Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 53. no. 4. P. 75 – 92. ISSN 2079-6641

INFORMATION AND COMPUTATIONAL TECHNOLOGY
https://doi.org/10.26117/2079-6641-2025-53-4-75-92
Research Article
Full text in Russian
MSC 65L05, 34A08, 34C23, 65Y05

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ABMSelkovFracSim 2.0 Software Package for Quantitative and Qualitative Analysis of the Selkov Fractional Oscillator

R. I. Parovik^{\ast}

Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, 684034, Mirnaya st., 7, Paratunka, Kamchatka, Russia

Abstract. This article presents a new version of the ABMSelkovFracSim software package (ABMSelkovFracSim 2.0), written in the Python programming language. This package includes a software module for constructing bifurcation diagrams for the qualitative analysis of the oscillatory modes of the Selkov fractional oscillator. The Selkov fractional oscillator is a Cauchy problem for a system of two coupled nonlinear ordinary differential equations with Gerasimov-Caputo derivatives of fractional order variables and non-constant coefficients. Using the Adams-Bashforth-Multon numerical algorithm, this software package not only constructs oscillograms and phase trajectories based on the values and functions of key parameters of the model equations, as previously implemented in the ABMSelkovFracSim software package, but also calculates bifurcation diagrams for the oscillatory modes of the Selkov fractional oscillator. The bifurcation diagram construction algorithm was implemented not only in a sequential version but also in a parallel version, leveraging the computing power of the central processing unit (CPU). The algorithm automatically determines the number of CPU threads, and the user can select the required number for faster bifurcation diagram construction. In this article, we present bifurcation diagrams constructed depending on the characteristic time scale of \theta. It is shown that, depending on this scale and the orders of fractional derivatives, various oscillatory modes can arise, transitioning from one to another. This makes it possible to determine the ranges of \theta parameter values within which a particular mode exists, which is important for solving specific applied problems.

Key words: software package, Selkov fractional oscillator, Adams-Bashforth-Multon method, bifurcation diagrams, oscillograms, phase trajectory.

Received: 20.11.2025; Revised: 04.12.2025; Accepted: 05.12.2025; First online: 06.12.2025

For citation. Parovik R. I. ABMSelkovFracSim 2.0 software package for quantitative and qualitative analysis of the Selkov fractional oscillator. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 53: 4, 75-92. EDN: CLATTO.
https://doi.org/10.26117/2079-6641-2025-53-4-75-92.

Funding. The work was carried out within the framework of the the state Assignment of IKIР FEB RAS (No. NIOKTR 124012300245-2).

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author participated in the writing of the article and is solely responsible for submitting the final version of the article for publication.

^{\ast}Correspondence: E-mail: parovik@ikir.ru

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

© Parovik R. I., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

References

1. Selkov E. E. Self-oscillations in glycolysis. I. A simple kinetic model. Eur. J. Biochem. 1968. no. 4. pp. 79–86.
2. Makovetsky V. I., Dudchenko I.P., Zakupin A. S. Auto oscillation model of microseism’s sources. Geosist. Pereh. Zon. 2017. no. 4. pp. 37–46 (In Russian).
3. Parovik R. I. Studies of the Fractional Selkov Dynamical System for Describing the Self-Oscillatory Regime of Microseisms. Mathematics. 2022. vol. 10, No. 22. 4208. DOI: 10.3390/math10224208.
4. Parovik R. I. Investigation of the Selkov fractional dynamical systems. Vestnik KRAUNC. Fiz.-mat. nauki. 2022. vol. 41. no. 4. pp. 146-166. DOI:10.26117/2079-6641-2022-41-4-146166 (In Russian).
5. Rabotnov Y. N. Elements of Hereditary Solid Mechanics, Mir. 1980.
6. Volterra V. Functional Theory, Integral and Integro-Differential Equations, Dover Publications. 2005.
7. Schroeder M.Fractals, chaos, power laws: Minutes from an infinite paradise. Courier Corporation. 2009.
8. Nakhushev A. M. Fractional calculus and its applications, FIZMATLIT, Moscow. 2003. 272 p. (In Russian).
9. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, The Netherlands. 2006.
10. Parovik R. I. Qualitative analysis of Selkov’s fractional dynamical system with variable memory using a modified Test 0-1 algorithm. Vestnik KRAUNC. Fiz.-mat. nauki. 2023. vol. 45. no. 4. pp. 9-23. DOI:10.26117/2079-6641-2023-45-4-9-23 (In Russian).
11. Parovik R.I. Study of bifurcation diagrams of Selkov’s fractional dynamic system to describe self-oscillatory modes of microseisms. Vestnik KRAUNC. Fiz.-mat. nauki. 2024. vol. 49. no. 4. pp. 24-35. DOI:10.26117/2079-6641-2024-49-4-24-35 (In Russian).
12. Shaw Z.A. Learn Python the Hard Way. Addison-Wesley Professional. 2024.
13. Van Horn B.M., Nguyen Q. Hands-On Application Development with PyCharm: Build Applications Like a Pro with the Ultimate Python Development Tool, Packt Publishing Ltd. 2023.
14. Parovik R. I. ABMSelkovFracSim — a software package for the qualitative and quantitative analysis of the Selkov fractional dynamic systemCertificate of state registration of computer program No. 2024681529 Russian Federation. Appl. 11.09.2024: Published 11.09.2024. Applicant: Federal State Budgetary Scientific Institution Institute of Cosmophysical Research and Radio Wave Propagation, Far Eastern Branch of the Russian Academy of Sciences. (In Russian).
15. Parovik R. Selkov’s Dynamic System of Fractional Variable Order with Non-Constant Coefficients. Mathematics. 2025. Vol. 13, No. 3. DOI: 10.3390/math13030372.
16. Aguilar J. F. G., Hern´andez M. M. Space-Time Fractional Diffusion-Advection Equation with Caputo Derivative. Abstract and Applied Analysis. 2014. vol. 2014. pp. 1–8. DOI: 10.1155/2014/283019
17. Novozhenova O. G. Life And Science of Alexey Gerasimov, One of the Pioneers of Fractional Calculus in Soviet Union. Fractional Calculus and Applied Analysis. 2017. vol. 20 (3). pp. 790–809. DOI: 10.1515/fca-2017-0040.
18. Caputo M., Fabrizio M. On the notion of fractional derivative and applications to the hysteresis phenomena. Meccanica. 2017. vol. 52 (13). pp. 3043–3052. DOI: 10.1007/s11012-017-0652-y.
19. Patnaik S., Hollkamp J.P., Semperlotti F. Applications of variable-order fractional operators: a review. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2020. vol. 476 (2234). 20190498. DOI: 10.1098/rspa.2019.0498.
20. Diethelm K., Ford N. J., Freed A. D. A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations. Nonlinear Dynamics. 2002. vol. 29 (1/4). pp. 3–22. DOI: 10.1023/A:1016592219341
21. Yang C., Liu F. A computationally effective predictor-corrector method for simulating fractional order dynamical control system. ANZIAM Journal. 2006. vol. 47. 168. DOI: 10.21914/anziamj.v47i0.1037.
22. Parovik R. I., Tverdyi D. Some Aspects of Numerical Analysis for a Model Nonlinear Fractional Variable Order Equation. Mathematical and Computational Applications. 2021. vol. 26, no. 3. 55. DOI: 10.3390/mca26030055
23. Gottwald G. A., Melbourne I. On the Implementation of the 0–1 Test for Chaos. SIAM Journal on Applied Dynamical Systems. 2009. 8 (1): 129–145. DOI: 10.1137/080718851
24. Armand Eyebe Fouda J. S., Bodo B., Sabat S. L., Effa J. Y. A Modified 0-1 Test for Chaos Detection in Oversampled Time Series Observations. International Journal of Bifurcation and Chaos. 2014. 24 (05): 1450063. DOI: 10.1142/S0218127414500631

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