Global dynamics of the generalized Lorenz systems having invariant algebraic surfaces

Abstract

The generalized Lorenz systems ẋ=a(y−x), ẏ=bx+cy−xz, ż=dz+xy are the unification of the classical Lorenz system, the Chen system and the Lü system. These systems all exhibit chaotic phenomena and are topologically different. Their global dynamics have not been fully characterized, and it seems to be a very difficult problem.In this paper we study the subclass of generalized Lorenz systems which have an invariant algebraic surface. Within this subclass we present their global dynamics via the blow up and Poincaré compactification. This approach may contribute to the understanding of the dynamics of the more general complex (chaotic) systems. Furthermore we prove that any system within this subclass has no limit cycles. This result is novel even for the classical Lorenz system which has an invariant algebraic surface.