Modeling and analysis of dynamics for a 3D mixed Lorenz system with a damped term

Abstract

Abstract The work in this paper consists of two parts. The one is modelling. After a method of classification for three dimensional (3D) autonomous chaotic systems and a concept of mixed Lorenz system are introduced, a mixed Lorenz system with a damped term is presented. The other is the analysis for dynamical properties of this model. First, its local stability and bifurcation in its parameter space are in detail considered. Then, the existence of its homoclinic and heteroclinic orbits, and the existence of singularly degenerate heteroclinic cycles, are studied by rigorous theoretical analysis. Finally, by using the Poincaré compactification for polynomial vector fields in R 3 ${\mathbb{R}}^{3}$ , a global analysis of this system near and at infinity is presented, including the complete description of its dynamics on the sphere near and at infinity. Simulations corroborate corresponding theoretical results. In particular, a possibly new mechanism for the creation of chaotic attractors is proposed. Some different structure types of chaotic attractors are correspondingly and numerically found.