×
Parovik, Roman Ivanovich

Qualitative analysis of Selkov’s fractional dynamical system with variable memory using a modified test 0-1 algorithm. (Russian. English summary) Zbl 07831234

Vestn. KRAUNTS, Fiz.-Mat. Nauki 45, No. 4, 9-23 (2023).
Summary: 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.

MSC:

65-XX Numerical analysis
34A08 Fractional ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations

Keywords:

mathematical modeling; Selkov fractional dynamic system; oscillogram; phase trajectory; test 0-1 algorithm; bifurcation diagrams; statistical characteristics; fractional derivatives of variable order; heredity

Software:

Matlab
Cite Review PDF
Full Text: DOI MNR
OA License

References:

[1] Rabotnov Yu. N., Elementy nasledstvennoi mekhaniki tverdykh tel, Nauka, M., 1977, 384 pp.
[2] Volterra V., Functional theory, integral and integro-differential equations, Dover Publications, New York, 2005, 288 pp.
[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. · Zbl 1092.45003
[5] Nigmatullin R. R., “Drobnyi integral i ego fizicheskaya interpretatsiya”, TMF, 90:3 (1992), 354-368
[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 · Zbl 1524.86005 · doi:10.26117/2079-6641-2022-41-4-146-166
[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
[9] Selkov E. E., “Self-oscillations in glycolysis. I. A simple kinetic model”, Eur. J. Biochem., 1968, no. 4, 79-86 · doi:10.1111/j.1432-1033.1968.tb00175.x
[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 · Zbl 1439.26028 · 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, 16:1 (1980), 9-20 · Zbl 0488.70015 · doi:10.1007/BF02128236
[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 · Zbl 0585.58037 · doi:10.1016/0167-2789(85)90011-9
[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 · Zbl 1009.65049 · 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, 47 (2005), 168-184 · doi:10.21914/anziamj.v47i0.1037
[16] Garrappa R., “Numerical solution of fractional differential equations: A survey and a software tutorial”, Mathematics, 6:2 (2018), 016 · Zbl 06916890 · 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, 8:1 (2009), 129-145 · Zbl 1161.37054 · 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, 24:5 (2014), 1450063 · Zbl 1296.37053 · doi:10.1142/S0218127414500631
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.