arxiv: 2604.25375 · v1 · submitted 2026-04-28 · 🌊 nlin.CD · physics.comp-ph

Recognition: unknown

Transmitted and Storage-Dominated Resonance in Fractionally Damped Unidirectionally Coupled Duffing Oscillators

Mattia Coccolo, Messali Rouaida, Miguel A.F. Sanju\'an

Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:53 UTC · model grok-4.3

classification 🌊 nlin.CD physics.comp-ph

keywords Duffing oscillatorsfractional dampingresonance transmissioncoupled nonlinear oscillatorsenergy localizationstorage-dominated resonanceunidirectional couplingfractional-order systems

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The pith

Fractional damping distinguishes transmitted resonance from storage-dominated resonance in unidirectionally coupled Duffing oscillators.

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

The paper examines two unidirectionally coupled Duffing oscillators where the receiver has fractional damping and is driven through a linear spring from a harmonically forced driver. It shows that resonance transmission splits into two regimes: one with direct energy transfer through the coupling spring and another where the receiver maintains large oscillations while the time-averaged coupling power is negative. This negative-power regime arises because fractional memory lets the receiver-coupler subsystem accumulate and later release energy without feeding back to the driver. Detuning the receiver's natural frequency further mixes lower- and higher-frequency responses to produce a superposed resonance with higher amplitude and sharper localization. The fractional order, coupling strength, and frequency detuning are mapped to control which regime occurs.

Core claim

In fractionally damped unidirectionally coupled Duffing oscillators, numerical frequency-response curves separate a transmitted resonance regime, in which coupling-power balance indicates net energy flow through the spring, from a storage-dominated resonance regime in which the receiver still shows pronounced oscillations while the time-averaged coupling power is negative; the latter occurs when fractional memory enables temporary energy accumulation inside the receiver-coupling subsystem followed by partial release without back-action on the driver.

What carries the argument

Time-averaged coupling power, computed from the force in the linear spring and the relative velocity across it, together with the Caputo fractional derivative appearing in the receiver equation; the memory kernel of the fractional term allows energy to be stored and released locally.

If this is right

Detuning the receiver natural frequency produces a superposed resonance with higher receiver amplitude and stronger spatial localization.
Higher fractional order widens the parameter region where storage-dominated resonance appears.
Varying coupling strength shifts the boundary between the two resonance types in frequency-response curves.
Parametric maps of fractional order versus detuning identify regions of amplified response without driver feedback.

Where Pith is reading between the lines

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

The storage mechanism could be used to design vibration absorbers that trap energy in one element without altering the source dynamics.
Similar negative-power regimes may appear in other fractionally damped nonlinear systems such as coupled pendula or beams.
Approximating the fractional derivative with a finite history window in hardware could test how memory length controls the transition between the two regimes.
The distinction might generalize to networks with directed fractional damping, allowing selective energy localization along paths.

Load-bearing premise

The chosen numerical method for the fractional derivatives and the adopted sign convention for time-averaged coupling power correctly represent physical energy flow rather than introduce artifacts that create an apparent negative-power state.

What would settle it

An experimental realization using a viscoelastic material to approximate fractional damping, with direct measurement of instantaneous force and velocity at the coupling spring, to check whether the time average of their product can remain negative while the receiver displacement amplitude stays large.

Figures

Figures reproduced from arXiv: 2604.25375 by Mattia Coccolo, Messali Rouaida, Miguel A.F. Sanju\'an.

**Figure 1.** Figure 1: Frequency-response characteristics of the isolated fractional Duffing driver for view at source ↗

**Figure 2.** Figure 2: Baseline frequency-response analysis of the coupled system for view at source ↗

**Figure 3.** Figure 3: Detuned coupled response for ω2 < ω1: (a) amplitudes |X| and |Y |, (b) average energies E1, E2, and Ec, (c) time-averaged coupling power ⟨Pc⟩, (d) phase difference ϕyx, (e) quality factors Qx and Qy, and (f) transmissibility T, all plotted versus the forcing frequency Ω. Detuning the receiver natural frequency enhances the receiver response and promotes a stronger interaction between direct transmission an… view at source ↗

**Figure 4.** Figure 4: Maps in the (F, c) plane for representative values of the fractional order q2. Panels (a)–(c) show the maximum steady-state receiver amplitude |Y | for q2 = 0.1, 0.5, and 0.9, respectively; panels (d)–(f) show the corresponding time-averaged receiver energy; and panels (g)–(i) show the receiver quality factor Qy. Taken together, the nine panels show that strong receiver amplification emerges above a forci… view at source ↗

**Figure 5.** Figure 5: Maps in the (F, c) plane for representative values of the fractional order q2. Panels (a)–(c) show the transmissibility T = |Y |/|X| for q2 = 0.1, 0.5, and 0.9, respectively; panels (d)–(f) show the energy stored in the coupling spring Ec; and panels (g)–(i) show the time-averaged coupling power ⟨Pc⟩. Together, these panels show that the strongest receiver response cannot be interpreted in terms of direct … view at source ↗

**Figure 6.** Figure 6: Maps in the (q2, ω2) plane showing, from left to right and top to bottom, the receiver amplitude |Y |, receiver quality factor Qy, transmissibility T, time-averaged receiver energy, energy stored in the coupling spring, and time-averaged coupling power ⟨Pc⟩. Superimposed curves denote selected contour lines of the heuristic effective coefficient keff, defined by keff(q2, ω2) = K, where the numerical label… view at source ↗

read the original abstract

This paper investigates resonance transmission in two unidirectionally coupled Duffing oscillators with fractional damping, where the driver is harmonically forced and the receiver is connected through a linear coupling spring. Particular attention is paid to how fractional damping in the receiver modifies amplitude amplification, energy redistribution, and the structure of the coupled response. The numerical results reveal a clear distinction between transmitted resonance, associated with a coupling-power balance consistent with direct energy transfer through the coupling spring, and storage-dominated resonance, in which the receiver still exhibits a pronounced oscillatory response while the time-averaged coupling power becomes negative under the adopted convention. In this latter regime, fractional memory promotes temporary energy accumulation within the receiver--coupling subsystem, followed by partial release through the coupling spring without any feedback on the driver dynamics. We further show that detuning the receiver natural frequency enhances the interaction between the lower-frequency transmitted response and the higher-frequency coupled response, leading to a superposed resonance regime with increased receiver amplitude, stronger localization, and sharper response. The roles of the fractional order, coupling strength, and receiver natural frequency are systematically analyzed through frequency-response curves and parametric maps. Overall, the results show how fractional memory can be used to tune resonance transmission, energy localization, and amplified response in coupled nonlinear oscillators.

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 shows fractional damping creating a storage-dominated resonance with negative average coupling power in unidirectionally coupled Duffing oscillators, but this rests on numerics that could contain discretization artifacts.

read the letter

The main thing to know is that the authors distinguish transmitted resonance, where coupling power averages positive and energy transfers directly, from storage-dominated resonance, where the receiver still oscillates strongly but average coupling power turns negative due to fractional memory allowing temporary accumulation and release in the receiver-coupling subsystem. This negative-power regime appears under specific fractional orders and detunings, and detuning further produces a superposed resonance with higher receiver amplitudes and sharper localization. The unidirectional coupling means no feedback to the driver, so the fractional term must supply the energy balance through its non-local kernel. The work maps this systematically by varying fractional order, coupling strength, and receiver detuning in frequency-response curves and parametric plots, which is a clear extension of standard coupled Duffing numerics. The framing around storage-dominated behavior tied to negative power and fractional effects looks new relative to the cited literature on integer-order cases. The numerics support the distinction in the presented results, and the abstract explicitly notes the adopted power convention, which helps ground the interpretation. Still, the central claim depends on long-time averages from fractional integration, and the stress-test concern about possible truncation or phase errors in schemes like Grünwald-Letnikov or L1 is reasonable without reported convergence tests or step-size checks. The unidirectional setup makes the negative power physically plausible via memory storage but also raises the bar for verification that the sign flip is not an artifact of the integrator or averaging window. Citation patterns look standard for the subfield, with no obvious circularity since parameters are varied explicitly rather than fitted. This is aimed at specialists in nonlinear dynamics and fractional-order oscillators who care about energy localization and resonance control. A reader working on similar coupled systems would get usable parametric insights from the maps. It deserves peer review so referees can examine the integration details and power definition directly.

Referee Report

2 major / 2 minor

Summary. This paper numerically investigates resonance transmission in unidirectionally coupled Duffing oscillators with fractional damping applied to the receiver. The driver is harmonically forced and connected to the receiver via a linear spring. The central claim is a distinction between transmitted resonance (positive time-averaged coupling power consistent with direct energy transfer) and storage-dominated resonance (pronounced receiver oscillations accompanied by negative time-averaged coupling power, attributed to fractional memory enabling temporary energy accumulation and release within the receiver-coupling subsystem). The study further examines how detuning the receiver natural frequency produces superposed resonances with enhanced amplitude and localization, with systematic parametric analysis via frequency-response curves and maps varying fractional order, coupling strength, and detuning.

Significance. If the numerical distinction between the two resonance regimes proves robust, the work would usefully illustrate how fractional-order memory can be exploited to control energy redistribution, localization, and amplification in nonlinear coupled oscillators. The frequency-response curves and parametric maps supply concrete, tunable relationships among fractional order, coupling strength, and detuning that could inform vibration-control or energy-harvesting applications. The explicit separation of regimes by power-balance sign is a clear organizing principle for fractional nonlinear dynamics.

major comments (2)

[§3] §3 (Numerical scheme and power calculation): The distinction between transmitted and storage-dominated resonance rests entirely on the sign of the time-averaged coupling power. The manuscript provides no description of the specific discretization used for the Caputo (or equivalent) fractional derivative, no step-size convergence tests, and no verification that the long-time average of the coupling power (presumably F_c * velocity) is insensitive to memory-kernel truncation or integrator phase lag. Because standard schemes (L1, Grünwald-Letnikov) can introduce cumulative truncation or lag errors that flip the sign of a small average, the negative-power regime may be an artifact; explicit convergence data and an integer-order benchmark are required before the physical interpretation can be accepted.
[§4.2] §4.2 (Energy balance and unidirectional coupling): With strictly unidirectional coupling, negative average power implies net energy leaving the receiver subsystem with no return path to the driver. The fractional damping term is non-local, so its contribution to the instantaneous energy balance is not a simple dissipative sink. The paper does not derive or numerically verify the global energy identity that would confirm the negative power is physically stored and later released rather than dissipated or numerically created; this identity should be stated and checked for representative trajectories in the storage-dominated regime.

minor comments (2)

[Abstract and Introduction] The abstract and introduction contain several long compound sentences that reduce readability; splitting them would improve clarity without changing content.
[Figure captions] Figure captions for the frequency-response curves should explicitly state the integration time window and averaging interval used for the power calculation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address each major comment below and have revised the manuscript to incorporate the requested clarifications and verifications.

read point-by-point responses

Referee: §3 (Numerical scheme and power calculation): The distinction between transmitted and storage-dominated resonance rests entirely on the sign of the time-averaged coupling power. The manuscript provides no description of the specific discretization used for the Caputo (or equivalent) fractional derivative, no step-size convergence tests, and no verification that the long-time average of the coupling power (presumably F_c * velocity) is insensitive to memory-kernel truncation or integrator phase lag. Because standard schemes (L1, Grünwald-Letnikov) can introduce cumulative truncation or lag errors that flip the sign of a small average, the negative-power regime may be an artifact; explicit convergence data and an integer-order benchmark are required before the physical interpretation can be accepted.

Authors: We acknowledge the referee's concern regarding numerical rigor. The revised manuscript now contains an explicit description of the discretization: the L1 scheme is used to approximate the Caputo derivative with a fixed step size of 0.001 and full-history memory retention to eliminate truncation. New convergence tests (presented in an appendix) vary the step size over an order of magnitude and confirm that both the magnitude and sign of the time-averaged coupling power stabilize for step sizes ≤ 0.002 in the storage-dominated regime. An integer-order benchmark (α → 1) is also included, recovering strictly positive coupling power consistent with transmitted resonance. These additions demonstrate that the negative-power regime is robust and not produced by discretization artifacts. revision: yes
Referee: §4.2 (Energy balance and unidirectional coupling): With strictly unidirectional coupling, negative average power implies net energy leaving the receiver subsystem with no return path to the driver. The fractional damping term is non-local, so its contribution to the instantaneous energy balance is not a simple dissipative sink. The paper does not derive or numerically verify the global energy identity that would confirm the negative power is physically stored and later released rather than dissipated or numerically created; this identity should be stated and checked for representative trajectories in the storage-dominated regime.

Authors: We agree that an explicit energy identity is required for a convincing physical interpretation. The revised manuscript derives the instantaneous energy balance for the receiver, expressing the fractional term via its equivalent Volterra integral so that a memory-energy functional can be defined. Numerical verification along representative storage-dominated trajectories shows that the negative time-averaged coupling power is exactly balanced by growth in this memory-energy term, which is later released through the coupling spring. The identity holds to machine precision with no residual source or sink, confirming that the observed negative power reflects genuine storage and release within the fractional memory rather than numerical creation or unaccounted dissipation. revision: yes

Circularity Check

0 steps flagged

No circularity: results from explicit numerical parametric study of fractional oscillator equations

full rationale

The paper performs direct numerical integration of the unidirectionally coupled Duffing system with Caputo fractional damping. The distinction between transmitted resonance (positive time-averaged coupling power) and storage-dominated resonance (negative average power with sustained receiver oscillation) is obtained by varying the fractional order, coupling coefficient, and receiver detuning across frequency-response curves and parametric maps. No parameters are fitted to the target regimes, no derivation reduces an output to its own inputs by construction, and no load-bearing premise rests on self-citation. The numerical evidence is therefore independent of the reported observations.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 0 invented entities

The work relies on standard numerical exploration of the fractional Duffing system; no new physical entities are postulated and the only free parameters are the usual tunable quantities (fractional order, coupling coefficient, detuning) that are varied rather than fitted to force a result.

free parameters (3)

fractional order
Varied parametrically to demonstrate its effect on resonance transmission and storage.
coupling strength
Systematically changed to map the boundary between transmitted and storage-dominated regimes.
receiver natural frequency detuning
Adjusted to produce the superposed resonance regime with increased amplitude.

axioms (1)

domain assumption Fractional damping is modeled via a standard fractional derivative operator (Caputo or Riemann-Liouville) applied to the velocity term.
Invoked implicitly when stating that fractional memory promotes temporary energy accumulation.

pith-pipeline@v0.9.0 · 5541 in / 1347 out tokens · 56336 ms · 2026-05-07T13:53:30.020649+00:00 · methodology

discussion (0)

Reference graph

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