Global Dynamics of the Lorenz System With Invariant Algebraic Surfaces

In this paper by using the Poincaré compactification of ℝ 3 we describe the global dynamics of the Lorenz system [Formula: see text] having some invariant algebraic surfaces. Of course (x, y, z) ∈ ℝ 3 are the state variables and (s, r, b) ∈ ℝ 3 are the parameters. For six sets of the parameter values, the Lorenz system has invariant algebraic surfaces. For these six sets, we provide the global phase portrait of the system in the Poincaré ball (i.e. in the compactification of ℝ 3 with the sphere 𝕊 2 of the infinity).