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 Vestnik KRAUNC. Fiz.-Mat. Nauki, 2018, Number 3(23), Pages 98–105 (Mi vkam260)

MATHEMATICAL MODELING OF DYNAMIC SYSTEMS

Calculation the maximum Lyapunov exponent for the oscillatory system of Duffing with a degree memory

V. A. Kima, R. I. Parovikba

a Vitus Bering Kamchatka State University, 683032, Kamchatskiy kray, 4, Pogranichnaya Str., Russia Petropavlovsk-Kamchatsky, Russia
b Institute of Cosmophysical Researches and Radio Wave Propagation, Far East Division, Russian Academy of Sciences

Abstract: In the study of nonlinear systems, one of the important problems is the determination of the type of oscillations-periodic, quasi-periodic, random, chaotic. It is especially difficult to distinguish between quasi-periodic oscillations from chaotic and random oscillations, since quasi-periodic oscillations often have a very complex shape, visually weakly distinguishable from «random». A feature of chaotic oscillations is their high sensitivity to small changes in the initial conditions. Therefore, one of the most reliable ways of detecting chaos is to determine the rate of run-off of trajectories, which is estimated using the Lyapunov exponent spectrum. Using the construction of the spectrum of Max Lyapunov exponents, depending on the values of the control parameters, chaotic regimes of the Duffing fractal oscillator with variable power memory were found, and its phase trajectories.

Keywords: spectrum of maximum Lyapunov exponents, Duffing fractal oscillator,  phase trajectories, limit cycle, chaotic attractor.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation ÌÊ-1152.2018.1ÀÀÀÀ-À17-117031050058-9 The work was supported by the grant of the President of the Russian Federation No. MK-1152.2018.1 and on the topic of research work at KamSU named after Vitus Bering “Application of fractional calculus in the theory of oscillatory processes” No. AAAA-A17-117031050058-9.

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DOI: https://doi.org/10.18454/2079-6641-2018-23-3-98-105

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UDC: 517.938
MSC: Primary 34A08; Secondary 34K28, 37N30

Citation: V. A. Kim, R. I. Parovik, “Calculation the maximum Lyapunov exponent for the oscillatory system of Duffing with a degree memory”, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2018, no. 3(23), 98–105

Citation in format AMSBIB
\Bibitem{KimPar18} \by V.~A.~Kim, R.~I.~Parovik \paper Calculation the maximum Lyapunov exponent for the oscillatory system of Duffing with a degree memory \jour Vestnik KRAUNC. Fiz.-Mat. Nauki \yr 2018 \issue 3(23) \pages 98--105 \mathnet{http://mi.mathnet.ru/vkam260} \crossref{https://doi.org/10.18454/2079-6641-2018-23-3-98-105} \elib{http://elibrary.ru/item.asp?id=35604475}