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

Llibre, Jaume; Valls, Clàudia

doi:10.1090/proc/13233

On the integrability of the $5$-dimensional Lorenz system for the gravity-wave activity
HTML articles powered by AMS MathViewer

by Jaume Llibre and Clàudia Valls PDF

Proc. Amer. Math. Soc. 145 (2017), 665-679 Request permission

Abstract:

We consider the $5$-dimensional Lorenz system \begin{align*} U’ &= -V W + b V Z, \\ V’ &= UW-b UZ, \\ W’&= -U V,\\ X’ &= -Z, \\ Z’&=b UV +X, \end{align*} 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.

References

Colin Christopher, Liouvillian first integrals of second order polynomial differential equations, Electron. J. Differential Equations (1999), No. 49, 7. MR 1729833
Colin Christopher, Jaume Llibre, and Jorge Vitório Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math. 229 (2007), no. 1, 63–117. MR 2276503, DOI 10.2140/pjm.2007.229.63
G. Darboux, Mémoire sur les équations différentielles du premier ordre et du premier degreé (Mélanges), Bull. Sci. Math. 2éme série 2 (1878), 60–96, 123–144, 151–200.
G. Darboux, De l’emploi des solutions particulières algébriques dans l’intégration des systèmes d’équations différentielles algébriques, C. R. Math. Acad. Sci. Paris 86 (1878), 1012–1014.
Freddy Dumortier, Jaume Llibre, and Joan C. Artés, Qualitative theory of planar differential systems, Universitext, Springer-Verlag, Berlin, 2006. MR 2256001
Jaume Llibre and Xiang Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity 15 (2002), no. 4, 1269–1280. MR 1912294, DOI 10.1088/0951-7715/15/4/313
Jaume Llibre and Xiang Zhang, Invariant algebraic surfaces of the Lorenz system, J. Math. Phys. 43 (2002), no. 3, 1622–1645. MR 1885027, DOI 10.1063/1.1435078
Andrzej J. Maciejewski, Maria Przybylska, Tomasz Stachowiak, and Marek Szydłowski, Global integrability of cosmological scalar fields, J. Phys. A 41 (2008), no. 46, 465101, 26. MR 2515846, DOI 10.1088/1751-8113/41/46/465101
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, DOI 10.1016/j.physleta.2010.09.060
Jaume Llibre, Radu Saghin, and Xiang Zhang, On the analytic integrability of the $5$-dimensional Lorenz system for the gravity-wave activity, Proc. Amer. Math. Soc. 142 (2014), no. 2, 531–537. MR 3133994, DOI 10.1090/S0002-9939-2013-11773-9
Edward N. Lorenz, On the existence of a slow manifold, J. Atmospheric Sci. 43 (1986), no. 15, 1547–1557. MR 855190, DOI 10.1175/1520-0469(1986)043<1547:OTEOAS>2.0.CO;2
S. I. Popov, W. Respondek, and J.-M. Strelcyn, On rational integrability of Euler equations on Lie algebra $\textrm {so}(4,\Bbb C)$, revisited, Phys. Lett. A 373 (2009), no. 29, 2445–2453. MR 2543752, DOI 10.1016/j.physleta.2009.04.075
M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279 (1983), no. 1, 215–229. MR 704611, DOI 10.1090/S0002-9947-1983-0704611-X
Michael F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), no. 2, 673–688. MR 1062869, DOI 10.1090/S0002-9947-1992-1062869-X
Xiang Zhang, Liouvillian integrability of polynomial differential systems, Trans. Amer. Math. Soc. 368 (2016), no. 1, 607–620. MR 3413876, DOI 10.1090/S0002-9947-2014-06387-3