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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2018, Volume 22, Number 2, Pages 364–379 (Mi vsgtu1611)

Mathematical Modeling, Numerical Methods and Software Complexes

Chaotic regimes of a fractal nonlinear oscillator

R. I. Parovikab

a Institute of Cosmophysical Researches and Radio Wave Propagation, Far East Division, Russian Academy of Sciences, Paratunka, Kamchatkiy kray, 684034, Russian Federation
b Vitus Bering Kamchatka State Univrsity, Petropavlovsk-Kamchatskiy, 683032, Russian Federation

Abstract: In the paper, a fractal nonlinear oscillator was investigated with the aim of identifying its chaotic oscillatory regimes. The measure of chaos for a dynamic system is the maximum Lyapunov exponents. They are considered as a measure of the dispersal of several phase trajectories constructed under different initial conditions. To determine the maximum Lyapunov exponents, algorithms are used which are related either to the study of time series (Benettin's algorithm) or to the direct solution of an extended dynamical system (Wolff's algorithm). In this paper, the Wolf algorithm with the Gram-Schmidt orthogonalization procedure was used as the method for constructing Lyapunov's maximum exponents. This algorithm uses the solution of the extended initial dynamical system in conjunction with the variational equations, and the Gram-Schmidt orthogonalization procedure makes it possible to level out the component of the maximum Lyapunov exponent when computing vectors along phase trajectories. Further, the Wolf algorithm was used to construct the spectra of Lyapunov exponents as a function of the values of the control parameters of the initial dynamical system. It was shown in the paper that certain spectra of Lyapunov exponents contain sets of positive values, which confirms the presence of a chaotic regime, and this is also confirmed by phase trajectories.It was also found that the fractal non-linear oscillator has not only oscillatory modes, but also rotations. These rotations can be chaotic and regular.

Keywords: maximum Lyapunov exponents, Wolf algorithm, chaotic attractor, limit cycle, spectrum of Lyapunov exponents, fractal nonlinear oscillator

 Funding Agency Grant Number Grant of the President of the Russian Federation for State Support of Young Russian Scientists --- Candidates of Science ÌÊ-1152.2018.1 This work was supported by the grant of the President of the Russian Federation for state support of scholarly research by young scholars no. MK-1152.2018.1.

DOI: https://doi.org/10.14498/vsgtu1611

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Bibliographic databases:

Document Type: Article
UDC: 517.938
MSC: 26A33, 34A08, 37N30, 74H60
Revised: April 26, 2018
Accepted: June 11, 2018
First online: June 28, 2018

Citation: R. I. Parovik, “Chaotic regimes of a fractal nonlinear oscillator”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 22:2 (2018), 364–379

Citation in format AMSBIB
\Bibitem{Par18} \by R.~I.~Parovik \paper Chaotic regimes of a fractal nonlinear oscillator \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2018 \vol 22 \issue 2 \pages 364--379 \mathnet{http://mi.mathnet.ru/vsgtu1611} \crossref{https://doi.org/10.14498/vsgtu1611} \elib{http://elibrary.ru/item.asp?id=35467735}