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


Authors: Jaume Llibre and Clàudia Valls
Journal: Proc. Amer. Math. Soc. 145 (2017), 665-679
MSC (2010): Primary 37J35, 37K10
DOI: https://doi.org/10.1090/proc/13233
Published electronically: July 28, 2016
MathSciNet review: 3577869
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Abstract: We consider the $ 5$-dimensional Lorenz system

$\displaystyle U'$ $\displaystyle = -V W + b V Z,$    
$\displaystyle V'$ $\displaystyle = UW-b UZ,$    
$\displaystyle W'$ $\displaystyle = -U V,$    
$\displaystyle X'$ $\displaystyle = -Z,$    
$\displaystyle Z'$ $\displaystyle =b UV +X,$    

where $ b \in \mathbb{R} \setminus \{0\}$ and the derivative is with respect to $ T$. This system describes coupled Rosby waves and gravity waves. First we prove that the number of functionally independent global analytic first integrals of this differential system is two. This solves an open question in the paper, On the analytic integrability of the $ 5$-dimensional Lorenz system for the gravity-wave activity, Proc. Amer. Math. Soc. 142 (2014), 531-537, where it was proved that this number was two or three. Moreover, we characterize all the invariant algebraic surfaces of the system, and additionally we show that it has only two functionally independent Darboux first integrals.

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Additional Information

Jaume Llibre
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra, Barcelona, Catalonia, Spain
Email: jllibre@mat.uab.cat

Clàudia Valls
Affiliation: Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal
Email: cvalls@math.ist.utl.pt

DOI: https://doi.org/10.1090/proc/13233
Keywords: Hamiltonian systems, weight-homogeneous differential systems, polynomial integrability, rational integrability, Darboux polynomials, exponential factors, Darboux first integrals.
Received by editor(s): February 9, 2016
Received by editor(s) in revised form: April 2, 2016
Published electronically: July 28, 2016
Additional Notes: The first author was partially supported by MINECO/FEDER grant number MTM2013-40998-P, an AGAUR grant number 2014SGR-568 and the grants FP7-PEOPLE-2012-IRSES 318999 and 316338
The second author was partially supported by FCT/Portugal through UID/MAT/04459/2013
Communicated by: Yingfei Yi
Article copyright: © Copyright 2016 American Mathematical Society

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