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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
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.

References [Enhancements On Off] (What's this?)

  • [1] Vladimir I. Arnold, Ordinary differential equations, Universitext, Springer-Verlag, Berlin, 2006. Translated from the Russian by Roger Cooke; Second printing of the 1992 edition. MR 2242407 (2007b:34001)
  • [2] Isaac A. García, Maite Grau, and Jaume Llibre, First integrals and Darboux polynomials of natural polynomial Hamiltonian systems, Phys. Lett. A 374 (2010), no. 47, 4746-4748. MR 2728364 (2011m:34001),
  • [3] Alain Goriely, Integrability and nonintegrability of dynamical systems, Advanced Series in Nonlinear Dynamics, vol. 19, World Scientific Publishing Co. Inc., River Edge, NJ, 2001. MR 1857742 (2002k:37001)
  • [4] Edward N. Lorenz, On the existence of a slow manifold, J. Atmospheric Sci. 43 (1986), no. 15, 1547-1557. MR 855190 (87h:76037),$ \langle $1547:OTEOAS$ \rangle $2.0.CO;2
  • [5] Andrzej J. Maciejewski and Maria Przybylska, Differential Galois obstructions for non-commutative integrability, Phys. Lett. A 372 (2008), no. 33, 5431-5435. MR 2439693 (2009j:37086),
  • [6] S. I. Popov, W. Respondek, and J.-M. Strelcyn, On rational integrability of Euler equations on Lie algebra $ {\rm so}(4,\mathbb{C})$, revisited, Phys. Lett. A 373 (2009), no. 29, 2445-2453. MR 2543752 (2010g:37092),
  • [7] Xiang Zhang, Local first integrals for systems of differential equations, J. Phys. A 36 (2003), no. 49, 12243-12253. MR 2025828 (2004i:34094),

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Additional Information

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

Radu Saghin
Affiliation: Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile

Xiang Zhang
Affiliation: MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

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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