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 Regul. Chaotic Dyn., 2015, Volume 20, Issue 2, Pages 123–133 (Mi rcd49)

Analytical Solutions of the Lorenz System

Nikolay A. Kudryashov

National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe Shosse 31, Moscow, 115409 Russia

Abstract: The Lorenz system is considered. The Painlevé test for the third-order equation corresponding to the Lorenz model at $\sigma \ne 0$ is presented. The integrable cases of the Lorenz system and the first integrals for the Lorenz system are discussed. The main result of the work is the classification of the elliptic solutions expressed via the Weierstrass function. It is shown that most of the elliptic solutions are degenerated and expressed via the trigonometric functions. However, two solutions of the Lorenz system can be expressed via the elliptic functions.

Keywords: Lorenz system, Painlevé property, Painlevé test, analytical solutions, elliptic solutions

 Funding Agency Grant Number Russian Science Foundation 14-11-00258 This research was supported by the Russian Science Foundation grant No. 14-11-00258.

DOI: https://doi.org/10.1134/S1560354715020021

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Document Type: Article
MSC: 01-00, 01A55, 01A60
Language: English

Citation: Nikolay A. Kudryashov, “Analytical Solutions of the Lorenz System”, Regul. Chaotic Dyn., 20:2 (2015), 123–133

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/rcd49
• http://mi.mathnet.ru/eng/rcd/v20/i2/p123

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. A. Kudryashov, “Refinement of the Korteweg-de Vries equation from the Fermi-Pasta-Ulam model”, Phys. Lett. A, 379:40-41 (2015), 2610–2614
2. A. K. Volkov, N. A. Kudryashov, “Nonlinear waves described by a fifth-order equation derived from the Fermi–Pasta–Ulam system”, Comput. Math. Math. Phys., 56:4 (2016), 680–687
3. N. A. Kudryashov, “On solutions of generalized modified Korteweg-de Vries equation of the fifth order with dissipation”, Appl. Math. Comput., 280 (2016), 39–45
4. N. A. Kudryashov, “From the Fermi-Pasta-Ulam model to higher-order nonlinear evolution equations”, Rep. Math. Phys., 77:1 (2016), 57–67
5. N. A. Kudryashov, Yu. S. Ivanova, “Painlevé analysis and exact solutions for the modified Korteweg-de Vries equation with polynomial source”, Appl. Math. Comput., 273 (2016), 377–382
6. Jaume Llibre, Clàudia Valls, “Darboux Polynomials, Balances and Painlevé Property”, Regul. Chaotic Dyn., 22:5 (2017), 543–550
7. L. Bougoffa, S. Al-Awfi, S. Bougouffa, “A complete and partial integrability technique of the Lorenz system”, Results Phys., 9 (2018), 712–716