Comments on “Invariant algebraic surfaces of the generalized Lorenz system”

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.