Geodesic
Study of the singular points of the fractional oscillator Van der Pol-Duffing
Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 27 (2019) no. 2, pp. 47-54
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A study is conducted on the asymptotic stability of the rest points of the fractional oscillator Van der Pol-Duffing. The fractional van der Pol-Duffing oscillator is an oscillatory system of two differential equations with fractional order derivatives in the sense of Gerasimov–Caputo. The orders of fractional derivatives characterize the properties of the medium (memory effects) in which the oscillatory process takes place and can be the same (commensurate) or different (incommensurable). Using theorems for commensurable and incommensurable systems, the asymptotic stability of the rest points of the fractional van der Pol-Duffing oscillator is investigated with concrete examples. The results of the studies were confirmed by constructing the appropriate waveforms and phase trajectories.
Keywords: Van der Pol's Duffing fractional oscillator, singular points, commensurate and incommensurate systems, asymptotic stability, oscillograms and phase trajectories. 
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E. R. Novikovа. Study of the singular points of the fractional oscillator Van der Pol-Duffing. Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 27 (2019) no. 2, pp. 47-54. http://geodesic.mathdoc.fr/item/VKAM_2019_27_2_a3/

[1] Vronskiy A. P., “Yavleniye posledeystviya v tverdom tele”, AN SSSR. Prikladnaya matematika i mekhanika, 5:1 (1941), 31–56

[2] Rabotnov Yu. N., “Ravnovesiye uprugoy sredy s posledeystviyem”, AN SSSR. Prikladnaya matematika i mekhanika, 12:1 (1948), 53–62 | MR | Zbl

[3] Gerasimov A. N., “Obobshcheniye lineynykh zakonov deformatsii i ikh prilozheniye k zadacham vnutrennego treniya”, AN SSSR. Prikladnaya matematika i mekhanika, 12:3 (1948), 251-260 | Zbl

[4] Volterra V., “Sur les 'equations int'egro-diff'erentiellesetleurs applications”, Acta Mathematica, 35:1 (1912), 295-356 | DOI | MR | Zbl

[5] Nakhushev A. M., Drobnoye ischisleniye i yego primeneniye, Fizmatlit, M., 2003, 272 pp.

[6] Petras I., Fractional-Order Nonlinear Systems. Modeling, Analysis and Simulation, Springer, Beijing; Springer-Verlag, Berlin–Heidelberg, 2011, 218 pp. | Zbl

[7] Parovik R. I., Matematicheskoye modelirovaniye nelineynykh ereditarnykh ostsillyatorov, KamGU im. Vitusa Beringa, Petropavlovsk-Kamchatskiy, 2017, 135 pp.

[8] Novikova Ye. R, Parovik R. I., “Issledovaniye tochek pokoya ereditarnoy dinamicheskoy sistemy Van der Polya-Duffinga”, Vestnik KRAUNTS. Fiz.-mat. nauki, 26:1 (2019), 71-77 | DOI

[9] Lipko O. D., “Issledovaniye ustoychivosti tochek pokoya drobnogo ostsillyatora FittsKH'yu-Nagumo”, Vestnik KRAUNTS. Fiz.-mat. nauki, 26:1 (2019), 63-70 | DOI

[10] Caputo M., Elasticit`a e dissipazione, Bologna, Zanichelli, 1969, 150 pp.

[11] Tavazoei M. S., Haeri M., “Chaotic attractors in incommensurate fractional order systems”, Physica D: Nonlinear Phenomena, 237:20 (2008), 2628-2637 | DOI | MR | Zbl