《Global dynamics of the generalized Lorenz systems having invariant》.pdf

Physica D 244 (2013) 25–35 Contents lists available at SciVerse ScienceDirect Physica D journal homepage: /locate/physd Global dynamics of the generalized Lorenz systems having invariant algebraic surfaces Kesheng Wu a , Xiang Zhang b,∗ a Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China b Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China a r t i c l e i n f o a b s t r a c t Article history: ˙ ˙ ˙ The generalized Lorenz systems x = a y − x , y = bx + cy − xz , z = dz + xy are the unification of the ( ) Received 8 May 2012 classical Lorenz system, the Chen system and the Lü system. These systems all exhibit chaotic phenomena Received in revised form and are topologically different. Their global dynamics have not been fully characterized, and it seems to 24 October 2012 be a very difficult problem. Accepted 25 October 2012 Available online 14 November 2012 In this paper we study the subclass of generalized Lorenz systems which have an invariant Communicated by K. Josic algebraic surface. Within this subclass we present their global dynamics via the blow up and Poincaré