below, providing clarifications where the manuscript already contains the required elements and committing to explicit additions or expansions where needed to improve rigor and reproducibility.

read point-by-point responses

  1. Referee: [Global lift and covering manifold construction] The global lift to a piecewise smooth Hamiltonian system on the covering manifold (central to §2-3) is load-bearing for the exact symplecticity claim. The manuscript must explicitly verify that this construction preserves the Filippov convexification at v=0 and correctly excises grazing trajectories from the non-sticking set; without this, the pulled-back symplectic form is not guaranteed to be preserved.

    Authors: We agree that explicit verification is desirable for clarity. The lift in Sections 2–3 is constructed precisely so that the vector field at v=0 lies in the Filippov convex hull, and the non-sticking set is defined as the maximal forward-invariant set excluding grazing points (where velocity reaches zero with vanishing acceleration). To make this fully transparent, we will insert a short lemma in Section 2 that verifies preservation of the convexification and excision of grazings, confirming that the pulled-back symplectic form remains well-defined and invariant under the lifted flow. revision: yes

  2. Referee: [Symplecticity of the stroboscopic map] The proof that the time-T stroboscopic map is exact symplectic on the maximal forward-invariant non-sticking set (following the lift construction) requires an explicit calculation showing preservation of the symplectic form under the piecewise flow; the current outline invokes the formalism but lacks the detailed pullback verification needed to apply Moser's twist theorem.

    Authors: We will add the requested explicit pullback calculation in the revised Section 3. On each smooth piece the flow is Hamiltonian, hence symplectic; across the switching surfaces the lift ensures that the differential of the stroboscopic map preserves the form because the impact map is exact symplectic in the covering coordinates. This detailed verification will be written out step by step, directly supporting the subsequent application of Moser's twist theorem. revision: yes

  3. Referee: [Computer-assisted verification] The computer-assisted proof of existence and ellipticity of a non-sticking periodic orbit (final section) reports no error estimates, interval arithmetic bounds, or the specific parameter values (F, ω, R) used. These are required to confirm the non-degeneracy condition for Moser's theorem and to make the verification reproducible.

    Authors: We accept this criticism. The revised final section will state the concrete parameter triple (F, ω, R) at which the orbit is computed, report the interval-arithmetic bounds employed, and supply rigorous a-posteriori error estimates. These estimates will be shown to guarantee a strictly positive lower bound on the twist coefficient, thereby confirming the non-degeneracy hypothesis of Moser's theorem and rendering the computer-assisted proof fully reproducible. revision: yes

  4. Referee: [Periodic orbits and bifurcation analysis] The closed-form existence equation for symmetric T-periodic orbits and the saddle-center bifurcation locus f_sc(F, ω, R) are derived from the system equations, but the manuscript should include a direct comparison showing how this corrects the universality claim in Gendelman et al. (2019) without hidden parameter restrictions.

    Authors: We will expand the bifurcation discussion to include an explicit side-by-side comparison. We will demonstrate that the universality statement in Gendelman et al. (2019) holds only when the forcing amplitude remains below the threshold that keeps all orbits non-sticking; our closed-form equation f_sc(F, ω, R) = 0 gives the exact locus in the full three-dimensional parameter space without that restriction. The added paragraph will contain both the analytical argument and a brief numerical illustration of the difference. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit construction and external theorems.

full rationale

The paper constructs a global lift of the Filippov flow to a piecewise smooth Hamiltonian system on a covering manifold, then directly proves that the time-T stroboscopic map restricts to an exact symplectic diffeomorphism on the maximal forward-invariant non-sticking set. The closed-form existence equation for symmetric T-periodic orbits and the saddle-center bifurcation locus are obtained by solving the piecewise linear system equations. Moser's twist theorem and Melnikov analysis are applied as external results to the constructed map, with no reduction of these claims to quantities defined by the authors' own prior fits or self-citations. The extension to multi-particle systems and computer-assisted verification of a specific orbit are likewise independent of the input data. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on standard results from nonsmooth dynamical systems and symplectic geometry rather than new fitted constants or postulated entities.

axioms (3)
  • standard math Moser's twist theorem applies to the exact symplectic stroboscopic map near elliptic periodic orbits
    Invoked directly to obtain Cantor families of KAM tori.
  • standard math Melnikov method detects transverse homoclinics in the hyperbolic dynamics near the saddle
    Used to conclude conjugacy to a Bernoulli shift.
  • domain assumption The Filippov regularization yields a globally well-posed flow for the Coulomb friction law
    Stated as proved in the paper and required for the lift construction.

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