Pith. sign in

REVIEW 3 major objections 5 minor 41 references

Parametric Autoresonance with Time-Delayed Control

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A delayed feedback switch controls whether parametric autoresonance grows or fades, with a sharp critical delay strength $k_{\mathrm{th}} = \gamma\omega/\sin(\omega\bar{\beta})$.

desk verdict A plausible incremental result with a useful threshold formula, but the paper needs to clean up the slow-flow derivation and soften its threshold claims. read the letter →

arxiv 2411.10105 v2 pith:OVTINYBD submitted 2024-11-15 nlin.CD cs.NAmath-phmath.MPmath.NAphysics.comp-phphysics.data-an

Somnath Roy , Mattia Coccolo , Miguel A.F. Sanjuán This is my paper · ORCID
classification nlin.CDcs.NAmath-phmath.MPmath.NAphysics.comp-phphysics.data-an MSC 34C1534K2070K28
keywords autoresonanceparametrictime-delayedfeedbackmultiple-scaleanalysisstabilitythresholdchirpedforcingDuffingoscillatoramplitudegrowth
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that a constant negative time delay in the feedback of a parametrically chirped nonlinear oscillator acts as a control switch for autoresonance. Using multiple-scale perturbation theory, it derives a critical delay strength $k_{\mathrm{th}} = \gamma\omega / \sin(\omega\bar{\beta})$: for $k>k_{\mathrm{th}}$ the phase-locked amplitude grows, while for $k

What carries the argument

The central object is the slow-flow system obtained by a two-time-scale expansion of the delayed parametric oscillator: two first-order equations, $D_1 A = F_1(A,\psi)$ and $D_1\psi = F_2(A,\psi)$, for the amplitude $A$ and the phase difference $\psi$. The quasi-steady fixed point of these equations gives an analytical amplitude formula, and the stability of that fixed point is analyzed through the Jacobian matrix $J$; the condition $\mathrm{Tr}(J)=0$, combined with the fixed-point condition $D_1 A=0$, produces the critical delay strength $k_{\mathrm{th}} = \gamma\omega / \sin(\omega\bar{\beta})$. This trace condition is the load-bearing step that turns the delay parameters $(k,\bar{\beta})$ into a predicted control threshold.

What would settle it

Numerically integrate the original delayed equation with $\gamma=0.002$, $\omega=0.5$, $\alpha=0.03$, $h=0.002$, and $\bar{\beta}=0.1$, scanning $k$ through $0.02$: if the fitted envelope exponent $b$ does not change sign near $k\approx0.0201$, or if amplitude growth appears for $k$ well below that value, the trace-condition threshold is not the actual onset. A second check is to evaluate $\det(J)$ along the $\mathrm{Tr}(J)=0$ curve to see whether the fixed point loses stability before the predicted threshold.

Watch

Extended reading notes

Core claim

The paper argues that for a parametrically chirped Duffing-type oscillator with negative delayed feedback, the slow-flow amplitude and phase equations admit a quasi-steady fixed point whose stability is controlled by the delay strength. Setting the trace of the linearized Jacobian to zero at this fixed point yields the critical value $k_{\mathrm{th}} = \gamma\omega / \sin(\omega\bar{\beta})$, below which the fixed point is unstable and the oscillation envelope decays, and above which autoresonant amplitude growth is sustained. The threshold is independent of the drive amplitude $h$ and of the nonlinearity coefficient $\alpha$, and it implies an inverse relation $\bar{\beta}_{\mathrm{th}} = (1/\omega)\arcsin(\gamma\omega/k)$ between delay time and delay strength. Numerical simulations confirm the predicted onset: the fitted envelope exponent $b$ changes from negative to positive at $k \approx 0.0201$ for $\bar{\beta}=0.1$ and at $k \approx 0.0068$ for $\bar{\beta}=0.3$, in good agreement with the analytical formula.

Load-bearing premise

The threshold formula rests on treating the slow-flow phase equation, which contains the fast time $\tau_0$ inside the term $\zeta\tau_0$, as a legitimate slow system, and on reading the stability boundary from the trace condition alone.

Editorial extensions

If this is right

  • If the threshold claim is correct, the delay strength $k$ can serve as an experimentally adjustable control for autoresonance, complementing the usual control through the chirp rate $\mu$.
  • The threshold is independent of the drive amplitude $h$, so the same onset value $k_{\mathrm{th}}$ should be observed for different forcing strengths, as the numerical results in Fig. 3 indicate.
  • For a fixed delay strength, the required time delay is $\bar{\beta}_{\mathrm{th}} = (1/\omega)\arcsin(\gamma\omega/k)$, giving a reciprocal design relation between delay time and delay strength.
  • The sign of the exponential envelope fit parameter $b$ provides a direct observable criterion for the onset of autoresonance: positive $b$ means growing amplitude, negative $b$ means decaying amplitude.
  • In mechanical and electrical systems where sensing and actuation introduce lag, the delay can be tuned to sustain or suppress resonance rather than being an uncontrolled disturbance.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper's own results, the stability threshold is derived from the trace condition alone; verifying whether $\det(J)>0$ also holds along the $\mathrm{Tr}(J)=0$ boundary could reveal a stricter onset or a Hopf bifurcation that the single-threshold formula does not capture.
  • The slow-flow phase equation contains the term $\zeta\tau_0$, which mixes the fast time $\tau_0$ into the slow dynamics; if this step is not rigorously justified, the amplitude formula and threshold may need modification at larger chirp rates.
  • The striped regions in the $(k,\bar{\beta})$ plane where autoresonance appears before the analytically predicted threshold suggest additional resonance tongues; mapping those tongues is a natural extension that could test the limits of the trace-condition prediction.
  • Because $k_{\mathrm{th}}$ depends only on damping, frequency, and delay time, one could design a gain-scheduled delay controller that switches resonance growth on and off; testing this at several damping values would directly probe the formula's validity.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript studies a parametrically excited Duffing-type oscillator with constant negative delayed feedback and a linearly chirped parametric drive. Using a two-timescale perturbation expansion, the authors derive a slow-flow system, a quasi-steady amplitude law (Eq. 13), and a critical delay strength kth = γω / sin(ωβ̄) (Eq. 18), claiming that autoresonant amplitude growth occurs only above this threshold. Numerical simulations with exponential envelope fits and parameter scans in k, β̄, and h are presented as corroboration. The paper concludes that time delay can serve as a control mechanism for parametric autoresonance.

Significance. The problem is timely: time-delayed control of autoresonance is much less studied than externally forced autoresonance, and the paper gives an explicit, falsifiable threshold formula that is empirically testable and appears to organize the numerical data in the two main cases examined. The numerical study is more thorough than a single-trajectory check, using envelope fits, maxima-minima diagrams, and two-parameter scans. The main analytical derivation, however, has a serious asymptotic inconsistency involving the fast-time variable in the slow-flow equation, and the stability analysis checks only one half of the necessary linear-stability conditions. If the asymptotic reduction can be repaired and the determinant condition verified, the threshold result would be a useful contribution; in its current form the analytical claim is not rigorously established.

major comments (3)
  1. [Section II.A, Eqs. (9)-(11)] The slow-flow equation for ψ contains the term ζτ0, where τ0 is the fast time variable of the multiple-scales expansion defined in Eq. (4). In a standard two-time reduction, the slow-flow right-hand sides may depend only on A, ψ, and the slow time τ1; a term linear in τ0 cannot be absorbed into a slow-flow fixed point. Indeed, because ψ is defined as η + ζτ0²/2, the derivative of ζτ0²/2 with respect to the independent slow variable τ1 is zero, while if one instead treats τ0 as the physical time, the term ζτ0 produces a non-uniformity on the long time scale. The paper never states the ordering of ζ = μ/ω² relative to ε; if ζ is O(ε), then ζτ0 is not a small uniformly slow correction, and the 'quasi-steady fixed point' D1A = D1ψ = 0 used for Eqs. (12)-(13) and (18) is not a true fixed point of a derived autonomous slow-flow system. A consistent derivation would introduce a slow phase variable that absorbs the chirp before constructing the slow flow. As it stands, the derivation of Eq. (13) and Eq. (18) is not asymptotically justified, so the central threshold claim lacks a rigorous analytical basis, regardless of the numerical agreement.
  2. [Section II.B, Eqs. (16)-(18)] The threshold is derived from Tr(J) = 0 alone, but linear stability of the quasi-steady branch also requires the determinant condition det(J) > 0 (in the notation of Eq. (16)). Using Eq. (10) at the fixed point, the paper's expression (17) for det(J) reduces to det(J) = (3/8)Λ h A² cos(2ψ). Thus det(J) > 0 imposes cos(2ψ) > 0, a condition that is neither derived nor checked in the numerical parameter range. Without this verification, the point k = kth is not shown to be the actual stability boundary of the quasi-steady branch; the correspondence between the sign change of the fitted exponent b and Eq. (18) is therefore empirical rather than established by the linear-stability calculation. In particular, the claimed h-independence of kth follows only from the trace condition and does not address the full stability boundary.
  3. [Section III, Figs. 1-4] The numerical validation uses the sign of the exponent b in an exponential envelope fit x = a e^{bt} as the criterion for autoresonance, but the analytical amplitude law (13) predicts algebraic growth proportional to t^{1/2}, not exponential growth. The fit exponent is therefore not a direct test of the predicted amplitude law, and no quantitative overlay of the numerical envelope on Eq. (13) is provided. Moreover, the text reports early autoresonance onset for k values below the predicted threshold, e.g., 'around k ≈ 0.015' and 'k ≈ 0.005 in the h = 0.002 case' in the Fig. 3 discussion; these pre-threshold events are acknowledged but not reconciled with the claim that kth marks a sharp transition. The paper should either justify the exponential fit as a valid proxy or compare the numerical envelope directly with Eq. (13).
minor comments (5)
  1. [Section II.B, Eq. (15)] The layout of the determinant in Eq. (15) is ambiguous: the bottom-right entry appears to combine -h/2 sin(2ψ) and -s, making it unclear whether the characteristic matrix is J - sI or J + sI. Please present the characteristic equation explicitly with a standard matrix form.
  2. [Section III, Fig. 3 discussion] The sentence describing pre-threshold autoresonance, 'there is a set of k values, around k ≈ 0.015, and the value k ≈ 0.005 in the h = 0.002 case', is unclear; please specify which panel and which parameter scan these values refer to.
  3. [Throughout] The paper uses 'stable autoresonance' to describe amplitude growth above the threshold, whereas the slow-flow fixed point itself would be unstable in the usual Lyapunov sense when the effective damping is negative. Please define what is meant by stability in this context.
  4. [References] Reference [19] duplicates Reference [7]; please remove the duplicate or cite distinct works.
  5. [Section II.A, Eq. (13)] Equation (13) contains the combination (ω² - ω₀²)/ε, which is dimensionally and asymptotically unclear; please explain the ordering and the precise meaning of ε in that expression.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the analytical threshold is derived from the slow-flow equations and corroborated by independent numerical integration of the original model.

full rationale

The paper's central claim is the delay-strength threshold kth = γω / sin(ωβ̄), obtained analytically in Eq. (18) from the slow-flow fixed-point condition and the trace of the Jacobian, not from fitting to the numerics. The subsequent numerical simulations integrate the original delayed oscillator (Eq. (2)) directly and independently detect the threshold by the sign change of an exponential envelope fit; agreement with Eq. (18) is a genuine comparison, not a construction. Self-citations ([38], [39], [42]) appear only as background on delay effects and are not load-bearing for the derivation. The potential asymptotic-ordering concern about the term ζτ0 in Eq. (11) is a correctness or validity issue, not a circularity: the threshold expression is still derived from the paper's own stated equations rather than being equivalent to its input by definition or by a fitted parameter renamed as a prediction. Therefore there is no identifiable circular step.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a multiple-scales perturbation analysis with a quasi-steady fixed point, plus a linear stability condition. The most fragile axiom is the treatment of the ζτ0 term in the slow-flow equation, which is not justified and could invalidate the derived amplitude and threshold expressions.

free parameters (1)
  • envelope fit exponent b = sign used; no tabulated values
    The exponential model x = a e^{bt} is fitted to oscillation maxima to classify autoresonance (b > 0) versus decay (b < 0). This fitted quantity is used as the diagnostic for autoresonance onset.
assumptions (6)
  • standard math Method of multiple scales is applicable to Eq. (2)
    Used to derive the slow-flow equations (10)-(11) in Section II-A.
  • domain assumption Parameters h, Γ, Λ, g are all O(ε)
    Required for the perturbation expansion; no physical scaling argument is given in Section II-A.
  • domain assumption A quasi-steady fixed point exists with D1A = D1ψ = 0
    This is used to derive the amplitude expression (13) and the threshold condition in Section II-B.
  • standard math Linearized stability with characteristic equation s^2 + Tr(J)s + det(J) = 0, stability iff Tr(J)>0 and det(J)>0
    Standard linear stability analysis, though the paper applies only the trace condition without verifying the determinant condition.
  • ad hoc to paper The term ζτ0 in Eq. (11) can be treated as a slowly varying detuning even though τ0 is the fast time scale
    This is not standard in multiple-scales analysis and is not justified; it underlies the derivation of the fixed point and threshold.
  • domain assumption The sign of the fitted exponential exponent b indicates autoresonance (b>0) versus decay (b<0)
    The predicted slow amplitude growth is algebraic (t^{1/2}), not exponential, so an exponential fit is a coarse diagnostic that may misclassify transients.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Parametric Autoresonance with Time-Delayed Control." pith.science (2026). https://pith.science/paper/OVTINYBD

@misc{pith2026241110105,
  author       = {Pith},
  title        = {Pith review of: Parametric Autoresonance with Time-Delayed Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVTINYBD}},
  note         = {Machine review of arXiv:2411.10105}
}
read the original abstract

We investigate how a constant time delay influences a parametric autoresonant system. This is a nonlinear system driven by a parametrically chirped force with a negative delay-feedback that maintains adiabatic phase locking with the driving frequency. This phase locking results in a continuous amplitude growth, regardless of parameter changes. Our study reveals a critical threshold for delay strength; above this threshold, autoresonance is sustained, while below it, autoresonance diminishes. We examine the interplay between time delay and autoresonance stability, using multi-scale perturbation methods to derive analytical results, which are corroborated by numerical simulations. Ultimately, the goal is to understand and control autoresonance stability through the time-delay parameters.

Figures

Figures reproduced from arXiv: 2411.10105 by the authors.

Figure 1
Figure 1. FIG. 1: Analytical plot of amplitude growth over time using Eq. (13) for parameters [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Panels (a) and (b) show the oscillations of the system with their maximum envelope. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The figure shows the maximum and minimum diagram, the oscillations amplitude, and [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Figures (a) and (b) show the maximum and minimum diagram, figures (c) and (d) the [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The figure shows the theoretical dependence of [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: In this figure, stripes and a yellow area in the top-right region of the panels indicate [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The figure shows the gradient plot in the [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    Rajasekar and M

    S. Rajasekar and M. A. F. Sanju´ an,Nonlinear Resonances(Springer International Publishing, Switzerland, 2016)

    work page 2016

  2. [2]

    Fajans and L

    J. Fajans and L. Friedland, Autoresonant (nonstationary) excitation of pendulums, plutinos, plasmas, and other nonlinear oscillators, Am. J. Phys. 69, 1096 (2001)

    work page 2001

  3. [3]

    M. S. Livingston, High-Energy Accelerators(Interscience, 1954)

    work page 1954

  4. [4]

    Meerson and L

    B. Meerson and L. Friedland, Strong autoresonance excitation of Rydberg atoms: The Ryd- berg accelerator, Phys. Rev. A 41, 5233 (1990)

    work page 1990

  5. [5]

    W.-K. Liu, B. Wu, and J.-M. Yuan, Nonlinear dynamics of chirped pulse excitation and dissociation of diatomic molecules, Phys. Rev. Lett. 75, 1292 (1995)

    work page 1995

  6. [6]

    Fajans, E

    J. Fajans, E. Gilson, and L. Friedland, Autoresonant (nonstationary) excitation of the dio- cotron mode in non-neutral plasmas, Phys. Rev. Lett. 82, 4444 (1999)

    work page 1999

  7. [8]

    G. B. Andresen et al., Autoresonant excitation of antiproton plasmas, Phys. Rev. Lett. 106, 025002 (2011)

    work page 2011

  8. [9]

    C. J. Baker, J. R. Danielson, N. C. Hurst, and C. M. Surko, Electron plasma dynamics during autoresonant excitation of the diocotron mode, Phys. Plasmas 22, 022 (2015)

    work page 2015

Show all 41 references
  1. [10]

    Friedland, Autoresonance of coupled nonlinear waves, Phys

    L. Friedland, Autoresonance of coupled nonlinear waves, Phys. Rev. E 57, 3494 (1998)

    1998

  2. [11]

    Friedland, Autoresonant solutions of the nonlinear Schr¨ odinger equation, Phys

    L. Friedland, Autoresonant solutions of the nonlinear Schr¨ odinger equation, Phys. Rev. E58, 3865 (1998)

    1998

  3. [12]

    Yaakobi, L

    O. Yaakobi, L. Caspani, M. Clerici, F. Vidal, and R. Morandotti, Complete energy conversion by autoresonant three-wave mixing in nonuniform media, Opt. Express 21, 1623 (2013)

    2013

  4. [13]

    Malhotra, Migrating planets, Sci

    R. Malhotra, Migrating planets, Sci. Am. 281, 56 (1999)

    1999

  5. [14]

    Friedland, Migration timescale thresholds for resonant capture in the Plutino problem, Astrophys

    L. Friedland, Migration timescale thresholds for resonant capture in the Plutino problem, Astrophys. J. 547, L75 (2001)

    2001

  6. [15]

    A. F. Lanza, Tidal excitation of auto-resonant oscillations in stars with close-by planets, Astron. Astrophys. 665, A47 (2022)

    2022

  7. [16]

    Friedland, Control of Kirchhoff vortices by a resonant strain, Phys

    L. Friedland, Control of Kirchhoff vortices by a resonant strain, Phys. Rev. E 59, 4106 (1999). 16

    1999

  8. [17]

    Friedland, Autoresonant excitation and evolution of nonlinear waves: The variational ap- proach, Phys

    L. Friedland, Autoresonant excitation and evolution of nonlinear waves: The variational ap- proach, Phys. Rev. E 55, 1929 (1997)

    1997

  9. [18]

    Fajans, E

    J. Fajans, E. Gilson, and L. Friedland, The effect of damping on autoresonant (nonstationary) excitation, Phys. Plasmas 8, 423 (2001)

    2001

  10. [19]

    Fajans, E

    J. Fajans, E. Gilson, and L. Friedland, Autoresonant (nonstationary) excitation of a collective nonlinear mode, Phys. Plasmas 6, 4497 (1999)

    1999

  11. [20]

    Fajans, E

    J. Fajans, E. Gilson, and L. Friedland, Second harmonic autoresonant control of the l = 1 diocotron mode in pure-electron plasmas, Phys. Rev. E 62, 4131 (2000)

    2000

  12. [21]

    Nakar and L

    E. Nakar and L. Friedland, Passage through resonance and autoresonance in x2n-type poten- tials, Phys. Rev. E 60, 5479 (1999)

    1999

  13. [22]

    Barth and L

    I. Barth and L. Friedland, Quantum phenomena in a chirped parametric anharmonic oscillator, Phys. Rev. Lett. 113, 040403 (2014)

    2014

  14. [23]

    L. A. Kalyakin, Asymptotic analysis of autoresonance models, Russ. Math. Surv. 63, 791 (2008)

    2008

  15. [24]

    L. A. Kalyakin and O. A. Sultanov, Stability of autoresonance models, Differ. Equ. 49, 267 (2013)

    2013

  16. [25]

    Kovaleva and L

    A. Kovaleva and L. I. Manevitch, Limiting phase trajectories and emergence of autoresonance in nonlinear oscillators, Phys. Rev. E 88, 024901 (2013)

    2013

  17. [26]

    L. I. Manevitch and A. Kovaleva, Autoresonant dynamics of weakly coupled oscillators, Non- linear Dyn. 84, 683 (2016)

    2016

  18. [27]

    Kovaleva, Autoresonance in a strongly nonlinear chain driven at one end, Phys

    A. Kovaleva, Autoresonance in a strongly nonlinear chain driven at one end, Phys. Rev. E 98, 052227 (2018)

    2018

  19. [28]

    Kovaleva and L

    A. Kovaleva and L. I. Manevitch, Autoresonance versus localization in weakly coupled oscil- lators, Physica D 320, 1 (2016)

    2016

  20. [29]

    Chac´ on, Energy-based theory of autoresonance phenomena: Application to Duffing-like systems, Europhys

    R. Chac´ on, Energy-based theory of autoresonance phenomena: Application to Duffing-like systems, Europhys. Lett. 70, 56 (2005)

    2005

  21. [30]

    Chac´ on, M

    R. Chac´ on, M. Y. Uleysky, and D. V. Makarov, Universal chaotic layer width in space- periodic Hamiltonian systems under adiabatic ac time-periodic forces, Europhys. Lett. 90, 40003 (2010)

    2010

  22. [31]

    Chac´ on, Breakdown of autoresonance due to separatrix crossing in dissipative systems: From Josephson junctions to the three-wave problem, Phys

    R. Chac´ on, Breakdown of autoresonance due to separatrix crossing in dissipative systems: From Josephson junctions to the three-wave problem, Phys. Rev. E 78, 066608 (2008). 17

    2008

  23. [32]

    Khain and B

    E. Khain and B. Meerson, Parametric autoresonance, Phys. Rev. E 64, 036619 (2001)

    2001

  24. [33]

    Assaf and B

    M. Assaf and B. Meerson, Parametric autoresonance in Faraday waves, Phys. Rev. E 72, 016310 (2005)

    2005

  25. [34]

    O. M. Kiselev and S. G. Glebov, The capture into parametric autoresonance, Nonlinear Dyn. 48, 217 (2007)

    2007

  26. [35]

    Friedland and A

    L. Friedland and A. G. Shagalov, Parametric autoresonant excitation of the nonlinear Schr¨ odinger equation, Phys. Rev. E94, 042216 (2016)

    2016

  27. [36]

    H. Hu, E. H. Dowell, and L. N. Virgin, Resonances of a harmonically forced Duffing oscillator with time delay state feedback, Nonlinear Dyn. 15, 311 (1998)

    1998

  28. [37]

    D. C. Mei, L. C. Du, and C. J. Wang, The effects of time delay on stochastic resonance in a bistable system with correlated noises, J. Stat. Phys. 137, 625 (2009)

    2009

  29. [38]

    Jeevarathinam, S

    C. Jeevarathinam, S. Rajasekar, and M. A. F. Sanju´ an, Theory and numerics of vibrational resonance in Duffing oscillators with time-delayed feedback, Phys. Rev. E 83, 066205 (2011)

    2011

  30. [39]

    Cantis´ an, M

    J. Cantis´ an, M. Coccolo, J. M. Seoane, and M. A. F. Sanju´ an, Delay-induced resonance in the time-delayed Duffing oscillator, Int. J. Bifurcation Chaos 30, 2030007 (2020)

    2020

  31. [40]

    Zakharova, N

    A. Zakharova, N. Semenova, V. Anishchenko, and E. Sch¨ oll, Time-delayed feedback control of coherence resonance chimeras, Chaos 27, 113119 (2017)

    2017

  32. [41]

    Maccari, Vibration control for the primary resonance of the Van der Pol oscillator by a time delay state feedback, Int

    A. Maccari, Vibration control for the primary resonance of the Van der Pol oscillator by a time delay state feedback, Int. J. Nonlinear Mech. 38, 123 (2003)

    2003

  33. [42]

    Chac´ on, F

    R. Chac´ on, F. Palmero, P. J. Mart ´ ınez, and S. Roy, Energy-based theory of autoresonance in chains of coupled damped-driven generic oscillators, Phys. Rev. E 110, 044203 (2024). 18

    2024

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.