On the analytic integrability of the $5$–dimensional Lorenz system for the gravity–wave activity
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- by Jaume Llibre, Radu Saghin and Xiang Zhang PDF
- Proc. Amer. Math. Soc. 142 (2014), 531-537 Request permission
Abstract:
For the $5$-dimensional Lorenz system \begin{eqnarray*} dU/dT &=& -VW+b VZ,\\ dV/dT &=& UW-b UZ,\\ dW/dT &=& -UV, \\ dX/dT &=& -Z,\\ dZ/dT &=& b UV+X \end{eqnarray*} (with $b\in \mathbb {R}$ a parameter), describing coupled Rosby and gravity waves, we prove that it has at most three functionally independent global analytic first integrals and exactly three functionally independent global analytic first integrals when $b=0$. In this last case the system is completely integrable with an additional functionally independent first integral which is not globally analytic.References
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Additional Information
- Jaume Llibre
- Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
- MR Author ID: 115015
- ORCID: 0000-0002-9511-5999
- Email: jllibre@mat.uab.cat
- Radu Saghin
- Affiliation: Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile
- Email: rsaghin@gmail.com
- Xiang Zhang
- Affiliation: MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China
- Email: xzhang@sjtu.edu.cn
- Received by editor(s): September 4, 2011
- Received by editor(s) in revised form: March 19, 2012
- Published electronically: October 22, 2013
- Communicated by: Yingfei Yi
- © Copyright 2013 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 531-537
- MSC (2010): Primary 34C05, 34A34, 34C14
- DOI: https://doi.org/10.1090/S0002-9939-2013-11773-9
- MathSciNet review: 3133994