Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 40. no. 3. pp. 179–198. ISSN 2079-6641

Contents of this issue

Read Russian Version US Flag

MSC 26A33, 34C15

Research Article

Implicit finite-difference scheme for a Duffing oscillator with a derivative of variable fractional order of the RiemannLiouville type

V. A. Kim¹, R. I. Parovik¹²

¹Vitus Bering Kamchatka State University, 683032, Petropavlovsk-Kamchatskiy, Pogranichnaya str., 4, Russia
²Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, 7, Mirnaya st., Kamchatka Krai, Yelizovsky district, 684034, c. Paratunka, Russia

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.

Key words: Duffing oscillator, Runge rule, Riemann-Liouville operator, Grunwald-Letnikov operator, amplitude-frequency response, phase-frequency response, Q-factor, Lyapunov exponents, Poincare sections, oscillogram

DOI: 10.26117/2079-6641-2022-40-3-179-198

Original article submitted: 24.11.2022

Revision submitted: 05.12.2022

For citation. Kim V. A., Parovik R. I. Implicit finite-difference scheme for a Duffing oscillator with a derivative of variable fractional order of the Riemann-Liouville type. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 40: 3, 179-198. DOI: 10.26117/2079-6641-2022-40-3-179-198

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

Funding. Financial support was provided within the framework of the grant of the President of the Russian Federation, No. MD-758.2022.1.1

The content is published under the terms of the Creative Commons Attribution 4.0 International License (

© Kim V. A., Parovik R. I., 2022


  1. Nakhushev A. M. Drobnoye ischisleniye i yego primineniye [Fractional calculus and its application], Moscow. Fizmatlit, 2003, 272 (In Russian).
  2. Petras I. Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. New York. Springer, 2010, 180.
  3. Kuznetsov S.P. Dinamicheskiy khaos [Dynamic chaos]. Moscow, Fizmatlit, 2001, 296 (In Russian).
  4. Zel’dovich B.Ya., N. V. Tabiryan, Optical bistability associated with orientational nonlinearity of liquid crystals, Quantum Electron., 1984, 14:12, 1599–1604, DOI:10.1070/QE1984v014n12ABEH006232.
  5. Eskov V. M., et al. Chaotic dynamics of the myograms, Journal of new medical technologies, eEdition, 2016, 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, 2018, 14:10, 1363–1388, ISSN 0973-1768.
  7. Syta A. Chaotic vibrations of the Duffing system with fractional damping, Chaos: An Interdisciplinary Journal of Nonlinear Science, 2014, 24:1, 10–16, DOI:10.1063/1.4861942.
  8. Xing W. Threshold for chaos of a duffing oscillator with fractional-order derivative, Shock Vib., 2019, 2019, 1–16, DOI:10.1155/2019/1230194.
  9. Shen Y., Li H., Yang S., Peng M., Han Y. Primary and subharmonic simultaneous resonance of fractional-order Duffing oscillator, Nonlinear Dyn., 2020, 102, 1485–1497, DOI:10.1007/s11071-020-06048-w.
  10.  El-Dib Y. O. Stability approach of a fractional-delayed Duffing oscillator, Discontinuity Nonlinearity Complex, 2020, 9, 367–376, DOI:10.5890/DNC.2020.09.003.
  11. Eze S. C. Analysis of fractional Duffing oscillator, Rev. Mex. F´ısica, 2020, 66, 187–191, DOI: 10.31349/revmexfis.66.187.
  12. Gouari Y., Dahmani Z., Jebril I. Application of fractional calculus on a new differential problem of Duffing type, Adv. Math. Sci. J., 2020, 9, 10989–11002, DOI: 10.37418/amsj.9.12.82.
  13. Syam M. I. The Modified Fractional Power Series Method for Solving Fractional Undamped Duffing Equation with Cubic Nonlinearity, Nonlinear Dyn. Syst. Theory, 2020, 20(5), 568–577.
  14. Barba-Franco J. J., Gallegos A., Jaimes-Re´ategui R., Pisarchik A. N. Dynamics of a ring of three fractional-order Duffing oscillators, Chaos, Solitons & Fractals, 2022, 155, 111–747 DOI: 10.1016/j.chaos.2021.111747.
  15. Kim V. A. Duffing oscillator with external harmonic action and variable fractional Riemann-Liouville derivative characterizing viscous friction, Bulletin KRASEC. Physical and Mathematical Sciences. 2016, 13:2, 46-49. DOI: 10.18454/2313-0156-2016-13-2-46-49.
  16. Kim V. А., Parovik R. I. Calculation the maximum Lyapunov exponent for the oscillatory system of Duffing with a degree memory, Vestnik KRAUNC. Fiz.-mat. nauki. 2018, 23: 3, 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, Fractal and Fractional, 2022, 6:5, 274–293, DOI: 10.3390/fractalfract6050274.
  18. Ким В. А., Паровик Р. И. Исследование вынужденных колебаний осциллятора Дуффинга с производной переменного дробного порядка Римана-Лиувилля, Известия Кабардино-Балкарского научного центра РАН, 2020, 93:1, 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, 2020, 8:11, 20–34, DOI: 10.3390/math8112063.

Kim Valentine Aleksandrovich – Junior researcher of the integrative Scientific Research Laboratory of Natural Disasters of Kamchatka — earthquakes and volcanic eruptions, Petropavlovsk-Kamchatskiy, Russia, ORCID 0000-0001-8895-6821.

Parovik Roman Ivanovich – D. Sci. (Phys. & Math.), Associate Professor, Prof., Dep. of Informatics and Mathematics, Vitus Bering Kamchatka State University, Petropavlovsk-Kamchatsky; Leading Researcher, Laboratory for Simulation of Physical Processes, Institute for Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russia, ORCID 0000-0002-1576-1860.