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REVIEW 3 major objections 4 minor 32 references

Chaotic magnetization dynamics in magnetic Duffing oscillator

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims the LLG equation for a ferromagnet with a perpendicular static field is a magnetic Duffing oscillator whose chaotic response to periodic drives can be tuned by the static field.

desk verdict A solid, honest toy model showing field-tunable Duffing-like chaos in magnetization dynamics, but the homoclinic-orbit mechanism is asserted heuristically and its own Fig. 7(b) contradicts the strict version of the claim. read the letter →

arxiv 2506.02762 v1 pith:R5XCXGWT submitted 2025-06-03 cond-mat.mes-hall cond-mat.mtrl-scinlin.CDphysics.app-ph

Ryo Tatsumi , Takahiro Chiba , Takashi Komine , Hiroaki Matsueda This is my paper
classification cond-mat.mes-hallcond-mat.mtrl-scinlin.CDphysics.app-ph
keywords Landau-Lifshitz-GilbertequationmagneticDuffingoscillatorhomoclinicorbitchaoticmagnetizationdynamicsLyapunovexponentspin-orbittorqueOerstedfieldpitchforkbifurcation
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

This paper proposes that a ferromagnet with uniaxial magnetic anisotropy, placed in a static magnetic field perpendicular to the anisotropy axis, is a magnetic analogue of the Duffing oscillator. The perpendicular field bends the magnetic potential into a double well, and the undamped Landau-Lifshitz-Gilbert phase portrait acquires butterfly-shaped homoclinic orbits whose topology matches the Duffing oscillator. When a periodic Oersted field or spin-orbit torque is added, the magnetization shows chaotic dynamics with positive Lyapunov exponents, and the static field acts as a control knob: as it approaches the anisotropy field, the drive amplitude needed for chaos shrinks toward zero, while above it the homoclinic orbits and the chaos disappear, apart from small low-damping exceptions near the pitchfork bifurcation. A sympathetic reader would care because this is a concrete, minimal mechanism connecting well-understood nonlinear dynamics to spintronic devices, with predictions about where to look for chaotic magnetization in experiments.

What carries the argument

The load-bearing object is the phase portrait of the undamped LLG equation on the sphere $S^2$, and in particular its homoclinic orbits---trajectories that leave a saddle point and return to the same saddle. Linear stability analysis of the equilibrium points provides the geometry: the Jacobian at the saddle has real eigenvalues of opposite sign, the centers have purely imaginary eigenvalues whose imaginary part defines the resonance frequency $\omega_0$, and the pitchfork bifurcation at $B_x = B_K$ marks where the double well collapses into one well. The paper then invokes the Smale-horseshoe heuristic: a homoclinic orbit in the undamped two-dimensional system is the qualitative precondition for chaos once damping and periodic forcing make the system three-dimensional, turning homoclinic tangles into a strange attractor. The maximum Lyapunov exponent, computed with the Shimada-Nagashima method, is the diagnostic that turns this geometric heuristic into a numerical statement about the driven system. The difference in torque direction is also part of the machinery: the Oersted field pushes the magnetization along isoenergetic curves, while SOT pushes it along the energy gradient, which the paper argues is why the two drives produce different chaotic parameter ranges.

What would settle it

A high-resolution scan of the maximum Lyapunov exponent across $B_x \in [195, 205]$ mT and $B_{ac} \in [0, 15]$ mT at $\alpha=0.01$ would settle whether the chaotic pockets above $B_K$ are real and how sharply the chaos boundary tracks the homoclinic-orbit condition.

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Extended reading notes

Core claim

The paper's central claim is that the LLG equation for a uniform ferromagnet with uniaxial anisotropy field $B_K$ and a static perpendicular field $B_x$ is a magnetic Duffing oscillator. For $B_x < B_K$, the equilibrium at $\phi=0$ is a saddle and the two minima at $\cos\varphi_0 = B_x/B_K$ are centers, and the phase space on the sphere $S^2$ contains two homoclinic orbits in a butterfly topology identical to the Duffing oscillator. With a periodic Oersted field or SOT-induced effective field at the linear resonance frequency, the authors compute bifurcation diagrams, Poincar\'e sections, and maximum Lyapunov exponents, finding positive Lyapunov exponents in finite windows of drive amplitude, with chaotic trajectories confined to one side of the double well or traversing both wells. The parameter range of the chaotic response expands as $B_x$ approaches $B_K$ from below, and the required drive amplitude tends to zero; for $B_x > B_K$ the homoclinic orbits disappear and the chaos vanishes, except for small chaotic pockets near the pitchfork bifurcation at low damping ($\alpha=0.01$).

Load-bearing premise

The load-bearing premise is that the existence of homoclinic orbits in the undamped, undriven LLG system predicts chaos in the damped, periodically driven system; the paper checks this numerically over parameter meshes, but the link is a heuristic, and the numerics themselves show exceptions near the pitchfork bifurcation at low damping.

Editorial extensions

If this is right

  • Tuning the static field $B_x$ toward $B_K$ lowers the Oersted-field amplitude needed for chaos to about 1 mT at $B_x=195$ mT, two orders of magnitude below $B_K$, making the chaotic regime experimentally reachable.
  • The same mechanism works for both Oersted and spin-orbit-torque drives, so chaotic magnetization dynamics does not rely on a specific driving force and should appear in various spintronic devices.
  • Above the pitchfork bifurcation ($B_x > B_K$), the absence of homoclinic orbits means chaos should disappear, giving a clear experimental signature to test the mechanism.
  • The double-well topology predicts two qualitatively different chaotic states---confined to one well and traversing both wells---linked to the two homoclinic orbits of the butterfly phase portrait.

Reading between the lines

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

  • The simulations fix the drive frequency at the linear resonance frequency computed at the center; sweeping the drive frequency could reveal chaotic regimes at smaller drive amplitudes or additional chaotic windows, since the resonance frequency itself changes with amplitude in the double-well regime.
  • The low-damping chaotic pockets above $B_x > B_K$ imply the homoclinic-orbit criterion is sufficient but not necessary; a second route to chaos exists near the pitchfork bifurcation and deserves its own analysis.
  • If the mechanism is robust, spin-torque ferromagnetic resonance measurements on ferromagnet/heavy-metal bilayers should show broadband voltage noise and fractal Poincar\'e structure in the predicted $B_x$ and drive-amplitude windows, offering a direct observable test.
  • Because the mechanism depends only on the double-well shape of the magnetic potential, the same chaotic response should be reproducible with voltage-controlled magnetic anisotropy or shaped demagnetizing fields, not only with a perpendicular static field.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a 'magnetic Duffing oscillator': a uniformly magnetized ferromagnet with uniaxial anisotropy under a static field perpendicular to the anisotropy axis. For B_x < B_K the undamped LLG flow has a double-well magnetic potential with two centers and a saddle, and the phase portrait contains butterfly-shaped homoclinic orbits. The authors use linear stability analysis to extract the resonance frequency from the Jacobian eigenvalues, then add a periodic Oersted field or spin-orbit torque and compute bifurcation diagrams, Poincaré sections, and the maximum Lyapunov exponent. They find chaotic regions in the (B_x, drive-amplitude) plane that shrink as B_x approaches B_K and mostly disappear for B_x > B_K, with low-damping exceptions near the pitchfork bifurcation. The paper argues that the homoclinic orbits are the origin of the chaos and that the static field can therefore control the parameter range of chaotic dynamics.

Significance. If the claims hold, this is a useful toy model that connects spintronic magnetization dynamics with the classical Duffing oscillator, and it makes falsifiable predictions about where chaotic magnetization dynamics should appear in ferromagnet/heavy-metal bilayers. The paper has clear strengths: physical parameters are used without fitting to the target behavior; the linear stability analysis is correct; the numerical evidence combines Lyapunov exponents, bifurcation diagrams, and Poincaré sections; and the comparison with the standard Duffing oscillator in Appendix A is helpful. The central numerical finding that periodically driven LLG dynamics can be chaotic in this geometry is credible and is supported by multiple diagnostics.

major comments (3)
  1. [II, Eq. (10)] Equation (10) is not exactly the polar-coordinate form of the LLG equation (1). Solving dm/dt = -γ m×B + α m×dm/dt for dm/dt introduces a factor 1/(1+α^2) multiplying all torque terms, but Eq. (10) omits this prefactor. For the values used (α=0.01 and 0.05) the numerical difference is at most 0.25%, so the qualitative conclusions are probably unaffected, but the manuscript should either include the prefactor or explicitly state that an approximate form is being simulated. As written, the simulations solve a slightly different equation from the one advertised.
  2. [IV.D, Fig. 7] The causal claim that the homoclinic orbit of the undamped (α=0) system is the origin of the driven chaos is not established. In a damped, periodically driven system, the relevant criterion is a transverse intersection of stable and unstable manifolds of the saddle-type periodic orbit, normally quantified by a Melnikov function; the undamped homoclinic orbit is neither necessary nor sufficient. No Melnikov calculation or invariant-manifold computation is presented. More importantly, Fig. 7(b) shows λ>0 for B_x>200 mT, where the undamped phase portrait has no homoclinic orbit. The statement in Sec. IV.D that 'chaotic dynamics disappears when the homoclinic orbit vanishes' is therefore literally false at low damping. The paper should either add a direct manifold/Melnikov check for both drive types or explicitly reframe the homoclinic-orbit link as a heuristic that is reliable for moderate damping but has documented exceptions near the pitchfork bifurcation.
  3. [IV.D, Fig. 7] The parameter-space maps in Fig. 7 change B_x and simultaneously change the drive frequency ω0, which is set to the linear resonance frequency of the center and vanishes as B_x→B_K. The shrinking and expansion of chaotic regions could therefore reflect the changing frequency detuning rather than only the changing homoclinic orbit geometry. To substantiate the control claim, the authors should provide at least one scan at fixed drive frequency, or otherwise demonstrate that the qualitative boundaries in Fig. 7 are not an artifact of the frequency choice.
minor comments (4)
  1. [IV.B] In the paragraph discussing Fig. 4(d), the phrase 'right panel of Fig. 4(c)' should refer to Fig. 4(d); the Fourier spectrum for B_ac = 9.425 mT is shown in panel (d), not panel (c).
  2. [IV.D] The text says 'the blight area indicates λ>0'; this should read 'bright area'.
  3. [II and IV] The simulation section does not state the initial condition m(0) or the precise number of periods used for the Lyapunov exponent and bifurcation diagrams. Adding these details would improve reproducibility.
  4. [V] There is a typo in 'spintoronics' in the first paragraph of Sec. V, and reference [28] contains 'Nonlineaar' instead of 'Nonlinear'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the static homoclinic-orbit analysis and the driven Lyapunov-exponent simulations are independent, with parameters fixed by physical constants and Jacobian linearization rather than fitted to the target chaotic regions.

full rationale

The derivation chain is self-contained. The undamped LLG phase portrait is analyzed directly from the potential V(θ,φ)=−γ(Bx sinθ cosφ + (1/2)BK sin²θ sin²φ) and the Jacobian at the equilibria; for Bx<BK the saddle at (π/2,0) has real eigenvalues ±γ√(Bx(BK−Bx)) and the centers have purely imaginary eigenvalues, and the homoclinic orbits are read off the phase portrait in Fig. 2(b). This static analysis does not use the chaotic Lyapunov data as an input. The driven simulations solve the full LLG equations with Oersted or SOT forcing, and chaos is diagnosed by the maximum Lyapunov exponent computed with the Shimada-Nagashima method. The drive frequency ω0 is fixed independently as the imaginary part of the Jacobian eigenvalues at the center, not extracted from the Lyapunov maps, so the parameter range of chaos is not fitted by construction. The paper explicitly acknowledges that the correspondence between homoclinic orbits and chaos is imperfect: for α=0.01, Fig. 7(b) shows λ>0 regions for Bx>BK where the undamped portrait has no homoclinic orbit, and the text attributes these to perturbation-sensitive dynamics near the pitchfork bifurcation. This is an honest numerical check rather than a circular confirmation. There is no load-bearing self-citation: the only author self-citation (Ref. [30], Chiba) appears in a peripheral ST-FMR discussion, while the central mechanism relies on standard textbook results (Wiggins, Strogatz) and on the paper's own numerical simulations. No uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in via citation. The homoclinic-orbit explanation is heuristic and not a rigorous Melnikov/manifold proof, but that is a rigor/correctness concern, not circularity: the predicted correlation is tested against an independently computed Lyapunov exponent and is found to hold broadly but not universally. Accordingly, the circularity score is 0.

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

No free parameters are fitted to data. The physical parameters (gamma, B_K, B_x, alpha) are standard material constants or chosen field values, and the scan in Fig. 7 explores the influence of B_x. The resonance frequency is derived from the linearized Jacobian, not fitted. No new entities are introduced; the 'magnetic Duffing oscillator' is a name for the existing model with a perpendicular field.

assumptions (6)
  • domain assumption Landau-Lifshitz-Gilbert equation with standard Gilbert damping describes the macrospin dynamics.
    Invoked in Sec. II, Eq. (1). The paper does not justify this beyond citing standard spintronics literature.
  • domain assumption Uniform magnetization (macrospin approximation); spatial modes, exchange, and dipolar fields beyond the uniaxial anisotropy are neglected.
    Sec. II states 'uniform ferromagnet'. This is a standard but strong simplification.
  • domain assumption Effective field contains only a static perpendicular field and a uniaxial anisotropy field; no demagnetization or thermal fluctuations.
    Sec. II, Eq. (2). The paper calls B_K magnetocrystalline or shape anisotropy, but does not include other effective-field terms.
  • domain assumption The AC drive (Oersted field or SOT) is applied exactly at the linear resonance frequency omega_0 computed from the undamped Jacobian at the center.
    Sec. IV: 'we assume the resonance condition, i.e., the angular frequency is equal to omega_0'. The frequency is derived from the model, not fitted.
  • standard math The undamped homoclinic orbit is used as a predictor of chaos in the damped, driven system (Smale horseshoe / homoclinic tangle heuristic).
    Sec. III and IV.D. This is a known criterion from nonlinear dynamics, but the paper does not prove it applies to the LLG system; it confirms numerically.
  • standard math Poincare-Bendixson theorem forbids chaos in 2D, motivating the z dimension for the autonomous system.
    Sec. IV opening. Standard theorem.

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Cite this review

Pith. "Pith review of Chaotic magnetization dynamics in magnetic Duffing oscillator." pith.science (2026). https://pith.science/paper/R5XCXGWT

@misc{pith2026250602762,
  author       = {Pith},
  title        = {Pith review of: Chaotic magnetization dynamics in magnetic Duffing oscillator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R5XCXGWT}},
  note         = {Machine review of arXiv:2506.02762}
}
read the original abstract

We propose a magnetic analogy of the Duffing oscillator--magnetic Duffing oscillator--which is characterized by a double-well magnetic potential of a ferromagnet with a uniaxial magnetic anisotropy. Based on the linear stability analysis of the Landau-Lifshitz-Gilbert equation, we show that an external magnetic field applied perpendicular to the magnetic anisotropy field creates an anharmonicity on the magnetic potential, generating homoclinic orbits in the phase space. By evaluating the Lyapunov exponent, we demonstrate that the magnetic Duffing oscillator exhibits chaotic behaviors in the presence of periodically oscillating external forces: Oersted field and spin-orbit torque by considering the ferromagnet/heavy-metal bilayer. We also show that the external magnetic field can be adjusted to generate or modify homoclinic orbits, thereby controlling the parameter range of the oscillating external forces that induce chaos. This work deepens our understanding of chaotic magnetization dynamics by bridging the fields of nonlinear dynamics and spintronics.

Figures

Figures reproduced from arXiv: 2506.02762 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Schematic illustration of the magnetic Duffing [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) The resonance (angular) frequency of the mag [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The illustration depicts a magnetic Duffing oscillator [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Oscillatory states of the magnetic Duffing oscillator driven by the Oersted field. (a) The bifurcation diagram as a [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The Poincar´e section for the magnetic Duffing oscil [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Oscillatory states of the magnetic Duffing oscillator driven by SOT. (a) The bifurcation diagram as a function of the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: (a) shows the Lyapunov exponent as functions of the static external magnetic field Bx and the Oersted field Bac for α = 0.05. In the context of λ > 0, the blight areas in [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. (a) The potential of the Duffing equation for [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Oscillatory states of the Duffing oscillator. (a) The bifurcation diagram as a function of the strength of the external [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Poincar´e section for the Duffing oscillator. The [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]

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Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    Chaotic neural networks,

    K. Aihara, T. Takabe, and M. Toyoda, “Chaotic neural networks,” Phys. Lett. A, vol.144, 333–340 (1990)

    work page 1990

  2. [2]

    Chaotic neural networks and the travelling salesman problem,

    T. Yamada, K. Aihara, and M. Kotani, “Chaotic neural networks and the travelling salesman problem,” in Proc. Int. Joint Conf. Neural Networks,2, 1549–1552 (1993)

    work page 1993

  3. [3]

    A novel memris- tive chaotic neuron circuit and its application in chaotic neural networks for associative memory,

    C. Pan, Q. Hong, and X. Wang, “A novel memris- tive chaotic neuron circuit and its application in chaotic neural networks for associative memory,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.,40, 521–532 (2021)

    work page 2021

  4. [4]

    Recent advances in physical reservoir comput- ing: a review,

    G. Tanaka, T. Yamane, J. B. H´ eroux, R. Nakane, N. Kanazawa, S. Takeda, H. Numata, D. Nakano and A. Hirose, “Recent advances in physical reservoir comput- ing: a review,” Neural Netw.115, 100–123 (2019)

    work page 2019

  5. [5]

    Optoelec- tronic Reservoir Computing,

    Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman and S. Massar, “Optoelec- tronic Reservoir Computing,” Sci. Rep.2, 287 (2012)

    work page 2012

  6. [6]

    Neuromorphic computing with nanoscale spin- tronic oscillators,

    J. Torrejon, M. Riou, F. A. Araujo1, S. Tsunegi, G. Khalsa, D. Querlioz, P. Bortolotti, V. Cros, K. yakushiji, A. Fukushima, H. Kubota, S. yuasa, M. D. Stiles and J. Grollier “Neuromorphic computing with nanoscale spin- tronic oscillators,” Nature547, 428–431 (2017)

    work page 2017

  7. [7]

    Legenstein and W, Maass., “Edge of chaos and pre- diction of computational performance for neural circuit 12 1 0 1 G 10 0 10§G FIG

    R. Legenstein and W, Maass., “Edge of chaos and pre- diction of computational performance for neural circuit 12 1 0 1 G 10 0 10§G FIG. 10. Poincar´ e section for the Duffing oscillator. The values of the parameter are the same as in Fig. 9 (f). models,” Neural Netw.20, 323-334(2007)

    work page 2007

  8. [8]

    Information processing in echo state networks at the edge of chaos,

    J. Boedecker, O. Obst, J. T. Lizier, N. M. Mayer, and M. Asada, “Information processing in echo state networks at the edge of chaos,” Theory Biosci.131, 205 (2012)

    work page 2012

Show all 32 references
  1. [9]

    Real-Time Com- putation at the Edge of Chaos in Recurrent Neural Net- works,

    N. Bertschinger and T. Natschl¨ ager, “Real-Time Com- putation at the Edge of Chaos in Recurrent Neural Net- works,” Neural computation.16, 1413 (2004)

    2004

  2. [10]

    Computational capability for physical reser- voir computing using a spin-torque oscillator with two free layers,

    T. Yamaguchi, S. Tsunegi, K. Nakajima and T. Taniguchi, “Computational capability for physical reser- voir computing using a spin-torque oscillator with two free layers,” Phys. Rev. B107, 054406 (2023)

    2023

  3. [11]

    Nonlinear magne- tization dynamics in nanosystems,

    I. Mayergoyz, G. Bertotti, C. Serpico, “Nonlinear magne- tization dynamics in nanosystems,” (Elsevier, New York, 2009)

    2009

  4. [12]

    Voltage-driven multistability and chaos in magnetic films,

    S. Contreras-Celada, M. G. Clerc, S. Coulibaly, R. G. Rojas, A. O. Leon, “Voltage-driven multistability and chaos in magnetic films,” J. Magn. Magn. Mater.562, 169793 (2022)

    2022

  5. [13]

    Non-periodic input-driven magnetization dynamics in voltage-controlled parametric oscillator,

    T. Taniguchi, “Non-periodic input-driven magnetization dynamics in voltage-controlled parametric oscillator,” J. Magn. Magn. Mater.563, 170009 (2022)

    2022

  6. [14]

    Dynamic magnetization states of a spin valve in the presence of dc and ac currents: Synchronization, modification, and chaos,

    Z. Li, Y. C. Li and S. Zhang, “Dynamic magnetization states of a spin valve in the presence of dc and ac currents: Synchronization, modification, and chaos,” Phys. Rev. B 74, 054417 (2006)

    2006

  7. [15]

    Synchronization and chaos in a spin-torque oscillator with a perpendicularly magne- tized free layer,

    T. Yamaguchi, N. Akashi, K. Nakajima, S. Tsunegi, H. Kubota, and T. Taniguchi, “Synchronization and chaos in a spin-torque oscillator with a perpendicularly magne- tized free layer,” Phys. Rev. B100, 224422 (2019)

    2019

  8. [16]

    Chaos in nanomagnet via feed- back current,

    T. Taniguchi, N. Akashi, H. Notsu, M. Kimura, H. Tsuka- hara, and K. Nakajima, “Chaos in nanomagnet via feed- back current,” Phys. Rev. B100, 174425 (2019)

    2019

  9. [17]

    Chaotic dynamics of a magnetic nanoparticle,

    J. Bragard, H. Pleiner, O. J. Suarez, P. Vargas, J. A. C. Gallas, and D. Laroze. “Chaotic dynamics of a magnetic nanoparticle,” Phys. Rev. E84, 037202 (2011)

    2011

  10. [18]

    The Duffing equation: non- linear oscillators and their behavior,

    I. Kovacic, M.J.Brennan “The Duffing equation: non- linear oscillators and their behavior,” John Wiley amp; Sons, New Jersey (2011)

    2011

  11. [19]

    Introduction to Applied Nonlinear Dynamical Systems and Chaos,

    S. Wiggins, M. Golubitsky, “Introduction to Applied Nonlinear Dynamical Systems and Chaos,” (Springer, New York, 2003)

    2003

  12. [20]

    Condensed Matter Field The- ory,

    A. Altland, B. Simons, “Condensed Matter Field The- ory,” (Cambridge University Press, Cambridge, England 2006)

    2006

  13. [21]

    Lyapunov Exponents: A Tool to Explore Complex Dynamics,

    A. Pikovsky and A. Politi, “Lyapunov Exponents: A Tool to Explore Complex Dynamics,” (Cambridge University Press, Cambridge, England, 2016)

    2016

  14. [22]

    A numerical approach to er- godic problem of dissipative dynamical systems,

    I. Shimada, T.Nagashima, “A numerical approach to er- godic problem of dissipative dynamical systems,” Prog. Theoret. Phys.61, 1605 (1979)

    1979

  15. [23]

    Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems,

    A. Manchon, J. ˇZelezn´ y, I. M. Miron, T. Jungwirth, J. Sinova, A. Thiaville, K. Garello, and P. Gambardella, “Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems,” Rev. Mod. Phys.91, 035004 (2019)

    2019

  16. [24]

    Transition to hyperchaos in chaotically forced coupled oscillators,

    T. Kapitaniak, “Transition to hyperchaos in chaotically forced coupled oscillators,” Phys. Rev. E47, R2975(R) (1993)

    1993

  17. [25]

    Transition to hyperchaos: Sudden expansion of attractor and inter- mittent large-amplitude events in dynamical systems,

    S. L.Kingston, T. Kapitaniak, S.K. Dana, “Transition to hyperchaos: Sudden expansion of attractor and inter- mittent large-amplitude events in dynamical systems,” Chaos32, 081106 (2022)

    2022

  18. [26]

    Noise-induced bifurcations, multiscaling and on-off intermittency,

    S. Auma ˆ ıtre, K. Mallick, and F. P´ etr´ elis, “Noise-induced bifurcations, multiscaling and on-off intermittency,” J. Stat. Mech. (2007) P07016

    2007

  19. [27]

    Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engi- neering,

    S. H. Strogatz, “Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engi- neering,” 1st ed. (Westview Press, Boulder, CO, 2001)

    2001

  20. [28]

    Tutorial: Nonlineaar magnonics,

    S. Zheng, Z. Wang, Y. Wang, F. Sun, Q. He, P. Yan, and H. Y. Yuan, “Tutorial: Nonlineaar magnonics,” J. Appl. Phys.134, 151101 (2023)

    2023

  21. [29]

    Spin-Torque Ferromagnetic Resonance Induced by the Spin Hall Effect,

    L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, “Spin-Torque Ferromagnetic Resonance Induced by the Spin Hall Effect,” Phys. Rev. Lett.106, 036601 (2011)

    2011

  22. [30]

    Current- Induced Spin-Torque Resonance of Magnetic Insulators,

    T. Chiba, G. E.W. Bauer, and S. Takahashi, “Current- Induced Spin-Torque Resonance of Magnetic Insulators,” Phys. Rev. Appl.2, 034003 (2014)

    2014

  23. [31]

    Anomalous Gilbert damping and Duffing features of the superconductor-ferromagnet- superconductorφ 0 Josephson junction,

    Y. M. Shukrinov, I. R. Rahmonov, A. Janalizadeh, and M. R. Kolahchi, “Anomalous Gilbert damping and Duffing features of the superconductor-ferromagnet- superconductorφ 0 Josephson junction,” Phys. Rev. B 104, 224511 (2021)

    2021

  24. [32]

    Duffing oscillation-induced reversal of magnetic vortex core by a resonant perpendicular magnetic field,

    K-W. Moon, B. S. Chun, W. Kim, Z. Q. Qiu, and C. Hwang. “Duffing oscillation-induced reversal of magnetic vortex core by a resonant perpendicular magnetic field,” Sci Rep.4, 6170 (2014)

    2014

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