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arxiv: 2605.03587 · v1 · submitted 2026-05-05 · 🌊 nlin.CD

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Gr\"unwald--Letnikov Memory Truncation in a Fractional Duffing Oscillator: Coherence Loss and Effective Delay Complexity

Mattia Coccolo
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Pith reviewed 2026-05-09 15:39 UTC · model grok-4.3

classification 🌊 nlin.CD
keywords Grünwald-Letnikov derivativefractional Duffing oscillatormemory truncationcoherence lossdelay complexityfractional calculusnonlinear dynamicsfinite memory kernel
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The pith

Truncating the Grünwald-Letnikov memory in a fractional Duffing oscillator produces effective delay dynamics whose spectral structure sets both coherence loss and delay complexity.

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

The paper treats truncation of the infinite Grünwald-Letnikov memory not as a numerical shortcut but as a genuine change to the dynamical system. It defines a coherence-loss time by comparing full-memory and truncated trajectories, then maps how the minimum memory length needed to keep the motion coherent varies with forcing amplitude and fractional order. The resulting thresholds turn out to be strongly non-monotonic. From the truncated kernel a local characteristic equation is derived; a one-delay trial yields a formal negative delay, which is structurally inadequate, so the kernel is instead expressed as a sum of positive-delay exponential modes. The smallest number of such modes that achieve a target spectral accuracy supplies an operational measure of delay complexity.

Core claim

The truncated GL kernel emerges as an intermediate object between distributed fractional memory and delay-type dynamics, with a local spectral structure that controls both coherence loss and effective delay complexity.

What carries the argument

The truncated Grünwald-Letnikov kernel, treated as a finite-memory modification whose local spectrum determines both the coherence-loss time and the minimal number of positive-delay modes (r_min) needed for spectral accuracy.

If this is right

  • The memory length required to preserve coherence depends non-monotonically on forcing amplitude, fractional order, and nonlinear sensitivity.
  • A single causal delay cannot represent the truncated kernel, because the minimal one-delay fit produces a negative delay.
  • A positive-delay exponential representation allows definition of r_min, the fewest modes needed to reach a prescribed spectral accuracy.
  • The local spectrum of the finite-memory kernel governs both the onset of coherence loss and the complexity of its delay approximation.

Where Pith is reading between the lines

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

  • If r_min can be computed cheaply for other fractional kernels, it could serve as a general diagnostic for when truncation is safe versus when it injects spurious delay-like behavior.
  • The non-monotonic memory thresholds suggest that adaptive, state-dependent truncation lengths might reduce computational cost while preserving long-term coherence in simulations.
  • Viewing truncation as a system modification rather than an error opens the possibility of designing finite-memory fractional controllers whose stability can be analyzed via the same local spectral tools.

Load-bearing premise

Direct trajectory comparisons produce a robust coherence-loss time that identifies critical truncation thresholds independently of numerical artifacts or initial conditions.

What would settle it

Repeating the coherence-loss extraction with a different integrator step size, a different history function, or a different numerical scheme for the same parameters and finding large shifts in the reported thresholds would falsify the robustness claim.

Figures

Figures reproduced from arXiv: 2605.03587 by Mattia Coccolo.

Figure 1
Figure 1. Figure 1: Direct comparison between the full-memory GL Duffing oscillator and its truncated view at source ↗
Figure 2
Figure 2. Figure 2: Amplitude–phase diagnostic of coherence loss for a representative sensitive case, with view at source ↗
Figure 3
Figure 3. Figure 3: Coherence-loss map and critical memory threshold in the ( view at source ↗
Figure 4
Figure 4. Figure 4: Complementary coherence-loss analysis in the ( view at source ↗
Figure 5
Figure 5. Figure 5: Local spectral interpretation of the truncated GL dynamics for view at source ↗
Figure 6
Figure 6. Figure 6: Positive-delay complexity of the truncated GL kernel in the ( view at source ↗
read the original abstract

We investigate the dynamical and analytical consequences of truncating the Gr\"unwald--Letnikov memory term in a fractional Duffing oscillator. The truncated memory is treated not merely as a computational approximation, but as a finite-memory modification of the underlying dynamical system. We define a coherence-loss time from direct comparisons between full-memory and truncated-memory trajectories, and use it to extract critical truncation thresholds in parameter planes involving the forcing amplitude and the fractional order. The results reveal strongly non-monotonic memory thresholds, showing that the retained memory required to preserve coherence depends on the forcing regime, the fractional order, and the nonlinear sensitivity of the dynamics. We also derive a local characteristic equation for the truncated GL kernel. A minimal one-delay approximation produces a formal negative delay, indicating that a single causal delay is structurally insufficient. This motivates a positive-delay exponential representation of the finite-memory kernel. The minimum number of positive-delay modes required to reach a prescribed spectral accuracy defines an operational delay-complexity measure, $r_{\min}$. Overall, the truncated GL kernel emerges as an intermediate object between distributed fractional memory and delay-type dynamics, with a local spectral structure that controls both coherence loss and effective delay complexity.

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 / 2 minor

Summary. The manuscript investigates the effects of truncating the Grünwald-Letnikov memory term in a fractional Duffing oscillator, treating the truncation as a finite-memory dynamical modification rather than a mere approximation. It defines a coherence-loss time extracted from direct comparisons of full-memory versus truncated trajectories and uses this to map critical truncation thresholds in the (forcing amplitude, fractional order) plane, reporting strongly non-monotonic behavior. Analytically, it derives a local characteristic equation for the truncated kernel; a minimal one-delay fit yields a negative delay, motivating a positive-delay exponential representation whose minimum mode count r_min quantifies effective delay complexity. The truncated kernel is positioned as an intermediate object whose local spectral structure governs both coherence loss and delay complexity.

Significance. If the central claims are substantiated, the work provides a concrete bridge between distributed-order fractional memory and finite-delay representations in nonlinear oscillators. The non-monotonic thresholds and the operational r_min measure could inform both numerical practice (how much memory must be retained) and model-reduction strategies. The explicit derivation of the local characteristic equation and the demonstration that a single causal delay is structurally inadequate are potentially useful technical contributions, provided the supporting numerics are shown to be robust.

major comments (2)
  1. [Section defining and applying the coherence-loss time] The coherence-loss time is defined via direct full-vs-truncated trajectory comparisons and is then used to extract critical truncation thresholds (abstract and the section presenting the (forcing amplitude, fractional order) maps). No systematic tests are reported for invariance under integrator choice, step size, solver tolerances, or initial conditions. Because these thresholds are load-bearing for the claim that spectral structure controls coherence loss, any numerical dependence would render the reported non-monotonicity an artifact rather than an intrinsic property.
  2. [Section deriving the local characteristic equation and r_min] The local characteristic equation for the truncated GL kernel (analytical section) leads to the conclusion that a single causal delay is structurally insufficient because the minimal one-delay approximation produces a negative delay. The derivation steps, the truncation levels at which this occurs, and the quantitative spectral error incurred by the subsequent positive-delay exponential representation are not shown in sufficient detail to confirm that r_min is a stable, reproducible complexity measure independent of fitting procedure.
minor comments (2)
  1. [Definition of r_min] The operational definition of r_min (minimum number of positive-delay modes for prescribed spectral accuracy) would benefit from an explicit formula or pseudocode so that the measure can be reproduced without ambiguity.
  2. [Figures showing non-monotonic thresholds] Figure captions and axis labels for the threshold maps should explicitly state the numerical scheme, tolerances, and initial-condition protocol used to compute the coherence-loss time.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important aspects of numerical robustness and analytical detail that we will address in the revision. Below we respond point by point.

read point-by-point responses

  1. Referee: [Section defining and applying the coherence-loss time] The coherence-loss time is defined via direct full-vs-truncated trajectory comparisons and is then used to extract critical truncation thresholds (abstract and the section presenting the (forcing amplitude, fractional order) maps). No systematic tests are reported for invariance under integrator choice, step size, solver tolerances, or initial conditions. Because these thresholds are load-bearing for the claim that spectral structure controls coherence loss, any numerical dependence would render the reported non-monotonicity an artifact rather than an intrinsic property.

    Authors: We agree that explicit tests for numerical invariance are essential to substantiate the robustness of the coherence-loss thresholds. In the revised manuscript, we will add a dedicated subsection or appendix presenting results from multiple integrators (e.g., explicit Runge-Kutta and linear multistep methods), varied step sizes and tolerances, and different initial conditions. These tests will demonstrate that the non-monotonic behavior in the (forcing amplitude, fractional order) plane persists across these variations, confirming it as an intrinsic dynamical feature rather than a numerical artifact. We will also discuss the specific choices used in the original computations. revision: yes

  2. Referee: [Section deriving the local characteristic equation and r_min] The local characteristic equation for the truncated GL kernel (analytical section) leads to the conclusion that a single causal delay is structurally insufficient because the minimal one-delay approximation produces a negative delay. The derivation steps, the truncation levels at which this occurs, and the quantitative spectral error incurred by the subsequent positive-delay exponential representation are not shown in sufficient detail to confirm that r_min is a stable, reproducible complexity measure independent of fitting procedure.

    Authors: We acknowledge that the analytical section would benefit from greater detail. In the revision, we will provide the complete step-by-step derivation of the local characteristic equation for the truncated kernel, explicitly state the truncation levels considered, and include quantitative plots or tables of the spectral error for the positive-delay exponential approximation. Additionally, we will test the stability of r_min by varying the fitting procedure (e.g., different optimization tolerances or basis functions) and report the resulting variations, thereby establishing r_min as a reproducible measure. This will strengthen the claim that the truncated kernel serves as an intermediate between fractional memory and delay dynamics. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation remains self-contained

full rationale

The paper defines coherence-loss time directly via trajectory comparisons between full and truncated memory, then extracts thresholds from those comparisons. Separately, it derives a local characteristic equation for the truncated GL kernel, obtains a negative-delay result from one-delay approximation, and defines r_min as the count of positive-delay modes needed for a target spectral accuracy. None of these steps reduce by construction to their own inputs, invoke self-citations as load-bearing premises, smuggle ansatzes, or rename known results. The interpretive claim that the truncated kernel acts as an intermediate object follows from the independent constructions rather than presupposing them.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

Relies on standard Grünwald-Letnikov definition and assumes truncation creates a modified dynamical system. Introduces coherence-loss time and r_min without external calibration.

axioms (2)
  • standard math The Grünwald-Letnikov fractional derivative is well-defined for the Duffing equation under chosen conditions.
    Used when comparing full and truncated trajectories.
  • domain assumption Trajectory divergence defines a scalar coherence-loss time independent of integrator details.
    Required to extract thresholds from parameter planes.
invented entities (2)
  • coherence-loss time no independent evidence
    purpose: Quantify duration of similarity between truncated and full-memory trajectories.
    New diagnostic from direct comparisons; no external falsifiable test shown.
  • r_min no independent evidence
    purpose: Count minimum positive-delay modes for spectral accuracy of truncated kernel.
    Operational measure from local characteristic equation; no external validation.

pith-pipeline@v0.9.0 · 8161 in / 1238 out tokens · 44687 ms · 2026-05-09T15:39:56.477475+00:00 · methodology

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discussion (0)

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Reference graph

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