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On the analytic integrability of the $ 5$-dimensional Lorenz system for the gravity-wave activity


Authors: Jaume Llibre, Radu Saghin and Xiang Zhang
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
Published electronically: October 22, 2013
MathSciNet review: 3133994
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Abstract: For the $ 5$-dimensional Lorenz system

$\displaystyle dU/dT$ $\displaystyle =$ $\displaystyle -VW+b\, VZ,$  
$\displaystyle dV/dT$ $\displaystyle =$ $\displaystyle UW-b\, UZ,$  
$\displaystyle dW/dT$ $\displaystyle =$ $\displaystyle -UV,$  
$\displaystyle dX/dT$ $\displaystyle =$ $\displaystyle -Z,$  
$\displaystyle dZ/dT$ $\displaystyle =$ $\displaystyle b\,UV+X$  

(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.

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Additional Information

Jaume Llibre
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
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

DOI: https://doi.org/10.1090/S0002-9939-2013-11773-9
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
Article copyright: © Copyright 2013 American Mathematical Society

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