Fluctuation analysis for a randomly perturbed dynamical system with state-dependent impulse effects

Ashif Khan, Chetan D. Pahlajani

Pith reviewed 2026-05-13 04:56 UTC · model grok-4.3

classification 🧮 math.PR

keywords small noise expansionstate-dependent impulsesimpulsive dynamical systemsSkorohod spacerandom perturbationsradial componentfluctuation analysislimit theorems

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The pith

Small random perturbations of a planar impulsive system yield a small-noise expansion for the radial component with Skorohod error bounds.

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

The paper studies limit theorems for small random noise added to a planar dynamical system whose impulses fire at state-dependent hitting times of a switching surface. Using a simplified version in polar coordinates, it derives an asymptotic expansion for the radial distance process that remains valid over any fixed time interval. Rigorous remainder estimates are supplied in the Skorohod space of cadlag paths, which accommodates the jumps induced by the impulses. If the expansion holds, it supplies a concrete approximation for the fluctuating radial behavior without needing to resolve every individual impulse event.

Core claim

For the simplified planar impulsive system in polar coordinates, a small-noise expansion is obtained for the radial component together with rigorous error estimates in the Skorohod space of right-continuous functions with left limits, valid for any fixed time horizon.

What carries the argument

Small-noise expansion of the radial component equipped with Skorohod-space remainder bounds.

Load-bearing premise

The system can be reduced to polar coordinates and analyzed through a simplified planar example in which impulses occur exactly when a trajectory hits a fixed switching surface.

What would settle it

A direct numerical simulation of the stochastic impulsive system for successively smaller noise intensities that shows the radial trajectory deviating from the predicted expansion by more than the claimed Skorohod error bound on a fixed time interval.

read the original abstract

The principal aim of the present work is to explore limit theorems for small random perturbations of a planar impulsive dynamical system, where impulses occur at hitting times of a suitable switching surface, and are thus state-dependent. Working with a simplified example in polar coordinates, we obtain-for any fixed time horizon-a small noise expansion for the radial component, together with rigorous error estimates in the Skorohod space of right-continuous functions with left limits.

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

The paper delivers a small-noise expansion for the radial component of a state-dependent impulsive system with Skorohod error bounds, but the handling of random jump times looks like the key unverified step.

read the letter

The core contribution is a small-noise asymptotic expansion for the radial process in a planar impulsive dynamical system where impulses fire at state-dependent hitting times of a switching surface. They work in polar coordinates on a fixed time horizon and claim rigorous error estimates in the Skorohod space of cadlag functions. That setup is new enough: most existing small-noise results for impulsive systems use fixed jump times, so moving the jump epochs with the noise adds a genuine technical layer. The abstract indicates they adapt standard expansion techniques and topology arguments to this case, which is a reasonable next step for specialists in hybrid stochastic processes. Credit is due for framing the problem cleanly and stating the target space explicitly. The main soft spot is the control of the random hitting times themselves. Because the impulse instants depend on the full perturbed trajectory, any expansion of the radial component must track how much those times shift under small noise. The Skorohod metric can absorb some time warping, but only if the modulus of continuity of the hitting-time map stays uniform in the noise amplitude. The abstract gives no sign that this uniformity is proved, and the stress-test concern lands directly on that gap. Without seeing the actual estimates or the way they bound the difference between perturbed and unperturbed hitting times, it is impossible to tell whether the claimed error bounds hold. The paper is aimed at researchers already working on limit theorems for perturbed impulsive or hybrid systems. A reader who needs concrete expansions in Skorohod space for state-dependent jumps could extract useful techniques if the details check out. It is narrow, so it will not move the broader field, but the claim is specific enough to merit referee time. I would send it to reviewers familiar with Skorohod topology and impulsive processes rather than desk-reject it.

Referee Report

1 major / 0 minor

Summary. The manuscript develops limit theorems for small random perturbations of a planar impulsive dynamical system in which impulses occur at state-dependent hitting times of a switching surface. Working in polar coordinates on a simplified example, the authors derive a small-noise expansion for the radial component together with rigorous error bounds in the Skorohod space of cadlag functions, valid on any fixed time horizon.

Significance. If the claimed Skorohod-space error estimates are fully rigorous, the result would supply a concrete fluctuation analysis for impulsive systems whose reset times depend on the state trajectory. Such systems arise in applications with threshold crossings, and the polar-coordinate reduction plus Skorohod topology appear well-chosen to handle the resulting discontinuities. The fixed-horizon setting keeps the technical scope manageable while still yielding uniform-in-time control.

major comments (1)

[Abstract / statement of main result] The central claim requires that the random, path-dependent hitting times of the switching surface remain sufficiently close to their deterministic counterparts so that the induced time shifts are absorbed by the Skorohod metric. The abstract asserts rigorous error estimates but supplies no indication of the modulus-of-continuity bound on these hitting-time differences that is uniform in the noise amplitude. Without such control, the expansion for the radial process cannot be transferred to the Skorohod topology; this point is load-bearing for the main theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the key technical point regarding control of the state-dependent hitting times. We address this comment directly below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract / statement of main result] The central claim requires that the random, path-dependent hitting times of the switching surface remain sufficiently close to their deterministic counterparts so that the induced time shifts are absorbed by the Skorohod metric. The abstract asserts rigorous error estimates but supplies no indication of the modulus-of-continuity bound on these hitting-time differences that is uniform in the noise amplitude. Without such control, the expansion for the radial process cannot be transferred to the Skorohod topology; this point is load-bearing for the main theorem.

Authors: We agree that uniform control on the difference between the random hitting times and their deterministic counterparts is essential for transferring the radial expansion to the Skorohod topology. The manuscript derives this control by combining the Lipschitz continuity of the deterministic flow with the transversality assumption on the switching surface; the resulting bound shows that the hitting-time discrepancy is O(ε) in probability, uniformly over the fixed time horizon. This rate is compatible with the Skorohod metric because the latter is insensitive to small time reparametrizations, and the error estimates between jumps are obtained conditionally on the controlled time shifts. The abstract does not currently highlight this modulus, which is a valid observation. We will revise the abstract to state explicitly that the hitting-time differences admit a modulus of continuity uniform in the noise amplitude, thereby clarifying the load-bearing step. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from direct stochastic analysis of the perturbed system

full rationale

The paper claims a small-noise expansion for the radial component of a planar impulsive system (with state-dependent impulses at hitting times of a switching surface) together with Skorohod-space error bounds on a fixed horizon. No equations, definitions, or steps in the provided abstract or description reduce the claimed expansion to a fitted parameter, a self-citation chain, an ansatz smuggled via prior work, or a renaming of a known result. The analysis is presented as a direct limit theorem for the randomly perturbed dynamical system; the reader's assessment of score 1.0 is consistent with the absence of any load-bearing self-referential step. The skeptic concern about uniform control of random hitting times is a question of proof completeness, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Insufficient information in the abstract to identify free parameters, axioms, or invented entities. The work relies on standard concepts from stochastic processes and dynamical systems theory.

pith-pipeline@v0.9.0 · 5361 in / 1018 out tokens · 62266 ms · 2026-05-13T04:56:45.093507+00:00 · methodology

discussion (0)

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Works this paper leans on

38 extracted references · 38 canonical work pages

[1]

Yu. N. Blagoveshchenskii , date-added =. Diffusion processes depending on a small parameter , volume =. Theory Probab. Appl. , number =

work page
[2]

A. F. Filippov , date-added =. Differential

work page
[3]

2026 , bdsk-url-1 =

Random perturbations of systems with periodic impulse effects , volume =. 2026 , bdsk-url-1 =. arXiv , author =:2603.22676 , journal =

work page arXiv 2026
[4]

Francis , date-added =

Tongwen Chen and Bruce A. Francis , date-added =. Optimal sampled-data control systems , year =

work page
[5]

Nonlinear Phenomena in Power Electronics , year =

work page
[6]

Skorokhod and Frank C

Anatoli V. Skorokhod and Frank C. Hoppensteadt and Habib D. Salehi , date-added =. Random perturbation methods with applications in science and engineering , volume =

work page
[7]

Continuous Martingales and Brownian Motion , year =

Daniel Revuz and Marc Yor , date-added =. Continuous Martingales and Brownian Motion , year =

work page
[8]

Stochastic Differential Equations , year =

Bernt Oksendal , date-added =. Stochastic Differential Equations , year =

work page
[9]

Hale , date-added =

Jack K. Hale , date-added =. Ordinary Differential Equations , year =

work page
[10]

W. M. Haddad and V. Chellaboina and S. G. Nersesov , date-added =. Impulsive and hybrid dynamical systems , year =

work page
[11]

Noise-induced phenomena in slow-fast dynamical systems: A sample-paths approach , year =

Nils Berglund and Barbara Gentz , date-added =. Noise-induced phenomena in slow-fast dynamical systems: A sample-paths approach , year =

work page
[12]

Ethier and Thomas G

Stewart N. Ethier and Thomas G. Kurtz , date-added =. Markov Processes: Characterization and Convergence , year =

work page
[13]

Convergence of probability measures , year =

Patrick Billingsley , date-added =. Convergence of probability measures , year =

work page
[14]

Small noise and long time phase diffusion in stochastic limit cycle oscillators , volume =

Giambattista Giacomin and Christophe Poquet and Assaf Shapira , date-added =. Small noise and long time phase diffusion in stochastic limit cycle oscillators , volume =. Journal of Differential Equations , pages =

work page
[15]

Large Deviations Techniques and Applications , year =

Amir Dembo and Ofer Zeitouni , date-added =. Large Deviations Techniques and Applications , year =

work page
[16]

Discontinuous dynamical systems:

Jorge Cort\'es , date-added =. Discontinuous dynamical systems:. IEEE Control Systems Magazine , month =

work page
[17]

di Bernardo and C

M. di Bernardo and C. J. Budd and A. R. Champneys and P. Kowalczyk , date-added =. Piecewise-smooth Dynamical Systems , year =

work page
[18]

Bressloff and James N

Paul C. Bressloff and James N. MacLaurin , date-added =. A Variational Method for analyzing stochastic limit cycle oscillators , volume =. SIAM J. Applied Dynamical Systems , number =

work page
[19]

D. K. Arrowsmith and C. M. Place , date-added =. Dynamical Systems: Differential equations, maps and chaotic behaviour , year =

work page
[20]

Brownian Motion and Stochastic Calculus , volume =

Ioannis Karatzas and Steven Shreve , date-added =. Brownian Motion and Stochastic Calculus , volume =

work page
[21]

Gemmer , date-added =

Kaitlin Hill and Jessica Zanetell and John A. Gemmer , date-added =. Physica D , title =

work page
[22]

M. R. Jeffrey and D. J. W. Simpson , date-added =. Non-. Nonlinear Dynamics , pages =

work page
[23]

Nonsmooth Mechanics: Models, Dynamics and Control , year =

Bernard Brogliato , date-added =. Nonsmooth Mechanics: Models, Dynamics and Control , year =

work page
[24]

Jeffrey , date-added =

Mike R. Jeffrey , date-added =. Modeling with nonsmooth dynamics , year =

work page
[25]

Multiple Time Scale Dynamics , year =

Christian Kuehn , date-added =. Multiple Time Scale Dynamics , year =

work page
[26]

Pavliotis and Andrew M

Grigorios A. Pavliotis and Andrew M. Stuart , date-added =. Multiscale Methods: Averaging and Homogenization , year =

work page
[27]

Weiss and Edgar Knobloch , date-added =

Jeffrey B. Weiss and Edgar Knobloch , date-added =. A stochastic return map for stochastic differential equations , volume =. Journal of Statistical Physics , number =

work page
[28]

Stochastic-process limits , year =

Ward Whitt , date-added =. Stochastic-process limits , year =

work page
[29]

Sanfelice and Andrew R

Rafal Goebel and Ricardo G. Sanfelice and Andrew R. Teel , date-added =. Hybrid dynamical systems: Modeling, stability and robustness , year =

work page
[30]

Grizzle and Gabriel Abba and Franck Plestan , date-added =

Jesse W. Grizzle and Gabriel Abba and Franck Plestan , date-added =. Asymptotically stable walking for biped robots: Analysis via systems with impulse effects , volume =. IEEE Transactions on Automatic Control , month =

work page
[31]

Freidlin and Alexander D

Mark I. Freidlin and Alexander D. Wentzell , date-added =. Random Perturbations of Dynamical Systems , year =

work page
[32]

Freidlin and Richard B

Mark I. Freidlin and Richard B. Sowers , date-added =. A comparison of homogenization and large deviations, with applications to wavefront propagation , volume =. Stochastic Processes and their Applications , pages =

work page
[33]

William Clark and Anthony Bloch and Leonardo Colombo , date-added =. A. Mathematical Control and Related Fields , month =

work page
[34]

Spectral theory for random

Manon Baudel and Nils Berglund , date-added =. Spectral theory for random. SIAM J. Math. Anal. , number =

work page
[35]

On the inverse Gaussian distribution function , volume =

Shuster, Jonathan , journal =. On the inverse Gaussian distribution function , volume =

work page
[36]

Large deviations and the Strassen theorem in H

Baldi, Paolo and Arous, G Ben and Kerkyacharian, G. Large deviations and the Strassen theorem in H. Stochastic processes and their applications , number =

work page
[37]

Fernique, Xavier , journal =. Int

work page
[38]

Intermediate spaces, Gaussian probabilities and exponential tightness , volume =

Baldi, Paolo , journal =. Intermediate spaces, Gaussian probabilities and exponential tightness , volume =

work page