Geodesic
Implicit finite-difference scheme for a Duffing oscillator with a derivative of variable fractional order of the Riemann-Liouville type
Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 40 (2022) no. 3, pp. 179-198 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The article considers an implicit finite-difference scheme for the Duffing equation with a derivative of a fractional variable order of the Riemann-Liouville type. The issues of stability and convergence of an implicit finite-difference scheme are considered. Test examples are given to substantiate the theoretical results. Using the Runge rule, the results of the implicit scheme are compared with the results of the explicit scheme. Phase trajectories and oscillograms for a Duffing oscillator with a fractional derivative of variable order of the Riemann-Liouville type are constructed, chaotic modes are detected using the spectrum of maximum Lyapunov exponents and Poincare sections. Q-factor surfaces, amplitude-frequency and phase-frequency characteristics are constructed for the study of forced oscillations. The results of the study showed that the implicit finite-difference scheme shows more accurate results than the explicit one.
Keywords: Duffing oscillator, Runge rule, Riemann-Liouville operator, Grunwald-Letnikov operator, amplitude-frequency response, phase-frequency response, Q-factor, Lyapunov exponents, Poincare sections
Mots-clés : oscillogram. 
@article{VKAM_2022_40_3_a14,
     author = {V. A. Kim and R. I. Parovik},
     title = {Implicit finite-difference scheme for a {Duffing} oscillator with a derivative of variable fractional order of the {Riemann-Liouville} type},
     journal = {Vestnik KRAUNC. Fiziko-matemati\v{c}eskie nauki},
     pages = {179--198},
     year = {2022},
     volume = {40},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VKAM_2022_40_3_a14/}
}
TY  - JOUR
AU  - V. A. Kim
AU  - R. I. Parovik
TI  - Implicit finite-difference scheme for a Duffing oscillator with a derivative of variable fractional order of the Riemann-Liouville type
JO  - Vestnik KRAUNC. Fiziko-matematičeskie nauki
PY  - 2022
SP  - 179
EP  - 198
VL  - 40
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VKAM_2022_40_3_a14/
LA  - ru
ID  - VKAM_2022_40_3_a14
ER  - 
%0 Journal Article
%A V. A. Kim
%A R. I. Parovik
%T Implicit finite-difference scheme for a Duffing oscillator with a derivative of variable fractional order of the Riemann-Liouville type
%J Vestnik KRAUNC. Fiziko-matematičeskie nauki
%D 2022
%P 179-198
%V 40
%N 3
%U http://geodesic.mathdoc.fr/item/VKAM_2022_40_3_a14/
%G ru
%F VKAM_2022_40_3_a14
V. A. Kim; R. I. Parovik. Implicit finite-difference scheme for a Duffing oscillator with a derivative of variable fractional order of the Riemann-Liouville type. Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 40 (2022) no. 3, pp. 179-198. http://geodesic.mathdoc.fr/item/VKAM_2022_40_3_a14/

[1] Nakhushev A. M., Drobnoe ischislenie i ego priminenie, Fizmatlit, M., 2003, 272 pp.

[2] Petras I., Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer, New York, 2010, 180 pp.

[3] Kuznetsov S. P., Dinamicheskii khaos, Fizmatlit, M., 2001, 296 pp.

[4] Zeldovich B. Ya., Tabiryan N. V., “Opticheskaya bistabilnost na orientatsionnoi nelineinosti zhidkikh kristallov”, Kvantovaya elektronika, 11:12 (1984), 2419–2426 DOI: 10.1070/QE1984v014n12ABEH006232

[5] Eskov V. V. i dr., “Khaoticheskaya dinamika miogramm”, Vestnik novykh meditsinskikh tekhnologii. Elektronnoe izdanie, 2016, no. 3 DOI: 12737/21668

[6] Ejikeme C. L., et al., “Solution to nonlinear Duffing oscillator with fractional derivatives using homotopy analysis method (HAM)”, Global Journal of Pure and Applied Mathematics, 14:10 (2018), 1363–1388 ISSN 0973-1768

[7] Syta A., “Chaotic vibrations of the Duffing system with fractional damping”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 24:1 (2014), 10–16 DOI: 10.1063/1.4861942 | DOI

[8] Xing W., “Threshold for chaos of a duffing oscillator with fractional-order derivative”, Shock Vib., 2019 (2019), 1–16

[9] Shen Y., Li H., Yang S., Peng M., Han Y., “Primary and subharmonic simultaneous resonance of fractional-order Duffing oscillator”, Nonlinear Dyn., 102 (2020), 1485–1497 | DOI

[10] El-Dib Y. O., “Stability approach of a fractional-delayed Duffing oscillator”, Discontinuity Nonlinearity Complex, 9 (2020), 367–376 | DOI

[11] Eze S. C., “Analysis of fractional Duffing oscillator”, Rev. Mex. Física, 66 (2020), 187–191 | DOI

[12] Gouari Y., Dahmani Z., Jebril I., “Application of fractional calculus on a new differential problem of Duffing type”, Adv. Math. Sci. J., 9 (2020), 10989–11002 | DOI

[13] Syam M. I., “The Modified Fractional Power Series Method for Solving Fractional Undamped Duffing Equation with Cubic Nonlinearity”, Nonlinear Dyn. Syst. Theory, 20 (2020), 568–577

[14] Barba-Franco J. J., Gallegos A., Jaimes-Reátegui R., Pisarchik A. N., “Dynamics of a ring of three fractional-order Duffing oscillators”, Chaos Solitons Fractals, 155 (2022), 111–747 | DOI

[15] Kim V. A., “Ostsillyator Duffinga s vneshnim garmonicheskim vozdeistviem i proizvodnoi peremennogo drobnogo poryadka Rimana-Liuvillya, kharakterizuyuschaya vyazkoe trenie”, Vestnik KRAUNTs. Fiz.-mat. nauki, 13:2 (2016), 46–49 DOI: 10.18454/2079-6641-2016-13-2-50-54

[16] Kim V. A., Parovik R. I., “Raschet maksimalnykh pokazatelei Lyapunova dlya kolebatelnoi sistemy Duffinga so stepennoi pamyatyu”, Vestnik KRAUNTs. Fiz.-mat. nauki, 23:3 (2018), 98–105 DOI: 10.18454/2079-6641-2018-23-3-98-105

[17] Kim V. A., Parovik R. I., “Application of the Explicit Euler Method for Numerical Analysis of a Nonlinear Fractional Oscillation Equation”, Fractional and fractals, 6:5 (2022), 274–293 DOI: 10.3390/fractalfract6050274 | DOI

[18] Kim V. A., Parovik R. I., “Issledovanie vynuzhdennykh kolebanii ostsillyatora Duffinga s proizvodnoi peremennogo drobnogo poryadka Rimana-Liuvillya”, Izvestiya Kabardino-Balkarskogo nauchnogo tsentra RAN, 93:1 (2020), 46–56 DOI: 10.35330/1991-6639-2020-1-93-46-56

[19] Kim V. A., Parovik R. I., “Mathematical model of fractional Duffing oscillator with variable memory”, Mathematics, 8:11 (2020), 20–34 DOI: 10.3390/math8112063