3-D Discrete Vortex and Convection for the Lorenz System

The Lorenz system is well known as a three-dimensional (3-D) mathematical model for atmospheric convection and is considered from the viewpoint of discrete chaos functions. Firstly, the 3-D generalized Lorenz map with system parameters is proposed, and the bifurcation diagrams are calculated to find discrete limit cycles. Then, the 3-D orbit solutions are shown to have the 2-D limit cycles with active dynamics on the x–y plane. Finally, the 3-D discrete vortices with distortion are numerically illustrated, depending on the system parameters and initial conditions of cells. In particular, the 3-D vortices with distortion and convection correspond to the 2-D limit cycles, which are briefly discussed on high or low atmospheric pressure of atmospheric convection and on 3-D climate change for the Lorenz system, as nonlinear dynamics of 3-D nonequilibrium open systems.