Geodesic
Investigation of forced oscillations
News of the Kabardin-Balkar scientific center of RAS, no. 1 (2020), pp. 46-56.

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A mathematical model of a Duffing type oscillator with a variable fractional derivative of the Riemann-Liouville derivative is studied. Using the harmonic balance method, algorithms for constructing amplitude-phase characteristics were found. The amplitude-frequency and phase-frequency characteristics were built. The inverse dependence of the Q factor on the order of the fractional derivative is shown. The Q-factor surfaces are constructed depending on the frequency and amplitude.
Keywords: Riemann-Liouville derivative, Grunwald-Letnikov derivative, oscillograms, phase trajectories, amplitude-frequency characteristic (AFC), phase-frequency characteristic (PFC). 
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V. A. Kim; R. I. Parovik. Investigation of forced oscillations. News of the Kabardin-Balkar scientific center of RAS, no. 1 (2020), pp. 46-56. http://geodesic.mathdoc.fr/item/IZKAB_2020_1_a2/

[1] A. Syta et al, “Chaotic vibrations of the Duffing system with fractional damping”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 24:1 (2014), 10–16 | DOI | MR

[2] S. P. Kuznetsov, Dinamicheskii khaos, Fizmatlit, M., 2001, 296 pp.

[3] V. V. Uchaikin, Metod drobnykh proizvodnykh, Artishok, Ulyanovsk, 2008, 512 pp.

[4] V. A. Kim, “Duffing oscillator with external harmonic action and variable fractional Riemann-Liouville derivative characterizing viscous friction”, Bulletin KRASEC. Physical and Mathematical Sciences, 13:2 (2016), 46–49 | MR

[5] R. I. Parovik, Khaoticheskie i regulyarnye rezhimy drobnykh ostsillyatorov, monografiya, KamGU im. Vitusa Beringa, Petropavlovsk-Kamchatskii, 2019, 132 pp.

[6] R. I. Parovik, “Matematicheskoe modelirovanie nelokalnoi kolebatelnoi sistemy Duffinga s fraktalnym treniem”, Vestnik KRAUNTs. Fiziko-matematicheskie nauki, 2015, no. 1 (10), 18–24 | Zbl

[7] V. A. Kim, “Modelnoe uravnenie ereditarnogo ostsillyatora Duffinga s proizvodnoi peremennogo drobnogo poryadka Rimana-Liuvillya, kharakterizuyuschei vyazkoe trenie”, Novyi universitet. Tekhnicheskie nauki, 2016, no. 4-5(50-51), 28–31

[8] V. A. Kim, “Matematicheskaya model nelineinogo ostsillyatora Duffinga s pamyatyu”, Aktualnye problemy differentsialnykh uravnenii i ikh prilozheniya, materialy Respublikanskoi nauchn. konf. s uchastiem zarubezh. uchenykh (g. Tashkent, 15-17 dekabrya 2017 g.), 253–254 | Zbl

[9] V. A. Kim, “Matematicheskoe modelirovanie ostsillyatora Duffinga s proizvodnoi peremennogo drobnogo poryadka”, Matematicheskoe modelirovanie i kraevye zadachi, m-ly XI vserossiiskoi nauchnoi konferentsii s mezhdunarodnym uchastiem (Samara, 27–29 maya 2019 g.), v. 2, SamGTU, Samara, 2019, 44–46

[10] V. A. Kim, “Matematicheskoe modelirovanie ostsillyatora Duffinga s proizvodnoi peremennogo drobnogo poryadka”, Nelokalnye kraevye zadachi i rodstvennye problemy matematicheskoi biologii, informatiki i fiziki, materialy V mezhdunarod. nauchn. konf. k 80-letiyu Adama Maremovicha Nakhusheva (g. Nalchik, Kabardino-Balkarskaya respublika, 4–7 dekabrya 2018 g.), Elbrus, Nalchik, 2018, 103–104

[11] V. A. Kim, R. I. Parovik, “Raschet maksimalnykh pokazatelei Lyapunova dlya kolebatelnoi sistemy Duffinga so stepennoi pamyatyu”, Vestnik KRAUNTs. Fiziko-matematicheskie nauki, 2018, no. 3 (23), 98–105 | Zbl

[12] V. A. Kim, “Khaoticheskie rezhimy ostsillyatora Duffinga s proizvodnoi peremennogo drobnogo poryadka Rimana-Liuvillya”, Aktualnye problemy prikladnoi matematiki, materialy IV mezhdunarod. nauchn. konf. (p. Terskol, Kabardino-Balkarskaya Respublika, 22-26 maya 2018 g.), Elbrus, Nalchik, 2018, 121–122

[13] I. G. Malkin, Metody Lyapunova i Puankare v teorii nelineinykh kolebanii, Izd. 2-e, M., 2004, 248 pp. | MR

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

[15] V. I. Oseledets, “Multiplikativnaya ergodicheskaya teorema. Kharakteristicheskie pokazateli Lyapunova dinamicheskikh sistem”, Tr. Moskovskogo matematicheskogo obschestva, 19, 1968, 197–231 | MR | Zbl

[16] R. I. Parovik, “Konechno-raznostnye skhemy dlya fraktalnogo ostsillyatora s peremennymi drobnymi poryadkami”, Vestnik KRAUNTs. Fiziko-matematicheskie nauki, 11:2 (2015), 88–95 | Zbl

[17] R. I. Parovik, “Ob issledovanii ustoichivosti ereditarnogo ostsillyatora Van-der-Polya”, Fundamentalnye issledovaniya, 2016, no. 3 (2), 283–287 | MR

[18] A. A. Petukhov, D. L. Reviznikov, “Algoritmy chislennykh reshenii drobno-differentsialnykh uravnenii”, Vestnik MAI, 16:6 (2009), 228–243

[19] X. Gao, J. Yu, “Chaos in the fractional order periodically forced complex Duffinges oscillators”, Chaos, Solitons Fractals, 24:4 (2005), 97–104

[20] I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, Springer, New York, 2010, 180 pp. | MR

[21] V. Volterra, Leçons sur les fonctions de lignes, professies a la Sorbonne en 1912, Gauthier-Villars, Paris, 1913, 240 pp.