Comments on ‘Global dynamics of the generalized Lorenz systems having invariant algebraic surfaces’

Abstract

In the recent paper ‘Global dynamics of the generalized Lorenz systems having invariant algebraic surfaces’ published in Physica D, the authors study the generalized Lorenz system, ẋ=a(y−x),ẏ=bx+cy−xz,ż=dz+xy, and consider the subclass of these systems which have an invariant algebraic surface. Within this subclass they present their global dynamics via the blow up and Poincaré compactification.In this Comment we argue that the results presented in the commented paper can be directly obtained from the well-known results on global dynamics of the Lorenz system having invariant algebraic surfaces. To achieve it, we use the equivalence existing, under generic conditions (c≠0), between the Lorenz system and the generalized Lorenz system by means of a linear scaling in time and coordinates.On the other hand, their study is not complete because a case is missed in the list of six invariant algebraic surfaces in the generalized Lorenz system considered by the authors. We also analyze here the global dynamics in this seventh situation.