# Vestnik КRAUNC. Fiz.-Mat. nauki. 2023. vol. 45. no. 4. P. 9-23. ISSN 2079-6641

MATHEMATICAL MODELLING
https://doi.org/10.26117/2079-6641-2023-45-4-9-23
Research Article
Full text in Russian
MSC 34A08, 34A34

Contents of this issue

Qualitative Analysis of Selkov’s Fractional Dynamical System with Variable Memory Using a Modified Test 0-1 Algorithm

R. I. Parovik^\ast

Institute for Cosmophysical Research and Radio Propagation FEB RAS, 684034, v. Paratunka, Mirnaya st., 7, Russia

Abstract. The article examines chaotic and regular modes of a fractional dynamic Selkov system with
variable memory. First, a numerical analysis is carried out using the Adams-Bashforth-Moulton method. Next, preliminary processing (modification) is carried out on the resulting solution, which consists of selecting from the given values the values corresponding to local extrema. Next, the set of values thinned out in this way is fed to the input of the Test 0-1 algorithm. The main idea of the Test 0-1 algorithm is to calculate the statistical characteristics of a discrete time series: the standard standard deviation, as well as its asymptotic growth rate through the correlation (covariance and variation) between the corresponding vectors. As a result, after repeatedly calculating the correlation coefficient, its median value is selected, which is the main criterion for choosing a dynamic mode scenario. If the median value is close enough to one, then we are dealing with a chaotic regime, and if it is close to zero, then with a regular regime. The Adams-Bashforth-Moulton numerical algorithm and the modified Test 0-1 algorithm were implemented in the computer mathematics system MATLAB, and the simulation results were visualized using bifurcation diagrams. In the work, it was shown using the modified Test 0-1 algorithm that a fractional dynamic system with variable memory can have chaotic modes. This is very important to know due to the fact that Selkov’s fractional dynamic system describes a self-oscillating regime, which, for example, can be used to describe the interaction of microseisms. In this case, chaotic modes must be eliminated by selecting appropriate values of system parameters.

Key words: mathematical modeling, Selkov fractional dynamic system, oscillogram, phase trajectory, Test 0-1 algorithm, bifurcation diagrams, statistical characteristics, fractional derivatives of variable order, heredity, MATLAB

Received: 02.11.2023; Revised: 16.11.2023; Accepted: 23.11.2023; First online: 11.12.2023

For citation. 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, 45: 4, 9-23. EDN: QBHJSG. https://doi.org/10.26117/2079-6641-2023-45-4-9-23.

Funding. The research was carried out within the framework of the Russian Science Foundation grant No. 22-11-00064 on the topic “Modeling of dynamic processes in the geospheres taking into account heredity”

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 fully responsible for submitting the final version of the article to the press.

^\astCorrespondence: E-mail: parovik@ikir.ru

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

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

References

1. Rabotnov Yu. N. Elementy nasledstvennoy mekhaniki tvordykh tel [Elements of hereditary mechanics of solids]. Мoscow. Nauka, 1977. 384 p.(In Russian).
2. Volterra V. Functional theory, integral and integro-differential equations. New York. Dover Publications, 2005. 288 p.
3. Nakhushev A. M. Drobnoye ischisleniye i yego primeneniye [Fractional calculus and its applications]. Moscow. Fizmatlit. 2003. 272 p.(In Russian).
4. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. Amsterdam. Elsevier. 2006. 523 p.
5. Nigmatullin R. R. Fractional integral and its physical interpretation. Theoret. and Math. Phys. 1992. vol. 90. no. 3. pp. 242–251.
6. Parovik R. I. Haoticheskiye i regulyarnyye rezhimy drobnykh ostsillyatorov [Chaotic and regular modes of fractional oscillators]. Petropavlovsk-Kamchatskiy. KAMCHATPRESS, 2019. 132 p.(In Russian).
7. Parovik R. I. Investigation of the Selkov fractional dynamical systems. Vestnik. KRAUNC. Fiz.-mat. nauki. 2022. vol. 41. no. 4. pp. 146-166. DOI: 10.26117/2079-6641-2022-41-4-146-166 (In Russian).
8. 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.
9. Selkov E. E. Self-oscillations in glycolysis. I. A simple kinetic model. Eur. J. Biochem. 1968. no. 4. pp. 79–86.
10. 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).
11. Patnaik S., Hollkamp J.P., Semperlotti F. Applications of variable-order fractional operators: A review. Proc. R. Soc. A R. Soc. Publ. 2020. vol. 476. 20190498. DOI: 10.1098/rspa.2019.0498.
12. Benettin G., Galgani L., Giorgilli A., Strelcyn J. M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica. 1980. vol. 16. no. 1. pp. 9-20.
13. Wolf A., Swift J. B., Swinney H. L., Vastano J. A. Determining Lyapunov exponents from a time series. Physica D: nonlinear phenomena. 1985. vol. 16. no. 3. pp. 285-317.
14. 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. no.1-4. pp. 3-22. DOI: 10.1023/A:1016592219341.
15. Yang C., Liu F. A computationally effective predictor-corrector method for simulating fractional order dynamical control system. ANZIAM Journal. 2005. vol. 47. pp. 168-184. DOI: 10.21914/anziamj.v47i0.1037.
16. Garrappa R. Numerical solution of fractional differential equations: A survey and a software tutorial. Mathematics. 2018. vol. 6. no. 2. 016. DOI: 10.3390/math6020016.
17. Gottwald G. A., Melbourne I. On the implementation of the 0–1 test for chaos. SIAM Journal on Applied Dynamical Systems. 2009. vol. 8. no. 1. pp. 129-145. DOI: 10.1137/080718851.
18. Fouda J. S.A.E., 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. vol. 24. no. 5.1450063. DOI: 10.1142/S0218127414500631.