G. A. Leonov, N. V. Kuznetsov, E. V. Kudryashova, “A direct method for calculating Lyapunov values of two-dimensional dynamical systems”, Trudy Inst. Mat. i Mekh. UrO RAN, 16, no. 1, 2010, 119–126; Proc. Steklov Inst. Math. (Suppl.), 272, suppl. 1 (2011), S119

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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2010, Volume 16, Number 1, Pages 119–126 (Mi timm532)

This article is cited in 17 scientific papers (total in 17 papers)

A direct method for calculating Lyapunov values of two-dimensional dynamical systems

G. A. Leonov, N. V. Kuznetsov, E. V. Kudryashova

Saint-Petersburg State University

Full-text PDF (164 kB) Citations (17)

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Abstract: A direct method is proposed for studying the behavior of two-dimensional dynamical systems in the critical case when the linear part of the system has two purely imaginary eigenvalues. This method allows one to construct approximations to solutions of the system and to the “turn-round” time of the trajectory in the form of a finite series in powers of the initial datum. With the help of symbolic computations and the proposed method, first approximations of a solution are constructed and expressions for the first three Lyapunov quantities of the Liénard system are written.

Keywords: Lyapunov quantities, limit cycle, symbolic computations.

Received: 04.03.2009

English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2011, Volume 272, Issue 1, Pages S119–S126
DOI: https://doi.org/10.1134/S008154381102009X

Bibliographic databases:

Document Type: Article

UDC: 517.977

Language: Russian

Citation: G. A. Leonov, N. V. Kuznetsov, E. V. Kudryashova, “A direct method for calculating Lyapunov values of two-dimensional dynamical systems”, Trudy Inst. Mat. i Mekh. UrO RAN, 16, no. 1, 2010, 119–126; Proc. Steklov Inst. Math. (Suppl.), 272, suppl. 1 (2011), S119–S126

Citation in format AMSBIB

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Linking options:

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This publication is cited in the following 17 articles:

Nikolay Kuznetsov, Timur Mokaev, Vladimir Ponomarenko, Evgeniy Seleznev, Nataliya Stankevich, Leon Chua, “Hidden attractors in Chua circuit: mathematical theory meets physical experiments”, Nonlinear Dyn, 111:6 (2023), 5859
Yumagulov M.G., Akmanova S.V., Kopylova N.A., “Lyapunov Quantities in the Problem About Local Bifurcations of Non-Autonomous Periodic Dynamical Systems”, Lobachevskii J. Math., 41:9, SI (2020), 1918–1923
N. I. Gusarova, S. A. Murtazina, M. F. Fazlytdinov, M. G. Yumagulov, “Operator methods for calculating Lyapunov values in problems on local bifurcations of dynamical systems”, Ufa Math. J., 10:1 (2018), 25–48
Czornik A., Jurgas P., “On the Lyapunov Exponents of Infinite-Dimensional Discrete Time-Varying Linear System”, 2016 21St International Conference on Methods and Models in Automation and Robotics (Mmar), IEEE, 2016, 496–499
Jerzy Klamka, Elżbieta Ferenstein, Artur Babiarz, Michał Niezabitowski, “On the Reducibility of the Discrete Linear Time-Varying Systems”, AMM, 789-790 (2015), 1027
S. Brezetskyi, D. Dudkowski, T. Kapitaniak, “Rare and hidden attractors in Van der Pol-Duffing oscillators”, Eur. Phys. J. Spec. Top., 224:8 (2015), 1459
N.V. Kuznetsov, G.A. Leonov, “Hidden attractors in dynamical systems: systems with no equilibria, multistability and coexisting attractors”, IFAC Proceedings Volumes, 47:3 (2014), 5445
Leonov G.A., Burova I.G., Aleksandrov K.D., “Visualization of Four Limit Cycles of Two-Dimensional Quadratic Systems in the Parameter Space”, Differ. Equ., 49:13 (2013), 1675–1703
Leonov G.A., Kuznetsov N.V., “Hidden Attractors in Dynamical Systems. From Hidden Oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits”, Int. J. Bifurcation Chaos, 23:1 (2013), 1330002
Gennady A. Leonov, Nikolay V. Kuznetsov, Advances in Intelligent Systems and Computing, 210, Nostradamus 2013: Prediction, Modeling and Analysis of Complex Systems, 2013, 5
Nikolay Kuznetsov, Olga Kuznetsova, Gennady Leonov, Vladimir Vagaitsev, Lecture Notes in Electrical Engineering, 174, Informatics in Control, Automation and Robotics, 2013, 149
Gennady A. Leonov, Nikolay V. Kuznetsov, Computational Methods in Applied Sciences, 27, Numerical Methods for Differential Equations, Optimization, and Technological Problems, 2013, 41
N. V. Kuznetsov, O. A. Kuznetsova, G. A. Leonov, “Visualization of Four Normal Size Limit Cycles in Two-Dimensional Polynomial Quadratic System”, Differ Equ Dyn Syst, 21:1-2 (2013), 29
G. A. Leonov, N. V. Kuznetsov, “Hidden oscillations in dynamical systems. 16 Hilbert's problem, Aizerman's and Kalman's conjectures, hidden attractors in Chua's circuits”, Journal of Mathematical Sciences, 201:5 (2014), 645–662
G. A. Leonov, N. V. Kuznetsov, E. V. Kudryashova, O. A. Kuznetsova, “Sovremennye metody simvolnykh vychislenii: lyapunovskie velichiny i 16-aya problema Gilberta”, Tr. SPIIRAN, 16 (2011), 5–36
G.A. Leonov, N.V. Kuznetsov, “Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems”, IFAC Proceedings Volumes, 44:1 (2011), 2494
Leonov G.A., Kuznetsov N.V., “Limit cycles of quadratic systems with a perturbed weak focus of order 3 and a saddle equilibrium at infinity”, Dokl. Math., 82:2 (2010), 693–696

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