Abstract
In a very recent paper by Deng (Z Angew Math Phys 64:1443–1449, 2013), the author claims to have successfully found all the invariant algebraic surfaces of the generalized Lorenz system, \({\dot{x} = a(y - x), \ \dot{y} = bx + cy - xz, \ \dot{z} = dz + xy}\). He provides six invariant algebraic surfaces, found according to the idea of the weight of a polynomial introduced by Swinnerton-Dyer (Math Proc Camb Philos Soc 132:385–393, 2002). Unfortunately, his result is incorrect because a seventh invariant algebraic surface is missed. Moreover, those six invariant algebraic surfaces can be obtained in a much simpler manner: Since the Lorenz system and the generalized Lorenz system are equivalent through a homothetic scaling in time and state variables (for c ≠ 0), it is trivial to obtain the corresponding results for the generalized Lorenz system from the well-known results on invariant algebraic surfaces of the Lorenz system.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Kús M.: Integrals of motion for the Lorenz system. J. Phys. A 16, L689–L691 (1983)
Llibre J., Zhang X.: Invariant algebraic surfaces of the Lorenz system. J. Math. Phys. 43, 1622–1645 (2002)
Swinnerton-Dyer P.: The invariant algebraic surfaces of the Lorenz system. Math. Proc. Camb. Philos. Soc. 132, 385–393 (2002)
Deng X.: Invariant algebraic surfaces of the generalized Lorenz systems. Z. Angew. Math. Phys. 64, 1443–1449 (2013)
Algaba, A., Fernández-Sánchez, F., Merino, M., Rodríguez-Luis, A.J.: On Darboux polynomials and rational first integrals of the generalized Lorenz system. Bull. Sci. Math. (2013).doi:10.1016/j.bulsci.2013.03.002
Algaba A., Fernández-Sánchez F., Merino M., Rodríguez-Luis A.J.: Comments on ‘Global dynamics of the generalized Lorenz systems having invariant algebraic surfaces’. Physica D 266, 80–82 (2014)
Algaba, A., Fernández-Sánchez, F., Merino, M., Rodríguez-Luis, A.J.: Comment on “A constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family” [Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 2886-2896], Commun. Nonlinear Sci. Numer. Simulat. 19, 758–761 (2014)
Algaba, A., Fernández-Sánchez, F., Merino, M., Rodríguez-Luis, A.J.: Comments on “Dynamics of the general Lorenz family” by Y. Liu and W. Pang, Nonlinear Dyn. 76, 887–891 (2014).doi:10.1007/s11071-013-1142-y
Lorenz E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)
Sparrow C.: The Lorenz Equations. Springer, New York (1982)
Wu K., Zhang X.: Darboux polynomials and rational first integrals of the generalized Lorenz systems. Bull. Sci. Math. 136, 291–308 (2012)
Chen G., Ueta T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999)
Lü J., Chen G.: A new chaotic attractor coined. Int. J. Bifurcation Chaos 12, 659–661 (2002)
Algaba A., Fernández-Sánchez F., Merino M., Rodríguez-Luis A.J.: Chen’s attractor exists if Lorenz repulsor exists: the Chen system is a special case of the Lorenz system. Chaos 23, 033108 (2013)
Algaba A., Fernández-Sánchez F., Merino M., Rodríguez-Luis A.J.: The Lü system is a particular case of the Lorenz system. Phys. Lett. A 377, 2771–2776 (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
The online version of the original article can be found under doi:10.1007/s00033-012-0296-7.
Rights and permissions
About this article
Cite this article
Algaba, A., Fernández-Sánchez, F., Merino, M. et al. Comments on “Invariant algebraic surfaces of the generalized Lorenz system”. Z. Angew. Math. Phys. 66, 1295–1297 (2015). https://doi.org/10.1007/s00033-014-0420-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-014-0420-y