Qualitative analysis of Selkov's fractional dynamical system with variable memory using a modified Test 0-1 algorithm

R. I. Parovik

Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 45 (2023) no. 4, pp. 9-23

Résumé

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.

Détail

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

Bibliographie

[1] Rabotnov Yu. N., Elementy nasledstvennoi mekhaniki tverdykh tel, Nauka, M., 1977, 384 pp. | MR

[2] Volterra V., Functional theory, integral and integro-differential equations, Dover Publications, New York, 2005, 288 pp. | MR

[3] Nakhushev A.M., Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003, 272 pp.

[4] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006, 523 pp. | MR | Zbl

[5] Nigmatullin R. R., “Drobnyi integral i ego fizicheskaya interpretatsiya”, TMF, 90:3 (1992), 354–368 | MR | Zbl

[6] Parovik R.I., Khaoticheskie i regulyarnye rezhimy drobnykh ostsillyatorov, KAMChATPRESS, Petropavlovsk-Kamchatskii, 2019, 132 pp.

[7] Parovik R.I., “Issledovanie drobnoi dinamicheskoi sistemy Selkova”, Vestnik KRAUNTs. Fiziko-matematicheskie nauki, 41:4 (2022), 146–166 DOI: 10.26117/2079-6641-2022-41-4-146-166 | MR

[8] Parovik R.I., “Studies of the Fractional Selkov Dynamical System for Describing the Self-Oscillatory Regime of Microseisms”, Mathematics, 10:22 (2022), 4208 DOI: 10.3390/math10224208 | DOI

[9] Selkov E. E., “Self-oscillations in glycolysis. I. A simple kinetic model”, Eur. J. Biochem., 1968, no. 4, 79–86 | DOI

[10] Makovetskii V. I., Dudchenko I. P., Zakupin A. S., “Avtokolebatelnaya model istochnikov mikroseism”, Geosistemy perekhodnykh zon, 2017, no. 4(1), 37–46

[11] Patnaik S., Hollkamp J. P., Semperlotti F., “Applications of variable-order fractional operators: A review”, Proc. R. Soc. A R. Soc. Publ., 2020, no. 476, 20190498 DOI: 10.1098/rspa.2019.0498 | DOI | MR | Zbl

[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, 16:1 (1980), 9-20 | DOI

[13] Wolf A., Swift J. B., Swinney H. L., Vastano J. A., “Determining Lyapunov exponents from a time series”, Physica D: nonlinear phenomena, 16:3 (1985), 285-317 | DOI | MR | Zbl

[14] Diethelm K., Ford N. J., Freed A. D., “A predictor-corrector approach for the numerical solution of fractional differential equations”, Nonlinear Dynamics, 29:1-4 (2002), 3-22 DOI: 10.1023/A:1016592219341 | DOI | MR | Zbl

[15] Yang C., Liu F., “A computationally effective predictor-corrector method for simulating fractional order dynamical control system”, ANZIAM Journal, 47 (2005), 168-184 DOI: 10.21914/anziamj.v47i0.1037 | DOI | MR

[16] Garrappa R., “Numerical solution of fractional differential equations: A survey and a software tutorial”, Mathematics, 6:2 (2018), 016 DOI: 10.3390/math6020016 | DOI

[17] Gottwald G. A., Melbourne I., “On the implementation of the 0–1 test for chaos”, SIAM Journal on Applied Dynamical Systems, 8:1 (2009), 129-145 DOI: 10.1137/080718851 | DOI | MR | Zbl

[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, 24:5 (2014), 1450063 DOI: 10.1142/S0218127414500631 | DOI | MR | Zbl