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arxiv: 2604.04769 · v1 · submitted 2026-04-06 · ❄️ cond-mat.stat-mech · nlin.AO

Analytical approach to subsystem resetting in generalized Kuramoto models

Rupak Majumder , Anish Acharya , Shamik Gupta This is my paper

Pith reviewed 2026-05-10 19:36 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech nlin.AO
keywords subsystem resettingKuramoto modelsynchronization transitioncontinued fractionorder parameternonequilibrium stationary statesphase transitionsstochastic resetting
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The pith

Subsystem resetting yields self-consistent equations for the order parameter of non-reset oscillators in generalized Kuramoto models.

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

The paper derives a continued-fraction method that produces closed equations for the stationary synchronization level when only a chosen subset of oscillators undergoes stochastic phase resetting. These equations hold for both noisy and deterministic dynamics and for couplings containing any number of harmonics. A reader would care because the results show that varying the reset rate, the fraction of reset oscillators, and the reset pattern can move, eliminate, or multiply the points at which collective order appears, including cases where order returns after disappearing. The framework recovers known special cases such as the noiseless model with Lorentzian frequencies while extending the analysis to broader settings.

Core claim

Using a continued-fraction expansion, we obtain self-consistent equations for the stationary-state order parameter of the non-reset subsystem in generalized Kuramoto models with subsystem resetting. These equations apply to both noisy and noiseless cases and arbitrary interaction harmonics, and they predict shifts, suppression, re-entrant behavior, and restructuring of phase boundaries in synchronization transitions.

What carries the argument

The continued-fraction closure applied to the stationary distribution of the non-reset oscillators, which closes the hierarchy and yields self-consistent equations for their collective order parameter under partial resetting.

If this is right

  • Subsystem resetting can shift or suppress the synchronization transition by tuning the resetting rate and the size of the reset subsystem.
  • Re-entrant synchronization appears for certain combinations of parameters, with order present in an intermediate range of coupling strengths.
  • The topology of phase boundaries in the space of coupling strength and frequency spread changes with the reset configuration.
  • The approach recovers exact known results for the noiseless Kuramoto model with Lorentzian frequencies as a special case while extending to noisy dynamics and general harmonics.

Where Pith is reading between the lines

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

  • The same partial-resetting construction could be tested on networks with heterogeneous reset rules to see whether additional collective states become accessible.
  • Engineering protocols that reset only a controllable fraction of units might stabilize or destabilize synchronization in applications such as power-grid models or biological oscillators.
  • Extending the continued-fraction method to time-periodic resetting schedules would reveal whether new dynamical regimes appear beyond the stationary analysis.

Load-bearing premise

The continued-fraction approximation that closes the equations for the phase distribution of the non-reset oscillators remains accurate when only a subset is reset and when frequencies and couplings take general forms.

What would settle it

Numerical integration of the underlying stochastic equations for a non-Lorentzian frequency distribution and multi-harmonic coupling, followed by direct comparison of the measured stationary order parameter against the value predicted by the self-consistent equations.

Figures

Figures reproduced from arXiv: 2604.04769 by Anish Acharya, Rupak Majumder, Shamik Gupta.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
read the original abstract

Stochastic resetting has emerged as a powerful mechanism for driving systems into nonequilibrium stationary states with tunable properties. While most existing studies focus on global resetting, where all degrees of freedom are simultaneously reset, recent work has shown that resetting only a subset of degrees of freedom (subsystem resetting) can qualitatively alter collective behavior in interacting many-body systems. In this work, we develop a general theoretical framework for analysing subsystem resetting in Kuramoto-type coupled-oscillator systems. Building on a continued-fraction approach, we derive self-consistent equations for the stationary-state order parameter of the non-reset subsystem, applicable to both noisy and noiseless dynamics and to models with arbitrary interaction harmonics. Using this framework, we systematically investigate how the stationary state and phase transitions depend on the resetting rate, the size of the reset subsystem, and the reset configuration. We show that subsystem resetting can shift or even suppress synchronization transitions, and can give rise to nontrivial features such as re-entrant behavior and restructuring of phase boundaries. In specific cases, including the noiseless Kuramoto model with a Lorentzian frequency distribution, our results recover known analytical predictions and extend them to more general settings. These results establish subsystem resetting as a versatile control protocol for engineering collective dynamics in nonequilibrium interacting systems.

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

2 major / 3 minor

Summary. The manuscript develops a continued-fraction analytical framework for subsystem resetting in generalized Kuramoto models. It derives self-consistent equations for the stationary order parameter of the non-reset subpopulation, applicable to noisy and noiseless dynamics with arbitrary interaction harmonics. The framework is then used to examine how resetting rate, reset-subsystem size, and reset configuration affect synchronization transitions, including shifts, suppression, re-entrant behavior, and restructuring of phase boundaries. Special cases such as the noiseless Kuramoto model with Lorentzian frequencies recover known analytical predictions.

Significance. If the derivations hold, the work supplies a general, extensible analytical tool for partial resetting in coupled-oscillator systems, extending global-resetting studies to tunable subsystems and arbitrary harmonics. This could enable systematic control of nonequilibrium collective states in physical and biological networks. Recovery of established limits provides partial validation, and the parameter-free structure in special cases is a methodological strength.

major comments (2)
  1. [§3] §3 (stationary Fokker-Planck equation for non-reset oscillators, after introduction of the resetting integral term): the continued-fraction closure for the Fourier coefficients is applied directly to the modified operator. The additional integral kernel arising from the reset-subsystem density does not obviously preserve the recursive structure that allows truncation or closure in the standard (non-reset) case; no explicit recursion relation or proof is supplied showing that the hierarchy remains closable. This assumption is load-bearing for the self-consistent order-parameter equations and is supported only by agreement in recovered limits.
  2. [§4] §4 (self-consistency condition for the order parameter): the final algebraic equations for the stationary r are obtained under the closed continued-fraction representation. Because the validity of that representation under partial resetting is not independently verified (e.g., by direct substitution back into the modified Fokker-Planck equation or by comparison with an alternative numerical solution of the hierarchy), the quantitative predictions for transition shifts and re-entrance rest on an unproven structural assumption.
minor comments (3)
  1. [Notation] Notation for the reset-subsystem density and the resetting kernel should be introduced once and used consistently; several symbols are redefined in passing between sections.
  2. [Figure 2] Figure 2 (phase diagrams): the re-entrant regions are visible but the boundaries are not labeled with the corresponding resetting-rate values; adding a few explicit curves or a legend would improve readability.
  3. [Abstract] The abstract states that 'self-consistent equations are derived' but does not display the key relation; inserting the final closed-form expression for the order parameter would make the central result immediately accessible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive evaluation of its significance. We address the two major comments point by point below, providing clarifications and committing to revisions that strengthen the presentation of the continued-fraction framework.

read point-by-point responses

  1. Referee: [§3] §3 (stationary Fokker-Planck equation for non-reset oscillators, after introduction of the resetting integral term): the continued-fraction closure for the Fourier coefficients is applied directly to the modified operator. The additional integral kernel arising from the reset-subsystem density does not obviously preserve the recursive structure that allows truncation or closure in the standard (non-reset) case; no explicit recursion relation or proof is supplied showing that the hierarchy remains closable. This assumption is load-bearing for the self-consistent order-parameter equations and is supported only by agreement in recovered limits.

    Authors: We thank the referee for identifying the need for an explicit demonstration that the resetting integral preserves the recursive structure. In the derivation, the reset-subsystem density is expressed via its own Fourier expansion in terms of the shared order parameter; the resulting integral kernel therefore contributes only to the effective drift term. When the full stationary Fokker-Planck operator (including this kernel) is projected onto the Fourier basis, the coefficients satisfy a three-term recurrence identical in form to the standard Kuramoto case, with the resetting rate and reset-subsystem size appearing as renormalized parameters in the recurrence coefficients. We will add the explicit recursion relation and the short algebraic steps leading to it in the revised §3, thereby removing reliance on limit-case agreement alone. revision: yes

  2. Referee: [§4] §4 (self-consistency condition for the order parameter): the final algebraic equations for the stationary r are obtained under the closed continued-fraction representation. Because the validity of that representation under partial resetting is not independently verified (e.g., by direct substitution back into the modified Fokker-Planck equation or by comparison with an alternative numerical solution of the hierarchy), the quantitative predictions for transition shifts and re-entrance rest on an unproven structural assumption.

    Authors: We agree that an independent check beyond recovery of known limits would increase in the quantitative predictions. In the revised manuscript we will (i) substitute the closed continued-fraction solution back into the stationary Fokker-Planck equation for the exactly solvable Lorentzian, noiseless case and verify that the residual vanishes identically, and (ii) add a short comparison, for a representative set of parameters, between the analytic order-parameter curves and a direct numerical truncation of the infinite hierarchy (retaining 20–30 modes). These additions will confirm that the closure remains valid under partial resetting. revision: yes

Circularity Check

0 steps flagged

Continued-fraction closure extended to subsystem resetting; no reduction to inputs by construction

full rationale

The derivation starts from the stationary Fokker-Planck equation augmented by a resetting term and applies the established continued-fraction representation for the non-reset subpopulation's stationary distribution. This representation is invoked as a building block from prior literature on standard Kuramoto models, then used to obtain self-consistent equations for the order parameter. Validation occurs through recovery of known analytical predictions in the noiseless Lorentzian limit and other special cases. No equation or step equates a derived quantity to a fitted parameter or prior result by definition; the framework remains self-contained once the closure assumption is granted, with the resetting modification treated as an additive integral operator whose effect is computed within the same recursive structure. Self-citations for the base method are not load-bearing in a circular sense, as the new protocol's consequences are independently explored via parameter variation and phase-boundary analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no explicit list of free parameters or invented entities; the approach implicitly relies on the validity of the continued-fraction representation for the stationary state under partial resetting.

axioms (1)
  • domain assumption The stationary distribution of the non-reset oscillators admits a continued-fraction expansion that closes into self-consistent equations for the order parameter.
    This is the central technical step stated in the abstract.

pith-pipeline@v0.9.0 · 5526 in / 1178 out tokens · 47086 ms · 2026-05-10T19:36:41.611705+00:00 · methodology

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Reference graph

Works this paper leans on

121 extracted references · 121 canonical work pages

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    (27) account for probability loss and gain due to resetting at rateλ

    The last two terms in Eq. (27) account for probability loss and gain due to resetting at rateλ. In the absence of theK 2 term, the only relevant order parameters arez 1,r ≡r 1,reiψ1,r andz 1,nr ≡r 1,nreiψ1,nr. Hence, until the end of this section, we will drop the subscript1for brevity. In the stationary state (st), we have∂P/∂t= 0in Eq. (27). To proceed,...

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    (63), we may compute the distribution of the oscillator angles (θ nr) of the 10 non-reset subsystem in the stationary state

    Stationary-State Distribution Once we obtain the solution ofγfrom Eq. (63), we may compute the distribution of the oscillator angles (θ nr) of the 10 non-reset subsystem in the stationary state. We start from the definition Pst(θnr) = Z +∞ −∞ dωrg(ωr) Z +∞ −∞ dωnrg(ωnr) Z 2π 0 dθr ×Pst(θr, ωr, θnr, ωnr).(64) Using the Fourier expansion ofP st(θr, ωr, θnr,...

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    Γ∗ 1,1 + Γ1,1Γ∗ 2,1 + 4π2 ∆∗ 1 + ∆1Γ∗ 2,1 1−Γ ∗ 2,1Γ2,1 # , (129) rst 2,rei2ψst 2,r = Z +∞ −∞ dωrg(ωr)

    Transition Points Let us now obtain the transition point of the order-disorder transition in presence of resetting. Note that Eq. (63) has the form of a self-consistent equationγ=F(γ). We are inter- ested in the following: Ifγ= 0is a solution of Eq. (63), does the equation also admit aγ̸= 0solution? Assuming there is only oneγ̸= 0solution possible in the ...

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    Before delving into the problem, let us first understand the situation physically

    Transition Points of Model II C We now move on to obtaining the transition points for the order-disorder transition corresponding to the two order pa- rametersz st 1,nr andz st 2,nr in the presence of subsystem resetting for the model II C. Before delving into the problem, let us first understand the situation physically. As it turns out that there is no ...

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    The threshold for the order-disorder transition inr st 1,nr is then determined by the conditionB 1 = 0

    +· · ·= 0, with the coefficients given by A1 = 2π2K1f b1 a∗ 1 ,(143) B1 = K1 2 " f a∗ 1 + ¯f c∗ 1 γ1 −4π 2f b2γ∗ 1 a∗ 1a∗ 2 + b3γ∗ 2 a∗ 1a∗ 3 + 4π2f b1 |a1|2 γ2 # .(144) Now, forα= 1/2, the quantityb 1 vanishes, leading to A1 = 0, and allowing for an incoherent solutionγ 1 = 0. The threshold for the order-disorder transition inr st 1,nr is then determined...

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    (143), here, the presence of a non-vanishing A2 term (here,b 2 ̸= 0forα= 1/2) indicates thatγ 2 = 0 is no longer a valid solution

    +· · ·= 0, with A2 = 2π2K2f b2 a∗ 2 ,(145) B2 =K 2 f a∗ 2 + ¯f c∗ 2 γ2 + 4π2f b1γ1 a∗ 1a∗ 2 − b3γ∗ 1 a∗ 2a∗ 3 − 4π2f b4 a∗ 2a∗ 4 γ∗ 2 .(146) In contrast to Eq. (143), here, the presence of a non-vanishing A2 term (here,b 2 ̸= 0forα= 1/2) indicates thatγ 2 = 0 is no longer a valid solution. This implies that for any finite λ, the quantityγ 2 can never be z...

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    Γ∗ 1,1 + Γ1,1Γ∗ 2,1 + 4π2 ∆∗ 1 + ∆1Γ∗ 2,1 1−Γ ∗ 2,1Γ2,1 # , (B2) Inr ≡Λ ∗ 1,2

    To obtain a nonzero solution ofγ 1 from Eq. (153), we must have det Kc 1 2 J−I = 0,(155) whereIis a2×2identity matrix. Now let the eigenvalues of the matrixJbeµ +(J)andµ −(J)withµ +(J)> µ −(J). As we increaseK 1, atK 1 =K 1,+ = 2/µ+(J), one eigenvalue of the matrix(K c 1J/2−I)becomes zero, thereby satisfying Eq. (155). From this point onward,γ 1 ̸= 0becom...

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